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This article is part of the series Jean Mawhin’s Achievements in Nonlinear Analysis.

Open Access Research

Homoclinic and heteroclinic solutions for a class of second-order non-autonomous ordinary differential equations: multiplicity results for stepwise potentials

Elisa Ellero and Fabio Zanolin*

Author Affiliations

Department of Mathematics and Computer Science, University of Udine, Via delle Scienze 206, Udine, 33100, Italy

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Boundary Value Problems 2013, 2013:167  doi:10.1186/1687-2770-2013-167

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2013/1/167


Received:30 November 2012
Accepted:11 June 2013
Published:15 July 2013

© 2013 Ellero and Zanolin; licensee Springer

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We prove some multiplicity results for a class of one-dimensional nonlinear Schrödinger-type equations of the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M1">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M2">View MathML</a> and the weight <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M3">View MathML</a> is a positive stepwise function. Instead of the cubic term, more general nonlinearities can be considered as well.

MSC: 34C37, 34B15.

Keywords:
homoclinic solutions; heteroclinic solutions; multiplicity; nonlinear Schrödinger equation; stepwise potential; topological methods

Dedication

Dedicated to Professor Jean Mawhin

1 Introduction

This paper deals with the study of homoclinic and heteroclinic solutions for a class of nonlinear second-order differential equations of the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M4">View MathML</a>

(1.1)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M5">View MathML</a> is a fixed (positive) coefficient and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M6">View MathML</a> is a bounded weight function. For the nonlinear term, which we split as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M7">View MathML</a>

we suppose that

(∗) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M8">View MathML</a>is a locally Lipschitz function which is even, strictly increasing on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M9">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M10">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M11">View MathML</a>.

Examples of functions f which are suitable for our considerations are, for instance, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M12">View MathML</a> as in [1], or <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M13">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M14">View MathML</a>) as in [2]. In both these cases, the function f is even. In Section 3 we briefly describe how this restriction could be avoided.

Equations of the form (1.1) naturally arise in the search of particular solutions for some classes of nonlinear Schrödinger equations (NLSE) with inhomogeneous nonlinearities. Typical examples are related to the NLSE

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M15">View MathML</a>

(1.2)

where ħ is the Planck constant, m is the particle’s mass, i is the imaginary unit, Δ is the Laplace operator and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M16">View MathML</a>. In literature, solutions of the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M17">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M18">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M19">View MathML</a> is a real-valued function, are called stationary waves. Their search leads to the equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M20">View MathML</a>

(1.3)

(see [3]). Usually, the Planck constant and the mass are omitted in (1.3) after rescaling.

In many significant models of NLSE, one-dimensional waves are considered. They are studied, for example, in nonlinear optics, in the theory of ocean rogue waves and for Bose-Einstein condensates (just to mention a few cases). For instance, in [1], Belmonte-Beitia and Torres analyzed the one-dimensional equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M21">View MathML</a>

(1.4)

which is a particular case of (1.3) with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M22">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M23">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M24">View MathML</a>. In equation (1.4) the nonlinearity is called inhomogeneous due to the presence of a non-constant weight <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M25">View MathML</a>, which in [1] is assumed to be positive. The sign condition on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M26">View MathML</a> implies that nontrivial bounded solutions of (1.4) can exist only for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M27">View MathML</a> (see [[1], Theorem 1]). For this reason, we prefer to set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M28">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M29">View MathML</a>.

In order to study equation (1.4) or its variants, we are going to follow a dynamical system approach, hence we choose to treat the independent variable (which in the applications has a spatial connotation) as a time variable, via the substitution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M30">View MathML</a>. Similarly, for the dependent variable, we make the substitution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M31">View MathML</a>. In this way, equation (1.4) reads as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M32">View MathML</a>

(1.5)

which belongs to the same class of (1.1).

In spite of the apparent simplicity of equation (1.5), a throughout study of its solutions may be a rather difficult task for a general nonconstant weight function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M3">View MathML</a>. Looking for homoclinic and heteroclinic solutions of (1.5), it will be natural to focus on the behavior of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M3">View MathML</a> at ±∞. In similar situations, various authors have confined their study to the case in which <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M3">View MathML</a> is asymptotically constant [4-6], or eventually constant [7,8]. These assumptions are also justified by the analysis of some physical underlying models, in which a layered structure is present. With this respect, see the introduction in [8], where different eventually autonomous cases are listed for related NLSEs arising in nonlinear optics. Examples in which the nonlinear term presents a piecewise constant weight function have been studied also in biological and chemical models. In particular, these situations occur in the theory of wave propagation for reaction-diffusion systems; see, for instance, [9-11]. In the context of equation (1.5), examples in which <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M3">View MathML</a> is a piecewise constant function have been considered as well (see [12]).

The aim of the present paper is to provide multiplicity results regarding homoclinic and heteroclinic solutions for equation (1.1) under particular assumptions on the weight function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M37">View MathML</a>. Actually, we suppose that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M37">View MathML</a> is an eventually constant piecewise function with only two steps.

Our approach combines phase-plane analysis with time-mapping estimates. As in [5,7,8,10,11,13], the solutions are obtained by connecting the unstable and stable manifolds of the equilibrium points of the asymptotically autonomous equations. Such connections are performed by means of orbit paths of an intermediate equation, which represents the behavior of the system during a suitable interval of transition <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M39">View MathML</a> between the asymptotic states. Multiple connecting solutions arise when such interval length is sufficiently large. Lower estimates for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M40">View MathML</a> will be provided in terms of time mappings, which can be expressed by Abelian integrals. This kind of approach is also reminiscent of some topological methods for the study of Sturm-Liouville boundary value problems. Indeed, a solution that satisfies the Sturm-Liouville boundary conditions can be interpreted as a trajectory in the phase-plane that connects two lines (see [14-17]). Generalized Sturm-Liouville solutions, which connect the graphs of two functions or given planar continua, have been considered as well (see [18-20]). Often these problems can be settled in the framework of the theory of ODEs with nonlinear boundary conditions (see [21,22]).

In order to make our approach more transparent, we are going to perform our analysis for equation (1.5). This choice is motivated by the sake of avoiding unnecessary technicalities. Our arguments can be modified in a straightforward manner in the case of more general equation (1.1), with f satisfying (∗) (see Section 3). Homoclinic and heteroclinic orbits can be interpreted as solutions for some boundary value problems on unbounded intervals. In the last section we also outline possible applications of our approach to boundary value problems on a compact interval (like the Sturm-Liouville one).

Besides his manifold achievements in different areas of mathematics, Professor Jean Mawhin is one of the pioneers in the study of topological methods for nonlinear boundary value problems. It is a pleasure and an honor to have the possibility to dedicate our work to his important contributions in this area.

2 Homoclinic and heteroclinic solutions: multiplicity results

2.1 General setting

We consider the second-order nonlinear differential equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M41">View MathML</a>

(2.1)

and its equivalent system in the phase-plane

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M42">View MathML</a>

(2.2)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M2">View MathML</a> is a fixed coefficient and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M44">View MathML</a> is a bounded measurable function. Solutions of (2.1) are meant in the generalized (i.e., Carathéodory) sense (see [23]). Actually, in our results a step function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M3">View MathML</a> with only two jumps is considered and the solutions are piecewise smooth.

In the particular case of a constant coefficient <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M46">View MathML</a>, an elementary analysis of the system

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M47">View MathML</a>

(2.3)

shows that the associated phase portrait presents a center <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M48">View MathML</a> and two saddle points <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M49">View MathML</a>. These saddles are connected by two heteroclinic solutions (one connecting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M50">View MathML</a> to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M51">View MathML</a> in the upper half-plane and a symmetric one from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M51">View MathML</a> to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M50">View MathML</a> in the lower half-plane). The heteroclinic orbits are described by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M54">View MathML</a>

(2.4)

In [1] Belmonte-Beitia and Torres supposed that the weight function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M3">View MathML</a> is even andT-periodic and proved the existence of a heteroclinic solution connecting two periodic stationary states. Such a result generalizes to the case of periodic coefficients the situation described above for the autonomous system.

As already observed in the introduction, equation (2.1) has been already studied by various authors for its relevance in many applicative models. The aim of the present paper is to provide multiplicity results for heteroclinic and also homoclinic solutions of equation (2.1). A possible way to obtain this goal is to assume that the weight function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M3">View MathML</a> has a different behavior at infinity and in some intermediate time interval. In this setting, a first step consists of analyzing the situation in which <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M3">View MathML</a> is a stepwise function assuming only two values. We believe that the study of such a simplified case may lead to more general considerations, in which, for instance, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M3">View MathML</a> is eventually periodic. In this way, multiplicity results for Belmonte-Beitia and Torres’s model can be obtained (see Section 3 for a brief overview concerning possible applications of our approach). Equation (2.1) with a periodic stepwise coefficient <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M3">View MathML</a> has recently been studied in [24] in the context of chaotic-like dynamics.

As a tool, in our paper, we use the Poincaré map associated to system (2.2). Given a fixed time interval <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M60">View MathML</a> and a point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M61">View MathML</a>, we indicate by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M62">View MathML</a> the solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M63">View MathML</a> of (2.2) satisfying the initial condition <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M64">View MathML</a>. The Poincaré map on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M60">View MathML</a> is defined as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M66">View MathML</a>

From the fundamental theory of differential equations it follows that the domain <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M67">View MathML</a> of Φ is an open subset of the plane and Φ is an orientation preserving homeomorphism of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M67">View MathML</a> onto its image <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M69">View MathML</a>.

2.2 Analysis of the equation

Let us consider system (2.2) for the stepwise weight function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M3">View MathML</a>, defined as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M71">View MathML</a>

(2.5)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M72">View MathML</a>

System (2.2) can be seen as the superposition of the autonomous systems

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M73">View MathML</a>

(2.6)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M74">View MathML</a>

(2.7)

The first one describes the asymptotic behavior of the equation. We are going to use the orbits of the second system for connecting unstable/stable manifolds of (2.6) during the time interval <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M39">View MathML</a>. Figure 1 shows the superposition of the phase portraits of the two systems.

thumbnailFigure 1. In the present figure we have considered the superposition of the phase portraits of systems (2.6) (in darker color) and (2.7) (in lighter color) with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M76">View MathML</a>,<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M77">View MathML</a>,<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M78">View MathML</a>. For graphical reasons, a slightly different x- and y-scaling has been used.

First of all, we briefly analyze the structure of system (2.6). Its equilibrium points are the origin (which is a center) and two saddle points <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M79">View MathML</a>, where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M80">View MathML</a>

The stable and unstable manifolds of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M81">View MathML</a> are illustrated in Figure 2. The sets <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M82">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M83">View MathML</a> are, respectively, the stable and the unstable manifolds of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M84">View MathML</a>. Symmetrically, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M85">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M86">View MathML</a> are the stable and the unstable manifolds of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M87">View MathML</a>.

thumbnailFigure 2. The stable and unstable manifolds of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M88">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M89">View MathML</a>. The arrows show the direction of the flow along the orbits.

In the case of the cubic nonlinearity, we find

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M90">View MathML</a>

The solutions of system (2.2) with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M3">View MathML</a> defined as in (2.5) will be obtained by connecting suitably chosen parts of stable and unstable manifolds of (2.6) with trajectories of (2.7). More in detail, we proceed as follows.

Due to the Hamiltonian nature of the equation under consideration, we can consider the ‘energy’ level lines of system (2.7), given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M92">View MathML</a>

(2.8)

We also set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M93">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M94">View MathML</a>

The curve <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M95">View MathML</a> is the part of the line at level <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M96">View MathML</a> in the strip <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M97">View MathML</a>. For every c such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M98">View MathML</a>

the set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M95">View MathML</a> is a periodic orbit of (2.7) run in the clockwise sense, which intersects the y-axis at the points <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M100">View MathML</a>. The (minimal) period <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M101">View MathML</a> of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M95">View MathML</a> can be expressed by the following time-mapping formula:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M103">View MathML</a>

(2.9)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M104">View MathML</a> is the potential associated to the equation and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M105">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M106">View MathML</a> are such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M107">View MathML</a>

In our case,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M108">View MathML</a>

and by symmetry <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M109">View MathML</a>. More in detail, we can express the period <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M101">View MathML</a> by means of an elliptic integral in the following way:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M111">View MathML</a>

(2.10)

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M112">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M113">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M114">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M115">View MathML</a>. Notice that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M116">View MathML</a> in (2.10). Therefore

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M117">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M118">View MathML</a> is the Jacobi elliptic sine function of modulus<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M119">View MathML</a> (see [25,26]).

The level line <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M120">View MathML</a> contains the saddle points <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M121">View MathML</a> of system (2.7) and their heteroclinic connections <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M122">View MathML</a> in the upper half-plane and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M123">View MathML</a> in the lower half-plane. The energy level

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M124">View MathML</a>

corresponds to the closed curve <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M125">View MathML</a>, which is tangent to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M126">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M127">View MathML</a>. Moreover, for

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M128">View MathML</a>

the energy level line <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M129">View MathML</a> contains the saddle points <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M81">View MathML</a> of system (2.6).

Using the parameters <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M131">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M132">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M133">View MathML</a>, we define the regions

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M134">View MathML</a>

Both ℰ and ℱ are invariant sets for (2.7); they are filled by periodic orbits. In the sequel, the orbits in the region ℰ will be called internal, while those in ℱ will be called external. The choice of these names is made in order to distinguish the trajectories with respect to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M129">View MathML</a>, which contains the saddle points <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M81">View MathML</a> of system (2.6). (See Figure 3.)

thumbnailFigure 3. The energy level lines<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M137">View MathML</a>,<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M138">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M139">View MathML</a>for system (2.7) superimposed over the stable and unstable manifolds of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M140">View MathML</a>for system (2.7). The colored regions are the invariant sets ℰ (the darker one) and ℱ (lighter). The figure has been plotted with the parameters <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M141">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M142">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M143">View MathML</a>. For graphical reasons, a slightly different x- and y-scaling has been used.

In order to obtain homoclinic or heteroclinic solutions for system (2.2) with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M3">View MathML</a> defined as in (2.5), we connect the unstable manifold of the point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M145">View MathML</a> to the stable manifold of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M146">View MathML</a> via an orbit path of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M95">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M148">View MathML</a>. Actually, this is the only way to get the desired solutions. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M149">View MathML</a>, such connection will follow a trajectory <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M150">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M151">View MathML</a>, so it will lie in the invariant annular region ℰ bounded by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M125">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M129">View MathML</a>. On the contrary, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M154">View MathML</a> the connection lies in the set ℱ between <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M129">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M120">View MathML</a>.

To provide more explicit details, we consider the case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M157">View MathML</a>. A solution of (2.2), which is homoclinic to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M51">View MathML</a>, can be produced in two ways. One consists in connecting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M159">View MathML</a> to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M160">View MathML</a>, using an orbit path of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M95">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M162">View MathML</a>. Another possibility is given by connecting a point of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M127">View MathML</a> to a point of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M126">View MathML</a> via an orbit path of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M95">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M166">View MathML</a>. For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M167">View MathML</a>, we can obtain our solution if and only if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M168">View MathML</a> is an integer multiple of the fundamental period <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M169">View MathML</a> of the orbit <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M129">View MathML</a>. Indeed, such a solution will be constantly equal to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M51">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M172">View MathML</a>, and for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M173">View MathML</a> it coincides with the periodic solution of (2.7) such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M174">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M175">View MathML</a>. This solution makes <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M176">View MathML</a> turns around the origin for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M177">View MathML</a>.

Similar considerations can be developed with respect to heteroclinic solutions. For instance, a heteroclinic orbit from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M51">View MathML</a> to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M50">View MathML</a> can be obtained as follows: by connecting a point of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M159">View MathML</a> with one of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M181">View MathML</a> using an orbit path of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M95">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M183">View MathML</a>, or by a connection of two different points of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M127">View MathML</a> via an orbit path of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M95">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M186">View MathML</a>. For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M187">View MathML</a>, we can obtain our solution if and only if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M168">View MathML</a> is an odd multiple of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M189">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M169">View MathML</a> is the period of the orbit <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M129">View MathML</a>. Indeed, such a solution will be constantly equal to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M51">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M193">View MathML</a> and constantly equal to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M50">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M195">View MathML</a>. Moreover, it coincides for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M173">View MathML</a> with the solution of (2.7) such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M174">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M175">View MathML</a>. This solution makes <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M199">View MathML</a> half-turns around the origin for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M173">View MathML</a>.

Figure 4, although not exhaustive of all the conceivable cases, summarizes several different possibilities. Indeed, a trajectory homoclinic to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M51">View MathML</a> can be obtained as follows: move by system (2.6) along <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M159">View MathML</a> from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M51">View MathML</a> to the intersection point of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M159">View MathML</a> with a closed curve external to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M87">View MathML</a>, namely δ (for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M206">View MathML</a>). This point is put in evidence with a small black circle. Next, follow δ (by system (2.7)) in order to reach the intersection of δ with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M160">View MathML</a> (for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M173">View MathML</a>). Such an intersection point is indicated by a grey square. Finally, switch to system (2.6) and move toward <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M51">View MathML</a> along <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M160">View MathML</a> (for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M211">View MathML</a>). With a similar procedure, we can obtain a heteroclinic trajectory from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M51">View MathML</a> to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M84">View MathML</a>. In fact, we can move by system (2.6) along <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M159">View MathML</a> from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M51">View MathML</a> to the intersection point of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M159">View MathML</a> with a closed curve external to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M87">View MathML</a>, namely γ (for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M206">View MathML</a>). Next, follow γ (by system (2.7)) to the intersection with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M181">View MathML</a> (for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M173">View MathML</a>). Finally, switch to system (2.6) and move toward <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M50">View MathML</a> along <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M181">View MathML</a> (for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M195">View MathML</a>).

thumbnailFigure 4. In the present figure we have considered the superposition of (2.6) and (2.7) for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M76">View MathML</a>,<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M77">View MathML</a>,<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M78">View MathML</a>. The closed curves α, β, γ, δ represent four different level lines of system (2.7): the curves α and β in ℰ correspond to a level <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M227">View MathML</a>, while γ and δ in ℱ refer to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M183">View MathML</a>. In order to describe a connection from an unstable manifold to a stable one of system (2.6) via an orbit path of (2.7), we have marked with a black circle possible starting points and with a grey square some available end points on the same level lines.

In general, the connections through the trajectories of system (2.7) will either involve only an arc of the closed curve <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M95">View MathML</a>, or they will require to perform a certain number of winds around the origin, depending on the available time. If the length of the time interval <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M39">View MathML</a> is small, only the first case is possible. However, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M168">View MathML</a> is large enough, more choices arise. For instance, looking at the orbit δ in Figure 4, we could make a certain number of loops from the black circle before reaching the grey square on the same orbit. Similar considerations apply to the heteroclinic trajectory described above with reference to γ.

Until now we have described the ‘external connections’, namely those which lie in the region ℱ. Further possibilities appear if we consider ‘internal connections’ by means of orbit paths contained in ℰ. For example, if we look at the orbit β in Figure 4, we can obtain trajectories homoclinic to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M51">View MathML</a> by choosing one of the two black circles as a starting point from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M127">View MathML</a> and one of the two grey squares as an end point on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M126">View MathML</a>. Loops along β will be permitted if the time is sufficiently large. The construction of heteroclinic orbits from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M51">View MathML</a> to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M50">View MathML</a> via the line α follows a similar procedure.

2.3 Study of the Poincaré map

Another possible point of view to describe the previous construction of homoclinic and heteroclinic solutions consists in considering the Poincaré map associated to system (2.7) for the time interval <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M39">View MathML</a>. Due to the autonomous nature of this system, we have that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M238">View MathML</a>, denoted simply by Φ when no confusion may occur. Moreover, the region

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M239">View MathML</a>

is a compact invariant set contained in the domain of Φ. Let us denote by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M240">View MathML</a> the part of the unstable manifold <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M159">View MathML</a> between <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M51">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M120">View MathML</a>, including the extreme points. Hence the external connections can be precisely described by looking at the intersection points of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M244">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M160">View MathML</a> (for the homoclinics) and with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M181">View MathML</a> (for the heteroclinics), as illustrated in Figure 5.

thumbnailFigure 5. In the present figure we show the transformation ofunder the Poincaré mapΦfor<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M248">View MathML</a>. The parameters used are <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M141">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M250">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M251">View MathML</a>. For graphical reasons a slightly different x- and y-scaling has been used.

With reference to Figure 5, we observe that the intersection point of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M159">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M120">View MathML</a> (indicated by a small circle) is moved by Φ along <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M120">View MathML</a> to a point very close to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M255">View MathML</a>. On the other hand, during the time interval <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M39">View MathML</a>, the point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M51">View MathML</a> makes a little more than a complete turn around the origin; the final position <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M258">View MathML</a> is indicated by a cross. The arc <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M244">View MathML</a> is a spiral-like curve with one end near <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M260">View MathML</a> and the other on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M129">View MathML</a>. For the time length <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M262">View MathML</a>, considered in Figure 5, there are precisely two intersections of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M244">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M160">View MathML</a>, corresponding to external homoclinic solutions. Meanwhile, we have also an external heteroclinic solution due to the intersection of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M244">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M181">View MathML</a>. All these three intersection points have been indicated by a grey square (following the same convention used in Figure 4).

In order to describe further these solutions, let us consider their energy at the time <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M267','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M267">View MathML</a> that we denote by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M268','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M268">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M269">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M270','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M270">View MathML</a>, with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M271','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M271">View MathML</a>. Let us define the points <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M272">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M273','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M273">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M274','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M274">View MathML</a>. The solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M63">View MathML</a> of (2.2) with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M276','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M276">View MathML</a> or <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M277','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M277">View MathML</a> has the following behavior: <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M278','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M278">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M279','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M279">View MathML</a>, with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M280','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M280">View MathML</a> and convex for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M281','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M281">View MathML</a>. Moreover, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M282','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M282">View MathML</a> is increasing on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M283','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M283">View MathML</a> and decreasing on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M284','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M284">View MathML</a>. On the interval <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M39">View MathML</a>, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M286','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M286">View MathML</a>, the solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M282','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M282">View MathML</a> is concave, while if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M288','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M288">View MathML</a>, the solution has two maxima and one (negative) minimum separated by two simple zeros. The solution of system (2.2) with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M289','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M289">View MathML</a> is such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M278','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M278">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M291','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M291">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M292','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M292">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M293','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M293">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M294','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M294">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M295','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M295">View MathML</a>; <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M282','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M282">View MathML</a> is convex for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M297','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M297">View MathML</a>. Moreover, the solution has one maximum and one minimum separated by one simple zero.

Even if the time interval length <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M298','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M298">View MathML</a> is small, we always have at least one intersection of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M244">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M160">View MathML</a> and thus a homoclinic solution. On the other hand, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M301','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M301">View MathML</a> grows, the spiral curve <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M244">View MathML</a> will wind more times around the origin, hence more homoclinic/heteroclinic solutions will appear.

If we look for the internal connections (made by orbit paths lying in the region ℰ), we proceed as follows.

Let us set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M303','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M303">View MathML</a> and denote by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M304','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M304">View MathML</a> its symmetric part with respect to the x-axis. Figure 6 illustrates the involved geometry from the point of view of the Poincaré map. We observe that all the points of ℒ are contained in ℰ, hence they are periodic points (of different periods). As before, during the time interval <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M39">View MathML</a>, the point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M51">View MathML</a> moves clockwise around the origin along <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M307','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M307">View MathML</a>; the final position <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M308','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M308">View MathML</a> is indicated by a cross. Symmetrically, the point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M50">View MathML</a> performs exactly the same angle around the origin, reaching the final position <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M310','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M310">View MathML</a>, indicated by a small circle. All the other points of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M127">View MathML</a> move, under the action of Φ, on the energy level lines <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M95">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M313','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M313">View MathML</a>. The arc <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M314','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M314">View MathML</a> is a curve contained in ℰ, connecting the two antipodal points <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M315','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M315">View MathML</a> and leaning on one point of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M125">View MathML</a>. This tangent point is the image through Φ of the point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M317','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M317">View MathML</a>. Such a property of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M314','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M314">View MathML</a> implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M314','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M314">View MathML</a> intersects both ℒ and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M320','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M320">View MathML</a> (at different points). These two points have been indicated by a grey square in the figure. Accordingly, we always find at least an internal homoclinic solution and an internal heteroclinic one.

thumbnailFigure 6. In the present figure we show the transformation of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M321','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M321">View MathML</a>under the Poincaré mapΦfor<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M322','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M322">View MathML</a>, the same time considered in Figure5. The parameters used are <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M141">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M250">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M325','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M325">View MathML</a>, as before. The point indicated by a cross is precisely the same point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M258">View MathML</a> of Figure 5. Notice that for this time interval the point makes a little more than a complete turn around the origin. For graphical reasons, a slightly different x- and y-scaling has been used.

Summarizing the above information, we conclude that, for every time interval <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M39">View MathML</a>, we have at least two solutions which are homoclinic to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M51">View MathML</a> (one external and one internal) and one internal heteroclinic solution from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M51">View MathML</a> to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M84">View MathML</a>.

For the external connections, if the time interval length <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M331','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M331">View MathML</a> grows, the situation can be summarized as follows. The number of external homoclinic/heteroclinic solutions increases, depending on the number of winds around the origin of the curve <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M244">View MathML</a>. Indeed, the end point of such a curve on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M120">View MathML</a> cannot go beyond <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M255">View MathML</a>, while the other end point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M258">View MathML</a> is free to move on the periodic orbit <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M307','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M307">View MathML</a>. If we denote by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M337','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M337">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M338','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M338">View MathML</a> the number of external homoclinic and heteroclinic solutions respectively, it holds that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M339','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M339">View MathML</a>. From the above discussion, we can conclude that if, for some nonnegative integer n, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M340','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M340">View MathML</a> (where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M169">View MathML</a> is the period of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M129">View MathML</a>), then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M343','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M343">View MathML</a>. An analogous lower bound can be provided for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M344','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M344">View MathML</a>. For a formal proof, see Theorem 2.1.

When <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M298','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M298">View MathML</a> grows, the situation for the internal homoclinic/heteroclinic solutions is more intriguing, as illustrated in Figure 7 and Figure 8. A twist effect depending on the different periods of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M125">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M129">View MathML</a> produces a double spiral-like curve <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M314','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M314">View MathML</a> when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M301','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M301">View MathML</a> is sufficiently large.

thumbnailFigure 7. In the present figure we show the transformation ofunder the Poincaré mapΦfor<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M351','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M351">View MathML</a>. The parameters used are <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M141">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M250">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M251">View MathML</a>, as before. For graphical reasons, a slightly different x- and y-scaling has been used.

thumbnailFigure 8. The present figure shows the evolution ofunder the Poincaré mapΦfor time intervals of different lengths. The parameters used are <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M141">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M250">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M325','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M325">View MathML</a>, as before, while for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M298','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M298">View MathML</a>, we have considered the cases 35, 60, 100, 150 (from the left to the right). For graphical reasons, a slightly different x- and y-scaling has been used.

The fact that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M314','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M314">View MathML</a> looks like a double spiral depends on the different velocities of the points <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M81">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M362','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M362">View MathML</a> (which is the intersection of ℒ with the negative y-axis). In fact, since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M363','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M363">View MathML</a>, the points on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M125">View MathML</a> move faster than those of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M129">View MathML</a>. By construction, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M366','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M366">View MathML</a>, while <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M367','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M367">View MathML</a>. Therefore, when the time gap <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M40">View MathML</a> is sufficiently large, the number of turns of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M369','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M369">View MathML</a> around the origin will exceed the number of turns of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M370','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M370">View MathML</a>. Accordingly, the image through Φ of the right part of ℒ (the sub-arc of ℒ connecting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M51">View MathML</a> to R) is a spiral that winds a certain number of times in the clockwise sense around the origin. Indeed, it connects the slower point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M258">View MathML</a> to the faster one <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M369','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M369">View MathML</a> inside the region ℰ, as illustrated in Figure 9. The number of half-turns of this spiral depends on a ‘rotational gap’ that, for our purposes, it will be convenient to define as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M374','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M374">View MathML</a>

(2.11)

Similarly, the image through Φ of the left part of ℒ (the sub-arc of ℒ connecting R to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M50">View MathML</a>) is a spiral-like curve winding a certain number of times in the counterclockwise sense around the origin. Indeed, it connects the points <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M369','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M369">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M377','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M377">View MathML</a> inside the region ℰ (the first point moves at a faster speed than the second one). Again, the number of half-turns of this second curve will depend on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M378','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M378">View MathML</a>. In the end, gluing together the two spiral-like curves, we conclude that, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M168">View MathML</a> is sufficiently large, the arc <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M314','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M314">View MathML</a> will appear like a double spiral with a central ‘hook’. To better describe the structure of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M314','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M314">View MathML</a>, we should observe that the ‘tip of the hook’ is not <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M369','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M369">View MathML</a>, but it is the image through Φ of a point (close to R) on the left part of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M383','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M383">View MathML</a> (we owe this remark to the referee). For a formal proof based on the argument outlined above, see Theorem 2.2.

thumbnailFigure 9. The present figure explains the twist effect due to the different periods of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M138">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M139">View MathML</a>. We have considered the evolution of the part of ℒ with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M386','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M386">View MathML</a> connecting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M51">View MathML</a> to the point R for two different time gaps: <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M262">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M389','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M389">View MathML</a>. The corresponding two images <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M258">View MathML</a> are indicated by a cross, while the images <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M369','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M369">View MathML</a> are marked by a black dot. The parameters used are <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M141">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M250">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M325','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M325">View MathML</a>, as before. Notice that the two images <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M258">View MathML</a>, although close to each other in the figure, have made a different number of turns around the origin. For graphical reasons, a slightly different x- and y-scaling has been used.

In preparation for this theorem, we introduce the following notation. Given a positive real number S and an energy level <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M396','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M396">View MathML</a>, we denote by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M397','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M397">View MathML</a> the lower integer part of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M398','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M398">View MathML</a>, and by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M399','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M399">View MathML</a> the upper integer part of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M398','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M398">View MathML</a>. By definition, during a time interval of length S, a point on the orbit <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M401','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M401">View MathML</a> makes more that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M402','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M402">View MathML</a> half-turns around the origin, but less that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M403','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M403">View MathML</a> half-turns. Thus the rotational gap defined above can be written as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M404','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M404">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M405','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M405">View MathML</a>. By definition, in the open interval <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M406','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M406">View MathML</a>, the smallest integer is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M407','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M407">View MathML</a>, while the largest integer is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M408','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M408">View MathML</a>. Hence, the open interval <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M409','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M409">View MathML</a> contains <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M410','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M410">View MathML</a> positive integers.

2.4 Conclusion

After the preliminary study of the previous subsections, we are now in a position to express our results for the equation

as statements with a formal proof. Recall that the weight function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M3">View MathML</a> is defined as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M413','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M413">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M414','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M414">View MathML</a>

We also set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M415','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M415">View MathML</a> and consider the Poincaré map <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M416','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M416">View MathML</a> associated to the system

In order to make the exposition more clear, we have decided to consider separately the cases of external and internal connections.

Theorem 2.1 (External connections)

Under the above assumptions, the following results hold for solutions which satisfy the energy condition

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M418','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M418">View MathML</a>

there always exist a (positive) solution homoclinic to<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M51">View MathML</a>and a (negative) solution homoclinic to<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M50">View MathML</a>;

if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M421','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M421">View MathML</a>, there always exist a heteroclinic solution from<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M51">View MathML</a>to<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M50">View MathML</a>and a heteroclinic solution from<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M50">View MathML</a>to<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M87">View MathML</a>, both with exactly one zero;

if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M426','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M426">View MathML</a> (for some integer<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M427','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M427">View MathML</a>) then, for each integerjwith<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M428','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M428">View MathML</a>, there exist at least one solution homoclinic to<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M51">View MathML</a>and one solution homoclinic to<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M84">View MathML</a>, both with exactly 2jzeros;

if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M431','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M431">View MathML</a> (for some integer<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M427','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M427">View MathML</a>) then, for each integerjwith<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M428','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M428">View MathML</a>, there exist at least one heteroclinic solution from<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M51">View MathML</a>to<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M50">View MathML</a>and one from<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M50">View MathML</a>to<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M87">View MathML</a>, both with exactly<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M438','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M438">View MathML</a>zeros.

Proof As a first step, we focus our attention on the search of solutions homoclinic to the point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M87">View MathML</a>. The case of homoclinics to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M50">View MathML</a> is analogous, thus it will be omitted.

Recalling that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M441','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M441">View MathML</a>, our goal is to find points on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M159">View MathML</a> which are moved by the Poincaré map <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M416','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M416">View MathML</a> on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M444','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M444">View MathML</a>.

We introduce in the phase-plane for (2.7), a system of polar coordinates <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M445','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M445">View MathML</a>, with center in the origin. The initial points in the arc (which is the closure of the intersection of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M159">View MathML</a> with the external region ℱ) are parameterized as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M448','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M448">View MathML</a>, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M449','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M449">View MathML</a>, with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M450','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M450">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M451','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M451">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M452','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M452">View MathML</a> strictly increasing with s. The target set is the symmetric of with respect to the x-axis. It will be denoted by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M454','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M454">View MathML</a> and parameterized by reversing the angle <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M455','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M455">View MathML</a>.

Using the same polar coordinates to represent the solutions of (2.7), we can express the final points <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M456','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M456">View MathML</a> as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M457','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M457">View MathML</a>

with

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M458','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M458">View MathML</a>

We notice that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M459','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M459">View MathML</a>

is the solution of (2.7) (at the time t), which departed from the point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M460','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M460">View MathML</a> at the time <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M461','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M461">View MathML</a>. Observe that, for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M449','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M449">View MathML</a>, the map

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M463','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M463">View MathML</a>

is strictly decreasing on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M9">View MathML</a> (this is an equivalent way to express the fact that the solutions turn around the origin in the clockwise sense).

With these positions, we obtain an external solution homoclinic to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M51">View MathML</a> if and only if there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M466','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M466">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M467','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M467">View MathML</a>. This happens if and only if

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M468','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M468">View MathML</a>

(2.12)

for some nonnegative integer j. In this case, if we denote by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M469','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M469">View MathML</a> the corresponding homoclinic solution of (2.1) such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M470','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M470">View MathML</a>, we have that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M471','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M471">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M472','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M472">View MathML</a>. Moreover, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M473','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M473">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M474','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M474">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M475','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M475">View MathML</a>, while <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M476','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M476">View MathML</a> has precisely 2j zeros in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M477','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M477">View MathML</a> if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M478','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M478">View MathML</a>.

Using the fact that the energy level lines of (2.7) in the region <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M479','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M479">View MathML</a> are strictly star-shaped with respect to the origin and symmetric with respect to the x-axis, we find that (2.12) holds (for some nonnegative integer j) if and only if

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M480','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M480">View MathML</a>

(2.13)

is satisfied (for the same j).

By virtue of (2.13), we can refer only to the angular coordinates, hence we will obtain solutions as follows. Since the point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M481','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M481">View MathML</a> lies on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M120">View MathML</a> (which is the trajectory of a heteroclinic solution of (2.7)), it can never reach the point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M255">View MathML</a>. Accordingly, the angle <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M484','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M484">View MathML</a> satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M485','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M485">View MathML</a>

On the other hand, as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M450','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M450">View MathML</a>, we have that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M487','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M487">View MathML</a>, therefore

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M488','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M488">View MathML</a>

As a consequence, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M489','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M489">View MathML</a> and the intermediate value theorem ensures the existence of an <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M490','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M490">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M491','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M491">View MathML</a>. In this way we have found a positive homoclinic solution to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M87">View MathML</a>, independently of the length T of the time interval <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M39">View MathML</a>.

Suppose now that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M494','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M494">View MathML</a> for some positive integer n. In this case,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M495','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M495">View MathML</a>

As a consequence, for every integer j with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M496','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M496">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M497','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M497">View MathML</a>, and again the intermediate value theorem ensures the existence of an <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M490','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M490">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M499','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M499">View MathML</a>. This ends the proof for homoclinic solutions.

For the search of heteroclinic solutions from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M51">View MathML</a> to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M50">View MathML</a>, we follow a similar procedure, choosing as a target the set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M502','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M502">View MathML</a>, which is the symmetric of with respect to the y-axis (here we exploit the oddness of the nonlinear term <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M504','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M504">View MathML</a>). In terms of polar coordinates, the desired solutions will be obtained if and only if there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M466','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M466">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M506','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M506">View MathML</a>

(2.14)

for some integer <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M478','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M478">View MathML</a>. Then the proof can be concluded as above, via the intermediate value theorem. We observe that the case of heteroclinic solutions from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M50">View MathML</a> to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M51">View MathML</a> is analogous, thus it will be omitted. □

The next theorem deals with the internal connections. For this result, it is useful to recall the rotational gap defined in (2.11).

Theorem 2.2 (Internal connections)

Under the above assumptions, the following results hold for solutions which satisfy the energy condition

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M510','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M510">View MathML</a>

there always exist a solution homoclinic to<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M51">View MathML</a>and a solution homoclinic to<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M50">View MathML</a>;

there always exist a heteroclinic solution from<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M51">View MathML</a>to<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M50">View MathML</a>and a heteroclinic solution from<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M50">View MathML</a>to<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M87">View MathML</a>;

if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M517','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M517">View MathML</a>, there exist at least<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M518','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M518">View MathML</a>solutions homoclinic to<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M51">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M518','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M518">View MathML</a>solutions homoclinic to<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M84">View MathML</a>;

if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M522','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M522">View MathML</a>, there exist at least<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M518','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M518">View MathML</a>heteroclinic solutions from<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M51">View MathML</a>to<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M50">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M518','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M518">View MathML</a>from<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M50">View MathML</a>to<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M87">View MathML</a>.

Proof As the first step, we focus our attention on the search of solutions homoclinic to the point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M87">View MathML</a>. The case of homoclinics to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M50">View MathML</a> is analogous, thus it will be omitted.

Our goal is to find points on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M531','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M531">View MathML</a> which are moved by the Poincaré map <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M416','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M416">View MathML</a> on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M533','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M533">View MathML</a>.

As a preliminary remark, we note that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M534','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M534">View MathML</a> is a simple arc contained in ℰ, connecting its end points <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M535','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M535">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M536','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M536">View MathML</a> on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M129">View MathML</a> through the ‘intermediate’ point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M538','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M538">View MathML</a> on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M539','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M539">View MathML</a>. Observe also that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M535','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M535">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M536','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M536">View MathML</a> are antipodal. In fact, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M50">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M51">View MathML</a> are antipodal and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M416','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M416">View MathML</a> is an odd map (indeed, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M150">View MathML</a> is a solution of (2.7) if and only if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M546','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M546">View MathML</a> is a solution of the same equation). Now, if the trivial situation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M547','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M547">View MathML</a> occurs, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M548','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M548">View MathML</a>. Otherwise, we find that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M536','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M536">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M535','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M535">View MathML</a> belong to the two different components of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M551','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M551">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M536','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M536">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M535','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M535">View MathML</a> are the end points of the arc <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M554','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M554">View MathML</a>, by an elementary connectivity argument, we conclude that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M555','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M555">View MathML</a>. This proves the first assertion of the theorem.

For the search of heteroclinic solutions from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M51">View MathML</a> to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M50">View MathML</a>, we adopt a similar procedure, choosing as a target the set ℒ itself. The fact that, for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M558','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M558">View MathML</a>, there is always at least one intersection point between <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M534','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M534">View MathML</a> and ℒ follows by the same argument developed for homoclinic solutions. Namely, if the trivial situation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M547','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M547">View MathML</a> occurs, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M561','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M561">View MathML</a>. Otherwise, we find that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M536','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M536">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M535','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M535">View MathML</a> belong to the two different components of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M564','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M564">View MathML</a>. Therefore, it follows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M565','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M565">View MathML</a>.

We study now the problem of multiplicity of solutions. As before, we consider at first the case of solutions homoclinic to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M87">View MathML</a>.

We introduce in the inner region ℰ of a phase-plane for (2.7) a system of modified polar coordinates <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M567','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M567">View MathML</a> with center in the origin. In this system, every point is determined by its angular coordinate θ and its energy E defined in (2.8). Since the energy level lines in ℰ are strictly star-shaped with respect to the origin, we obtain a coordinate system equivalent to the polar one.

Observe that the set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M568','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M568">View MathML</a> intersects every half-line from the origin exactly in one point. Accordingly, we can parameterize those points using the angular coordinate. With this convention, the initial points in the arc ℒ are parameterized as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M460','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M460">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M570','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M570">View MathML</a> (s is the angle) with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M571','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M571">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M572','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M572">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M573','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M573">View MathML</a>. As a consequence, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M452','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M452">View MathML</a> is strictly decreasing for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M575','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M575">View MathML</a> and strictly increasing in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M576','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M576">View MathML</a>. The target set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M577','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M577">View MathML</a> (the symmetric of ℒ with respect to the x-axis) is parameterized with the angle <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M578','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M578">View MathML</a>.

For the specific case of system (2.6), an analytic expression for ℒ as a graph in the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M567','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M567">View MathML</a>-plane, is given as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M580','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M580">View MathML</a>

(2.15)

with

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M581','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M581">View MathML</a>

Hence, the natural parametrization of ℒ in the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M567','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M567">View MathML</a>-plane is given by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M583','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M583">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M584','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M584">View MathML</a>.

Using the same modified polar coordinates to represent the solutions of (2.7), we can express the final points <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M534','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M534">View MathML</a> in the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M567','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M567">View MathML</a>-plane by means of their angular coordinate <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M587','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M587">View MathML</a> and energy <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M588','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M588">View MathML</a>. As in the previous proof, we observe that, for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M570','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M570">View MathML</a>, the map <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M590','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M590">View MathML</a> is strictly decreasing on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M9">View MathML</a>, while the energy is constant with respect to t. Observe also that in the new coordinates <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M567','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M567">View MathML</a>, system (2.7) becomes

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M593','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M593">View MathML</a>

(2.16)

with

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M594','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M594">View MathML</a>

In the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M567','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M567">View MathML</a>-plane, the points of ℒ move with a negative angular speed along the lines of constant energy. Thus, under the action of the flow associated to (2.16), the points of ℒ shift from the right to the left in the strip

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M596','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M596">View MathML</a>

As we have explained before, we obtain an internal solution homoclinic to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M51">View MathML</a> whenever there is a point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M598','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M598">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M599','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M599">View MathML</a>, with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M405','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M405">View MathML</a>. In the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M567','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M567">View MathML</a>-plane, this target set is expressed as the union of the graphs

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M602','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M602">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M603','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M603">View MathML</a>

with

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M604','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M604">View MathML</a>

(see Figures 10, 11 for a graphical description).

thumbnailFigure 10. The initial setand the target set<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M606','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M606">View MathML</a>represented in the<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M607','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M607">View MathML</a>-plane. The present figure is drawn for the parameters <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M141">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M609','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M609">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M251">View MathML</a>. The internal region ℰ corresponds to the strip <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M611','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M611">View MathML</a>. Since we are interested in the evolution of the set ℒ through the Poincaré map, we consider only the half-strip <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M612','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M612">View MathML</a>. For the flow associated to (2.16), all the points of ℒ move from the right to the left on lines parallel to the θ-axis. The point R (as well as the points on the line <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M613','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M613">View MathML</a>) moves faster than the points <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M81">View MathML</a> which are on the line <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M615','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M615">View MathML</a>. During all the evolution, the distance between the images of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M50">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M51">View MathML</a> remains constantly equal to π.

thumbnailFigure 11. The present figure describes the same situation of Figure7in the angle-energy coordinates (with the same choice of the coefficientsk,<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M618','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M618">View MathML</a>,<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M619','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M619">View MathML</a>and time gap<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M620','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M620">View MathML</a>).

In this setting, we obtain an internal solution homoclinic to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M51">View MathML</a> if and only if there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M622','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M622">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M623','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M623">View MathML</a>, where we have denoted by Φ the Poincaré map for (2.16) in the time interval <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M624','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M624">View MathML</a>. Of course Φ is exactly <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M416','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M416">View MathML</a> (which was the Poincaré map associated to (2.7)) in the new <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M567','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M567">View MathML</a>-coordinates. Using the parameterized curve <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M460','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M460">View MathML</a> for the initial points, we can express Φ as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M628','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M628">View MathML</a>

(recall that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M583','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M583">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M630','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M630">View MathML</a>).

The points <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M50">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M51">View MathML</a> are antipodal, lie on the same energy line <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M129">View MathML</a> and move with the same angular speed. In the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M567','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M567">View MathML</a>-plane, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M50">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M51">View MathML</a> are expressed by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M637','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M637">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M638','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M638">View MathML</a>. Therefore,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M639','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M639">View MathML</a>

(2.17)

Moreover, for every integer <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M640','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M640">View MathML</a>, we have that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M641','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M641">View MathML</a>

(this follows from the fact that the orbit <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M129">View MathML</a> has a period <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M169">View MathML</a> and is symmetric with respect to the x-axis).

In the same plane, the point R is indicated by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M644','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M644">View MathML</a>. Using the fact that the orbit <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M125">View MathML</a> has a period <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M646','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M646">View MathML</a> and is symmetric with respect to the y-axis, we find that, for every integer <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M647','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M647">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M648','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M648">View MathML</a>

Suppose now that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M649','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M649">View MathML</a>

(2.18)

Then, the open interval <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M409','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M409">View MathML</a> contains at least one odd integer.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M651','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M651">View MathML</a> be the largest odd integer contained in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M406','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M406">View MathML</a>. From <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M653','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M653">View MathML</a>, we have that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M654','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M654">View MathML</a>

(2.19)

Suppose also that j is a positive integer with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M655','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M655">View MathML</a>. In this case, from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M656','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M656">View MathML</a>, we have that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M657','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M657">View MathML</a>

(2.20)

Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M658','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M658">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M659','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M659">View MathML</a>

with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M660','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M660">View MathML</a> be the left and the right parts of ℒ in the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M567','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M567">View MathML</a>-plane. Similarly, we define

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M662','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M662">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M663','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M663">View MathML</a>

By definition, for any nonnegative integer i, the set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M664','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M664">View MathML</a> is a simple arc connecting the points <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M665','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M665">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M666','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M666">View MathML</a> in the rectangle <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M667','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M667">View MathML</a> and, similarly, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M668','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M668">View MathML</a> is a simple arc connecting the points <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M666','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M666">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M670','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M670">View MathML</a> in the rectangle <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M671','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M671">View MathML</a>. On the other hand, the set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M672','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M672">View MathML</a> is a simple arc connecting the points <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M673','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M673">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M674','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M674">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M675','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M675">View MathML</a> and, similarly, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M676','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M676">View MathML</a> is a simple arc connecting the points <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M674','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M674">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M678','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M678">View MathML</a> in the same strip.

Suppose now that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M679','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M679">View MathML</a> (in this case, j is odd). In such a situation, from (2.19) and (2.20) we have that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M680','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M680">View MathML</a>

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M681','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M681">View MathML</a> an even integer. Hence the points <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M674','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M674">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M673','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M673">View MathML</a> are separated by the arc <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M684','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M684">View MathML</a>, while the points <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M674','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M674">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M686','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M686">View MathML</a> are separated by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M684','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M684">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M688','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M688">View MathML</a>. As a consequence, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M689','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M689">View MathML</a> contains at least three points, precisely the nonempty intersections of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M690','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M690">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M691','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M691">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M692','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M692">View MathML</a>.

Suppose that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M693','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M693">View MathML</a> (in this case, j is even). In such a situation, from (2.19) and (2.20) we have that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M694','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M694">View MathML</a>

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M695','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M695">View MathML</a> an even integer. Hence the points <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M674','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M674">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M673','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M673">View MathML</a> are separated by the arcs <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M684','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M684">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M688','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M688">View MathML</a>. The points <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M674','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M674">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M686','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M686">View MathML</a> are separated by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M684','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M684">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M688','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M688">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M704','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M704">View MathML</a>. As a consequence, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M689','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M689">View MathML</a> contains at least five points: two points coming from the intersections of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M672','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M672">View MathML</a> with the target set Ξ and three from the intersections of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M676','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M676">View MathML</a>.

Proceeding by induction, with the same argument, the following result is obtained.

Claim 1<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M708','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M708">View MathML</a>has at least<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M709','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M709">View MathML</a>solutions, where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M710','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M710">View MathML</a>is the number of integers less than or equal to<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M651','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M651">View MathML</a>which are contained in the open interval<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M406','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M406">View MathML</a>.

A completely symmetric argument leads to the same multiplicity result for solutions which are homoclinic to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M84">View MathML</a>.

At last, we look for a multiplicity result for heteroclinic solutions from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M51">View MathML</a> to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M84">View MathML</a>. We follow step by step the argument described in the part of the proof devoted to the search of multiple internal homoclinic solutions and, therefore, we transform our equation in the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M567','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M567">View MathML</a>-coordinates. As we have explained before, we obtain an internal heteroclinic (from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M51">View MathML</a> to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M50">View MathML</a>) whenever there is a point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M598','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M598">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M720','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M720">View MathML</a>, with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M721','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M721">View MathML</a>. In the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M567','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M567">View MathML</a>-plane, this target set is expressed as the union of the graphs

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M723','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M723">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M724','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M724">View MathML</a> is the function defined in (2.15).

As before, we assume the validity of condition (2.18). Then the open interval <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M725','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M725">View MathML</a> contains at least one even integer.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M726','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M726">View MathML</a> be the largest even integer contained in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M406','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M406">View MathML</a>. From <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M728','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M728">View MathML</a>, we have that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M729','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M729">View MathML</a>

For any positive integer j with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M730','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M730">View MathML</a>, we have that (2.20) holds too. At this point, we have simply to repeat (with obvious changes due to the fact that the target set is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M731','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M731">View MathML</a> which is a shift of Ξ by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M732','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M732">View MathML</a>) the argument presented above for the case of homoclinic connections and obtain the following conclusion.

Claim 2<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M733','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M733">View MathML</a>has at least<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M734','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M734">View MathML</a>solutions, where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M735','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M735">View MathML</a>is the number of integers less than or equal to<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M726','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M726">View MathML</a>which are contained in the open interval<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M406','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M406">View MathML</a>.

A completely symmetric argument leads to a multiplicity result for heteroclinic solutions from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M50">View MathML</a> to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M87">View MathML</a>.

As final step, we have just to show how the condition on the rotational gap allows us to achieve the conclusion from Claim 1 and Claim 2 that we have obtained along the proof. Now, it is sufficient to observe that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M740','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M740">View MathML</a> implies (2.18) and, therefore, the open interval <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M409','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M409">View MathML</a> contains at least two integers. More precisely, by the definition of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M742','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M742">View MathML</a>, the interval <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M409','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M409">View MathML</a> contains at least <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M410','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M410">View MathML</a> integers and, therefore,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M745','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M745">View MathML</a>

This concludes the proof of the theorem.  □

3 Remarks and related results

We end the paper with a list of remarks about possible variants and extensions of the main results obtained for equation

1. A problem which naturally arises from the analysis that we have performed concerns what happens if, for a stepwise weight function satisfying (2.5), we suppose that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M747','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M747">View MathML</a>. Repeating the preliminary phase-plane analysis of Section 2.2, one can easily check that for any gap <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M748','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M748">View MathML</a>, there always exist a solution homoclinic to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M51">View MathML</a> and another homoclinic to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M84">View MathML</a>, as well as a heteroclinic from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M51">View MathML</a> to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M50">View MathML</a> and another one from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M50">View MathML</a> to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M87">View MathML</a>. However, in general, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M755','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M755">View MathML</a>, one cannot obtain multiplicity results like those achieved in Section 2 without some extra assumptions on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M756','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M756">View MathML</a>. This is the reason for which, in the study of equation (2.1), we have considered only the case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M757','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M757">View MathML</a>. For different examples on related equations in which a weight coefficient can be above or below its limits at infinity, see, for instance, [4-6].

On the other hand, if we assume that the weight function changes its sign, with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M758','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M758">View MathML</a>, some interesting multiplicity results could be produced. In fact the phase portrait of (1.1) in the time interval <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M39">View MathML</a> shows a global center, hence if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M40">View MathML</a> is large, a lot of connections between the unstable and stable manifolds of the points <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M81">View MathML</a> can be obtained. For the sake of conciseness, we omit the study of this latter situation, which is beyond the goal of the present paper.

2. We notice that the same argument of the proofs applies to an equation of the form

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M5">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M764','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M764">View MathML</a>, and with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M765','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M765">View MathML</a> satisfying condition (∗), if we assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M37">View MathML</a> is a stepwise weight function, playing the same role of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M3">View MathML</a> in (2.1). In such a case, the saddle points become <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M768','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M768">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M769','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M769">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M770','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M770">View MathML</a> is the inverse of f restricted to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M9">View MathML</a>. In this way, we can apply our results to nonlinearities like those considered in [2,27]. More precisely, Theorem 2.1 holds without any further assumption, while for Theorem 2.2 we need to require a gap between the periods of the orbits <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M125">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M129">View MathML</a>. In order to obtain this gap, we can apply (for instance) some results ensuring the monotonicity of the time-map (like [[28], Theorem A]).

3. Due to the special form of the weight coefficient, it is standard to verify (via a simple rescaling procedure) that (1.1) is equivalent to an equation of the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M774','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M774">View MathML</a>

(3.1)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M775','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M775">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M776','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M776">View MathML</a> is a stepwise coefficient. In this manner, one can deal with some nonlinear Schrödinger-type equations related to the case of potential wells or potential walls [29,30].

4. The approach used in the proofs, based on the properties of the Poincaré map, guarantees that our results are stable with respect to small perturbations. More precisely, fixed a suitable length <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M777','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M777">View MathML</a> for the time interval, Theorems 2.1 and 2.2 provide a lower bound for the number of solutions. We can state that the same lower bound persists for a small perturbation of the coefficient in the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M778','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M778">View MathML</a>-norm on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M39">View MathML</a>. Therefore, the assumption that the weight <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M37">View MathML</a> in (1.1) or <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M776','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M776">View MathML</a> in (3.1) are stepwise functions can be slightly relaxed, so we can ‘smooth’ them.

5. With reference to equations (1.1) or (3.1) with stepwise coefficients, we observe that our approach can be adapted to boundary value problems on a compact interval <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M624','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M624">View MathML</a> like, e.g., the Dirichlet (two-point) or the Neumann problem. In these cases, we have to find solutions connecting given lines which depart from the origin. For the sake of conciseness, we cannot describe the most general situation, but we just outline a possible application for the Neumann problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M783','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M783">View MathML</a>

(3.2)

with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M3">View MathML</a> a stepwise function such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M785','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M785">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M786','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M786">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M787','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M787">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M788','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M788">View MathML</a>, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M789','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M789">View MathML</a>. Let us denote by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M790','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M790">View MathML</a> the Poincaré map associated to system (2.6) on the time interval <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M791','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M791">View MathML</a>, and by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M792','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M792">View MathML</a> the Poincaré map associated to system (2.7) on the time interval <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M793','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M793">View MathML</a>. We observe that there exists a maximal compact interval <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M794','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M794">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M795','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M795">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M796','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M796">View MathML</a> for all the points <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M797','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M797">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M798','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M798">View MathML</a>. In this manner <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M799','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M799">View MathML</a>. The curve

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M800','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M800">View MathML</a>

represents the set of all the points in the region <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M801','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M801">View MathML</a>, which are images (by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M790','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M790">View MathML</a>) of initial points of the x-axis (therefore <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M803','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M803">View MathML</a>). Figure 12 illustrates the curve <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M804','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M804">View MathML</a> for a short time interval (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M805','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M805">View MathML</a>). For a larger τ, the line <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M804','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M804">View MathML</a> becomes a double spiral with a certain number of turns around the origin, while the part of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M804','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M804">View MathML</a> contained in the region ℱ gets very close to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M159">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M809','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M809">View MathML</a>. Then we can repeat the same argument developed in the previous sections by looking for the intersections of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M810','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M810">View MathML</a> with the x-axis.

thumbnailFigure 12. In the present figure the line<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M811','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M811">View MathML</a>is the transformation of the segment<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M812','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M812">View MathML</a>under the Poincaré map<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M813','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M813">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M814','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M814">View MathML</a>. The parameters used are <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M141">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M250">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M251">View MathML</a>. For graphical reasons, a slightly different x- and y-scaling has been used.

6. As a final remark, we mention the fact that combining our technique with Ważewski’s method [31], following the approach developed by Conley in [18], one can deal with some more general classes of weight functions. For example, one could tackle with these techniques the cases of asymptotically constant or asymptotically periodic coefficients. These extensions, however, need a more delicate analysis and they are beyond the goals of the present paper.

We hope that the abundance of multiplicity results found in the present work (in the special case of stepwise coefficients) may suggest possible directions for extending Theorem 2.1 and Theorem 2.2 to more general weight functions. This will be our goal for a future investigation of the problem.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors contributed equally to the manuscript and read and approved the final manuscript.

Acknowledgements

The authors are deeply indebted with the referee for the careful checking of the manuscript and for his/her remarks, including a correction to an erroneous argument in the previous version of the proof. This research was partially supported by the project PRIN-2009 Equazioni Differenziali Ordinarie e Applicazioni.

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