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

Open Access Research

On Massera’s theorem concerning the uniqueness of a periodic solution for the Liénard equation. When does such a periodic solution actually exist?

Lilia Rosati and Gabriele Villari*

Author Affiliations

Dipartimento di Matematica ‘U. Dini’, Università degli Studi di Firenze, viale Morgagni 67/A, Firenze, 50134, Italy

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Boundary Value Problems 2013, 2013:144  doi:10.1186/1687-2770-2013-144

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2013/1/144


Received:14 December 2012
Accepted:25 May 2013
Published:11 June 2013

© 2013 Rosati and Villari; 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

In this note we consider the classical Massera theorem, which proves the uniqueness of a periodic solution for the Liénard equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M1">View MathML</a>

and investigate the problem of the existence of such a periodic solution when f is monotone increasing for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M2">View MathML</a> and monotone decreasing for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M3">View MathML</a> but with a single zero, because in this case the existence is not granted. Sufficient conditions for the existence of a periodic solution and also a necessary condition, which proves that with this assumptions actually it is possible to have no periodic solutions, are presented.

MSC: 34C05, 34C25.

Keywords:
Liénard equation; limit cycles; Massera theorem

1 Preliminaries and well-known results

The problem of the existence and uniqueness of a periodic solution for the Liénard equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M4">View MathML</a>

(1.1)

has been widely investigated in the literature. Among the uniqueness results, a special place is, without any doubt, taken by the classical Massera theorem. This is due to the simple geometrical ideas and the fact that this result, despite several efforts, is in most cases no more valid for the generalized Liénard equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M5">View MathML</a>

For related results concerning the more general equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M6">View MathML</a>

we refer to [1].

Equation (1.1) is equivalent to the phase-plane system

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M7">View MathML</a>

(1.2)

and to the Liénard system

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M8">View MathML</a>

(1.3)

We present the classical Massera theorem which is a milestone among the results of limit cycles uniqueness for system (1.2).

Theorem 1.1 (Massera [2])

System (1.2) has at most one limit cycle which is stable, and hence equation (1.1) has at most one nontrivial periodic solution which is stable, provided thatfis continuous and

1. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M9">View MathML</a>is monotone decreasing for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M3">View MathML</a>,

2. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M9">View MathML</a>is monotone increasing for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M2">View MathML</a>.

The theorem of Massera improved a previous result due to Sansone [3] in which there was the additional assumption <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M13">View MathML</a>. This assumption comes from the fact that Sansone was using the polar coordinates. Such a strong restriction on f is clearly not satisfied in the polynomial case, and hence Massera’s result is much more powerful. We recall the recent papers [4] and [5] in which a discussion concerning these two results, as well as related results, may be found.

We observe that in his paper Massera was proving the uniqueness of limit cycles, regardless of the existence, because only the monotonicity properties and the continuity were required. It is well known that in order to fulfill the necessary conditions for the existence of limit cycles, the only two cases to be considered are the following:

A <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M9">View MathML</a> has two zeros <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M15">View MathML</a>. In this case the existence of limit cycles is granted (see, for instance, [6]). Moreover, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M16">View MathML</a> has three zeros at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M17">View MathML</a>, 0 and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M18">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M19">View MathML</a> is negative for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M20">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M21">View MathML</a>) and positive outside this interval, and F is monotone increasing for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M22">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M23">View MathML</a>.

B <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M9">View MathML</a> remains negative for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M3">View MathML</a> (or for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M2">View MathML</a>), while it intersects the x-axis once in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M2">View MathML</a> (or for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M3">View MathML</a>).

This is because if we consider the distance from the origin, namely <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M29">View MathML</a>, and evaluate <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M30">View MathML</a> in systems (1.2) and (1.3), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M31">View MathML</a>

Hence the distance from the origin increases if f is negative and decreases if f is positive, and a necessary condition for the existence of limit cycles is that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M9">View MathML</a> changes sign.

As for [A], in a very recent paper [5], the monotonicity assumptions, which are required in the whole line, were relaxed to the interval <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M33">View MathML</a>, where α and δ can be easily determined. In particular, the following result holds.

Theorem 1.2 ([5])

The Liénard system (1.3) has exactly one limit cycle, which is stable, provided thatfis continuous and

1. if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M34">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M9">View MathML</a>is monotone decreasing for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M36">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M9">View MathML</a>is monotone increasing for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M38">View MathML</a>;

2. if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M39">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M9">View MathML</a>is monotone decreasing for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M41">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M9">View MathML</a>is monotone increasing for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M43">View MathML</a>with

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M44">View MathML</a>

wherea, bare the zeros of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M9">View MathML</a>andα, βare the nontrivial zeros of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M16">View MathML</a>. (If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M47">View MathML</a>, the uniqueness result comes from classical results due to Levinson-Smith[7]and Sansone[3]and no monotonicity assumptions are required.)

Let us consider case [B]. Now the existence of the limit cycle is no more granted, and, as far as we know, this case was not explicitly investigated in the literature.

The aim of this note is to discuss such a situation. In the next section, we present some sufficient conditions for the existence, and hence the uniqueness, of the limit cycle as well as a necessary condition. Such a condition shows that actually there are situations in which there is no limit cycle despite the monotonicity properties and the fact that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M9">View MathML</a> changes sign.

2 Existence of the limit cycle

We consider the case

B <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M9">View MathML</a> remains negative for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M3">View MathML</a>, while it intersects the x-axis once in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M2">View MathML</a>.

The case ‘<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M9">View MathML</a> is negative for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M2">View MathML</a>, while it intersects the x-axis once in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M3">View MathML</a>’ is treated in exactly the same way.

At first we work in the phase-plane

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M55">View MathML</a>

and we give a constructive result based on the methods used in [8].

Theorem 2.1The phase-plane system (1.2) has exactly one limit cycle, which is stable, provided thatfis continuous and

1. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M9">View MathML</a>is monotone decreasing for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M3">View MathML</a>,

2. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M9">View MathML</a>is monotone increasing for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M2">View MathML</a>and has a single zero at<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M60">View MathML</a>,

3.

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M61">View MathML</a>

(2.1)

4.

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M62">View MathML</a>

(2.2)

Proof We just notice that the continuity of f guarantees the property of uniqueness for the solutions to the Cauchy problem associated to system (1.2), and therefore the trajectories of such a system cannot intersect. Consider the 0-isocline of system (1.2), that is, the points in the phase-plane were <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M63">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M64">View MathML</a>.

For these points, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M65">View MathML</a>. Let Δ be the graph of the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M66">View MathML</a>. Clearly, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M67">View MathML</a>

(2.3)

and that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M68">View MathML</a> is bounded as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M69">View MathML</a>. Consider the slope of the trajectories of system (1.2) given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M70">View MathML</a>

(2.4)

In <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M71">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M68">View MathML</a> is negative and the slope at the points below Δ is negative. Hence we can select a point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M73">View MathML</a>, with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M74">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M75">View MathML</a> arbitrarily small, such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M76">View MathML</a> is smaller than all the other values of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M68">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M78">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M79">View MathML</a> be a negative semi-trajectory which starts at P. Inspection of the slopes shows that such a semi-trajectory does not intersect Δ, and therefore is bounded away from the x-axis for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M78">View MathML</a>. We now consider the positive semi-trajectory <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M81">View MathML</a> which starts at P. At first we observe that being monotone, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M9">View MathML</a> has a limit, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M83">View MathML</a>, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M84">View MathML</a>. Hence, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M85">View MathML</a>, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M86">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M87">View MathML</a> for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M88">View MathML</a>. It is trivial to see that in the strip <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M89">View MathML</a> the slope is bounded for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M90">View MathML</a>, and therefore <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M81">View MathML</a> intersects the line <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M92">View MathML</a> at a point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M93">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M94">View MathML</a>, or intersects the x-axis in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M3">View MathML</a>. Clearly, we must consider the first situation being our goal to reach the x-axis. For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M88">View MathML</a>, we get

(2.5)

(2.6)

(2.7)

This means that in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M88">View MathML</a>, Δ lies between the lines <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M101">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M102">View MathML</a> and hence the 0-isocline is bounded away from the line <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M103">View MathML</a>. Such a line has the slope <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M104">View MathML</a>. On the other hand, in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M88">View MathML</a> the slope of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M81">View MathML</a> is

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M107">View MathML</a>

(2.8)

Therefore, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M108">View MathML</a>, eventually <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M109">View MathML</a> will intersect the line <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M110">View MathML</a> at some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M111">View MathML</a>, and hence also Δ at some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M112">View MathML</a>. A straightforward calculation yields that such an inequality holds if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M113">View MathML</a>, and being ϵ arbitrary, this gives <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M114">View MathML</a>, that is, the value of the limit for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M115">View MathML</a> in the assumptions. Now we recall that in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M3">View MathML</a> the slope at the points above Δ is negative, and it is easy to see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M81">View MathML</a> eventually will intersect the x-axis in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M3">View MathML</a>. In fact, any trajectory intersecting the x-axis in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M3">View MathML</a> intersects the y-axis in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M120">View MathML</a>. This is because the slope being bounded for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M121">View MathML</a>, no vertical asymptote is possible. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M81">View MathML</a> will intersect the x-axis in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M2">View MathML</a>. This proves that the trajectory <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M124">View MathML</a> is winding (see Figure 1). Being <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M125">View MathML</a>, the origin is a source and the existence of a limit cycle is granted by the Poincaré-Bendixson theorem, while the uniqueness is given by monotonicity properties in virtue of the Massera theorem.  □

thumbnailFigure 1. Trajectory<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M126">View MathML</a>in Theorem 2.1.

We observe that if we keep all the sign assumptions and the limits at infinity, but not the monotonicity, clearly the uniqueness is no more granted, but the existence result still holds, and one can relax the assumption

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M127">View MathML</a>

(2.9)

with

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M128">View MathML</a>

(2.10)

as in [8].

Hence we have the following corollary.

Corollary 2.2The phase-plane system (1.2) has at least a stable limit cycle provided thatfis continuous and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M129">View MathML</a>, with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M60">View MathML</a>, and

1.

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M131">View MathML</a>

(2.11)

2.

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M132">View MathML</a>

(2.12)

As already mentioned, the first result is based on the direct construction of a winding trajectory.

A more powerful approach can be adopted in the Liénard plane, and it is based on the results presented in [9], where the problem of intersection with the vertical isocline <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M133">View MathML</a> is investigated in full for the more general Liénard system

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M134">View MathML</a>

(2.13)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M135">View MathML</a>

(2.14)

where as usual <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M136">View MathML</a>.

The following result holds.

Theorem 2.3The Liénard system (1.3) has exactly one limit cycle, which is stable, provided thatfis continuous and

1. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M9">View MathML</a>is monotone decreasing for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M3">View MathML</a>,

2. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M9">View MathML</a>is monotone increasing for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M2">View MathML</a>and has a single zero at<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M60">View MathML</a>,

3.

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M142">View MathML</a>

(2.15)

4.

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M143">View MathML</a>

(2.16)

Proof As already mentioned, the continuity of f guarantees the property of uniqueness for the solutions to the Cauchy problem associated to system (1.3), and therefore the trajectories of such a system cannot intersect.

As the proof is based on different applications of both necessary and sufficient conditions for the intersection of a semi-trajectory with the vertical isocline <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M133">View MathML</a> (2.14) given in [9], we are going to introduce the same notation as in [9] and to recall the theorems we are going to use here.

As well as in the phase-plane, for any point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M145">View MathML</a>, we denote by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M81">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M79">View MathML</a> the positive, respectively negative, semi-trajectory of the Liénard system (2.13) passing through P. With respect to the intersection of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M81">View MathML</a> with the vertical isocline (2.14), we observe that by a standard inspection of the direction of the field, it suffices to consider the cases

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M149">View MathML</a>

(2.17)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M150">View MathML</a>

(2.18)

With respect to the intersection of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M79">View MathML</a> with the vertical isocline (2.14), it suffices to consider the cases

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M152">View MathML</a>

(2.19)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M153">View MathML</a>

(2.20)

As in [9], we define

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M154">View MathML</a>

and, finally, we introduce the function

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M155">View MathML</a>

For this particular case, we have the following result.

Theorem 2.4 ([[9], Theorem 2.5.])

If, for every<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M156">View MathML</a>verifying (2.19), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M79">View MathML</a>intersects the curve<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M133">View MathML</a>at<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M159">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M78">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M161">View MathML</a>

(2.21)

Theorem 2.5 ([[9], Theorem 2.3.])

If there is a constant<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M162">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M163">View MathML</a>

(2.22)

then, for every<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M156">View MathML</a>verifying (2.18), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M165">View MathML</a>intersects the curve<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M133">View MathML</a>at<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M159">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M168">View MathML</a>.

Let us see that hypothesis (2.15) implies that there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M162">View MathML</a> such that (2.22) is verified. First of all, we observe that in the case we are dealing with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M170">View MathML</a>. By hypothesis (2.15), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M171">View MathML</a> there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M86">View MathML</a> such that, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M88">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M174">View MathML</a>. It is not restrictive to suppose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M175">View MathML</a>. This is because we are studying the case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M176">View MathML</a>, being the result already proved for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M114">View MathML</a>, in virtue of the previous theorem. Then, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M178">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M179">View MathML</a>

In our assumptions, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M3">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M181">View MathML</a>. Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M182">View MathML</a>

A straightforward calculation yields

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M183">View MathML</a>

therefore, choosing <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M184">View MathML</a>, we can find <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M185">View MathML</a> such that for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M186">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M187">View MathML</a>

Then, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M188">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M189">View MathML</a>

At last

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M190">View MathML</a>

Then limit (2.22) is verified provided that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M191">View MathML</a>

ϵ and η being arbitrary, the existence of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M192">View MathML</a> such that the last inequality holds is ensured by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M193">View MathML</a>. With a very similar calculation (with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M194">View MathML</a>, where α is the non-null zero of F), hypothesis (2.16) yields to prove that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M195">View MathML</a>

(2.23)

As the necessary condition for the intersection of the negative semi-trajectory with the vertical isocline is denied, there exists a point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M196">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M197">View MathML</a> is not intersecting the curve <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M133">View MathML</a> defined in (2.14). By a standard inspection of the direction of the field, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M199">View MathML</a> crosses the negative y-semiaxis and reaches a point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M200">View MathML</a> verifying (2.18). By the existence of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M162">View MathML</a> such that the sufficient condition (2.22) holds, the positive semi-trajectory <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M81">View MathML</a> intersects at first the y-axis because again there are no vertical asymptotes and then the curve <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M133">View MathML</a> because the sufficient condition for intersection is fulfilled. By the uniqueness for the solutions to the Cauchy problem associated to system (1.3), trajectories cannot intersect and hence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M204">View MathML</a> is a winding trajectory (see Figure 2). Being <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M125">View MathML</a>, the origin is a source and the existence of a limit cycle is granted by the Poincaré-Bendixson theorem, while the uniqueness is given by monotonicity properties in virtue of the Massera theorem.  □

thumbnailFigure 2. Trajectory<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M206">View MathML</a>in Theorem 2.3.

If we relax the monotonicity assumptions, we lose uniqueness, but existence still holds. Hence we have the following corollary.

Corollary 2.6The Liénard system (1.3) has at least a stable limit cycle provided thatfis continuous and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M129">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M60">View MathML</a>, and

1.

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M209">View MathML</a>

(2.24)

2.

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M210">View MathML</a>

(2.25)

Now we consider the case of non-existence of limit cycles. We consider again the results of [9] and, in particular, the case in which the semi-trajectory <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M79">View MathML</a>, starting at a point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M156">View MathML</a> verifying (2.19), intersects the vertical isocline in the positive x-half plane, but <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M81">View MathML</a> does not intersect the same curve in the negative x-half plane, and we get the following necessary condition.

Theorem 2.7The Liénard system (1.3) has no limit cycles provided thatfis continuous and

1. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M9">View MathML</a>is monotone decreasing for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M3">View MathML</a>,

2. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M9">View MathML</a>is monotone increasing for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M2">View MathML</a>and has a single zero at<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M60">View MathML</a>,

3.

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M219">View MathML</a>

(2.26)

4.

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M220">View MathML</a>

(2.27)

Proof At first we need the following results.

Theorem 2.8 ([[9], Theorem 2.6.])

If there is a constant<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M162">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M222">View MathML</a>

(2.28)

then, for every<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M156">View MathML</a>verifying (2.19), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M224">View MathML</a>intersects the curve<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M133">View MathML</a>at<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M159">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M78">View MathML</a>.

Theorem 2.9 ([[9], Theorem 2.2])

If, for every<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M156">View MathML</a>verifying (2.18), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M81">View MathML</a>intersects the curve<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M133">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M231">View MathML</a>

(2.29)

As in the previous theorem, the assumptions (2.28) and (2.29) become respectively

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M232">View MathML</a>

where α is the non-null zero of F.

With the same calculation, which we omit for the sake of simplicity, we get that if assumptions hold, we can provide a trajectory which is unwinding and goes to infinity without intersecting the curve <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M133">View MathML</a> for x negative (see Figure 3).

thumbnailFigure 3. Trajectory<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M206">View MathML</a>in Theorem 2.7.

Now assume, by contradiction, that there exists a limit cycle. The monotonicity property of the Massera theorem guarantees that such a limit cycle is unique and stable. Therefore there are trajectories surrounding the limit cycle which are winding and going to such a cycle. But with the same calculation already applied, which we omit for the sake of simplicity, if assumptions (2.26) and (2.27) hold, we can provide a large trajectory which is unwinding and goes to infinity without intersecting the curve <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M235">View MathML</a> for x negative. We can apply again the Poincaré-Bendixson theorem and get the existence of a second limit cycle which is unstable.

But as already mentioned, the monotonicity properties of the Massera theorem give the uniqueness, and hence in this situation we can conclude that there is no limit cycle and all trajectories are unwinding.  □

We observe that there are cases which still are not investigated. For instance, if we consider <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M9">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M237">View MathML</a>

we can produce a trajectory which lies entirely below the curve <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M133">View MathML</a> in the Liénard plane (or below the x-axis in the phase-plane) and therefore is not winding or unwinding.

In order to describe in full the possible configurations, we consider the following example.

Example 2.10 Let f be continuous, monotone decreasing for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M3">View MathML</a> and monotone increasing for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M2">View MathML</a> with a single zero at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M60">View MathML</a>.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M242">View MathML</a>, and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M243">View MathML</a>

Define <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M244">View MathML</a> and consider the system

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M245">View MathML</a>

(2.30)

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M246">View MathML</a> or <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M247">View MathML</a>, system (2.30) has no limit cycles, while if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M248">View MathML</a>, system (2.30) has exactly one limit cycle which is stable.

This is because if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M249">View MathML</a>, we can apply Theorem 2.7, while if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M250">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M251">View MathML</a> is negative, and if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M252">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M251">View MathML</a> is positive (for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M254">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M251">View MathML</a> is positive for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M256','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M256">View MathML</a>). And as already mentioned, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M257">View MathML</a> does not change sign, there is no limit cycle. On the other hand, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M248">View MathML</a>, then we can apply Theorem 2.3, and in this case there is a unique limit cycle which is stable.

We can describe the movement of such a cycle. At <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M259">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M260">View MathML</a> changes sign and we have a standard Hopf bifurcation, which gives a small amplitude limit cycle bifurcating from the origin. When ϵ decreases, such a limit cycle enlarges and eventually collapses to a separatrix at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M261','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M261">View MathML</a>. For a complete description of the bifurcation from a separatrix, we recall [10]. For the bifurcation of limit cycles we recall [11].

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

Both authors contributed equally in writing this paper.

Acknowledgements

We thank the anonymous referees for their valuable suggestions and Dr. Francesco Della Santa for the figures.

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