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Attractor bifurcation for the extended Fisher-Kolmogorov equation with periodic boundary condition

Qiang Zhang1* and Hong Luo2

Author Affiliations

1 College of Computer Science, Civil Aviation Flight University of China, Guanghan, Sichuan, 618307, P.R. China

2 College of Mathematics and Software Science, Sichuan Normal University, Chengdu, Sichuan, 610066, P.R. China

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Boundary Value Problems 2013, 2013:169  doi:10.1186/1687-2770-2013-169

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2013/1/169


Received:24 October 2012
Accepted:5 July 2013
Published:19 July 2013

© 2013 Zhang and Luo; 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 paper, we study the bifurcation and stability of solutions of the extended Fisher-Kolmogorov equation with periodic boundary condition. We prove that the system bifurcates from the trivial solution to an attractor as parameter crosses certain critical value. The topological structure of the attractor is also investigated.

MSC: 35B32, 35K35, 37G35.

Keywords:
extended Fisher-Kolmogorov equation; periodic boundary condition; attractor bifurcation; center manifold

1 Introduction

In this paper we work with the extended Fisher-Kolmogorov type equation with periodic boundary condition, which reads

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M1">View MathML</a>

(1.1)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M2">View MathML</a> is an unknown function, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M3">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M4">View MathML</a> are constants, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M5">View MathML</a> is the system parameter. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M6">View MathML</a> is a polynomial on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M7">View MathML</a>, which is given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M8">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M9">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M10">View MathML</a> are given constants.

The extended Fisher-Kolmogorov (EFK) equation has been proposed as a model for phase transitions and other bistable phenomena [1-3]. It has been extensively studied during past decades. Kalies and van der Vorst [4] considered the steady-state problem; by analyzing the variational structure, they proved the existence of heteroclinic connections, which are the critical points of a certain functional. Also, by the variational method, Tersian and Chaparova [5] derived the existence of periodic and homoclinic solutions. Peletier and Troy [6] were interested in the stationary spatially periodic patterns and showed that the structure of the solutions is enriched by increasing the coefficient of the fourth-order derivative term. The structure of the solution set was also investigated by van den Berg [7], who enumerated all the possible bounded stationary solutions provided this coefficient is small. Rottschäer and Wayne [8] showed that for every positive wavespeed there exists a traveling wave. And they also found the critical wavespeed to discriminate the monotonic solution from the oscillatory one. By an iteration procedure, Luo and Zhang [9] proved that equation (1.1) possesses a global attractor in the Sobolev space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M11">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M12">View MathML</a> provided that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M13">View MathML</a> and p is an odd number. We refer the interested readers to the references in [4-9] for other results on the EFK equation; see also, among others, [10-13].

Returning to problem (1.1), our main interest in the present paper is the bifurcation and stability of solutions. By using a notion of bifurcation called attractor bifurcation developed by Ma and Wang in [14,15], a nonlinear attractor bifurcation theory for this problem is established. Work on the topic of attractor bifurcation also can be seen in [16,17].

The main objectives of this theory include:

(1) existence of attractor bifurcation when the system parameter crosses some critical number,

(2) dynamic stability of bifurcated solutions, and

(3) the topological structure of the bifurcated attractor.

Our main results can be summarized as follows.

1. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M14">View MathML</a>, the steady state <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M15">View MathML</a> is locally asymptotically stable.

2. As λ crosses <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M16">View MathML</a>, i.e., there exists an <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M17">View MathML</a> such that for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M18">View MathML</a>, system (1.1) bifurcates from the trivial solution to an attractor <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M19">View MathML</a>.

3. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M19">View MathML</a> is homeomorphic to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M21">View MathML</a> and consists of exactly one cycle of steady solutions of (1.1).

Moreover, we apply this theory to a model of the population density for single-species and derive biological results.

This article is organized as follows. The preliminaries are given in Section 2. The mathematical setting is presented in Section 3. The mathematical results are given in Section 4. In Section 5 we apply mathematical results to a model of the population density for single-species and derive biological results. In Section 6 we discuss some existing results and compare them with ours. Finally, Section 7 is devoted to the conclusions.

2 Preliminaries

We begin with the definition of attractor bifurcation which was first proposed by Ma and Wang in [14,15].

Let H and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M22">View MathML</a> be two Hilbert spaces, and let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M23">View MathML</a> be a dense and compact inclusion. We consider the following nonlinear evolution equations

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M24">View MathML</a>

(2.1)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M25">View MathML</a> is the unknown function, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M26">View MathML</a> is the system parameter, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M27">View MathML</a> are parameterized linear completely continuous fields depending continuously on λ, which satisfy

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M28">View MathML</a>

(2.2)

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M29">View MathML</a> is a sectorial operator which generates an analytic semigroup <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M30">View MathML</a> for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M31">View MathML</a>, we can define fractional power operators <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M32">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M33">View MathML</a> with domain <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M34">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M35">View MathML</a> if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M36">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M37">View MathML</a> (see [18,19]).

In addition, we assume that the nonlinear terms <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M38">View MathML</a> for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M39">View MathML</a> are a family of parameterized <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M40">View MathML</a> bounded operators (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M41">View MathML</a>) such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M42">View MathML</a>

(2.3)

Definition 2.1[15]

A set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M43">View MathML</a> is called an invariant set of (2.1) if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M44">View MathML</a> for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M45">View MathML</a>. An invariant set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M43">View MathML</a> of (2.1) is said to be an attractor if Σ is compact, and there exists a neighborhood of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M47">View MathML</a> of Σ such that for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M48">View MathML</a> we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M49">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M50">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M51">View MathML</a>.

Definition 2.2[15]

(1) We say that the solution to equation (2.1) bifurcates from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M52">View MathML</a> to an invariant set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M19">View MathML</a> if there exists a sequence of invariant sets <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M54">View MathML</a> of (2.1) such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M55">View MathML</a>, and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M56">View MathML</a>

(2) If the invariant sets <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M19">View MathML</a> are attractors of (2.1), then the bifurcation is called attractor bifurcation.

To prove the main result, we introduce an important theorem.

Let the eigenvalues (counting multiplicity) of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M29">View MathML</a> be given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M59">View MathML</a>

and the principle of exchange of stabilities holds true:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M60">View MathML</a>

(2.4)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M61">View MathML</a>

(2.5)

Let the eigenspace of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M29">View MathML</a> at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M63">View MathML</a> be

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M64">View MathML</a>

It is known that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M65">View MathML</a>.

The following attractor bifurcation theorem can be found in [15].

Theorem 2.1Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M66">View MathML</a>, conditions (2.4), (2.5) hold true, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M15">View MathML</a>is a locally asymptotically stable equilibrium point of (2.1) at<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M63">View MathML</a>. Then the following assertions hold true:

(1) Equation (2.1) bifurcates from<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M52">View MathML</a>to attractors<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M19">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M71">View MathML</a>, with dimension<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M72">View MathML</a>, which is connected as<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M73">View MathML</a>.

(2) The attractor<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M19">View MathML</a>is a limit of a sequence ofm-dimensional annuli<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M75">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M76">View MathML</a>; especially, if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M19">View MathML</a>is a finite simplicial complex, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M19">View MathML</a>has the homology type of the<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M79">View MathML</a>-dimensional sphere<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M80">View MathML</a>.

(3) For any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M81">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M82">View MathML</a>can be expressed as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M83">View MathML</a>

(4) If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M84">View MathML</a>is globally asymptotically stable for (2.1) at<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M63">View MathML</a>, then for any bounded open set<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M86">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M87">View MathML</a>, there is an<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M88">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M89">View MathML</a>, the attractor<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M19">View MathML</a>attracts<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M91">View MathML</a>inH, where Γ is the stable manifold of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M15">View MathML</a>with codimensionm. In particular, if (2.1) has a global attractor for allλnear<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M93">View MathML</a>, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M94">View MathML</a>.

Remark 2.1 As <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M22">View MathML</a> and H are infinite dimensional Hilbert spaces, if (2.1) satisfies conditions (2.2)-(2.5) and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M84">View MathML</a> is a locally (global) asymptotically stable equilibrium point of (2.1) at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M63">View MathML</a>, then the assertions (1)-(4) of Theorem 2.1 hold; see [14,15].

To get the structure of the bifurcated solutions, we introduce another theorem.

Let v be a two-dimensional <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M40">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M41">View MathML</a>) vector field given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M100">View MathML</a>

(2.6)

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M101">View MathML</a>. Here

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M102">View MathML</a>

(2.7)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M103">View MathML</a> is a k-multilinear field, which satisfies the inequality

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M104">View MathML</a>

(2.8)

for some constants <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M105">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M106">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M107">View MathML</a>.

Theorem 2.2 (Theorem 5.10 in [15])

Under conditions (2.7), (2.8), the vector field (2.6) bifurcates from<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M108">View MathML</a>to an attractor<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M19">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M110">View MathML</a>, which is homeomorphic to<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M21">View MathML</a>. Moreover, one and only one of the following conclusions is true:

(1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M19">View MathML</a>is a period orbit.

(2) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M19">View MathML</a>consists of infinitely many singular points.

(3) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M19">View MathML</a>contains at most<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M115">View MathML</a>singular points and has<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M116">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M117">View MathML</a>) singular points, 2Nof which are saddle points, 2Nof which are stable node points (possibly degenerate), andnof which have index zero.

3 Mathematical setting

Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M118">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M119">View MathML</a>

We define <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M120">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M121">View MathML</a> by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M122">View MathML</a>

(3.1)

Consequently, we have an operator equation which is equivalent to problem (1.1) as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M123">View MathML</a>

(3.2)

4 Mathematical results

As mentioned in the introduction, we study in this manuscript attractor bifurcation of the EFK equation under the periodic boundary condition. Then we have the following bifurcation theorem.

Theorem 4.1For problem (1.1), if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M124">View MathML</a>is satisfied, then the following assertions hold true:

(1) If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M14">View MathML</a>, the steady state<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M15">View MathML</a>is locally asymptotically stable.

(2) If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M127">View MathML</a>, system (1.1) bifurcates from the trivial solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M15">View MathML</a>to an attractor<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M19">View MathML</a>.

(3) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M19">View MathML</a>is homeomorphic to<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M21">View MathML</a>and consists of exactly one cycle of steady solutions of (1.1).

(4) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M19">View MathML</a>can be expressed as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M133">View MathML</a>

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M134">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M135">View MathML</a>), or<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M136">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M137">View MathML</a>), and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M138">View MathML</a>, ϵis sufficiently small.

Proof of Theorem 4.1 We shall prove Theorem 4.1 in four steps.

Step 1. In this step, we study the eigenvalue problem of the linearized equation of (3.2) and find the eigenvectors and the critical value of λ.

Consider the eigenvalue problem of the linear equation,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M139">View MathML</a>

(4.1)

It is not difficult to find that the eigenvalues and the normalized eigenvectors of (4.1) are

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M140">View MathML</a>

(4.2)

under condition that we get the principle of exchange of stabilities

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M141">View MathML</a>

Step 2. We verify that for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M26">View MathML</a>, operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M143">View MathML</a> satisfies conditions (2.2) and (2.3).

Thanks to the results in [9,18,19], we know that the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M144">View MathML</a> is a sectorial operator which implies that condition (2.2) holds true.

It is easy to get the following inequality:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M145">View MathML</a>

which implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M146">View MathML</a>, where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M147">View MathML</a>

then condition (2.3) holds true.

Step 3. In this part, we prove the existence of attractor bifurcation and analyze the topological structure of the attractor <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M19">View MathML</a>.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M149">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M150">View MathML</a>. Let Φ be the center manifold function, in the neighborhood of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M151">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M152">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M153">View MathML</a>.

Then the reduction equations of (3.2) are as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M154">View MathML</a>

(4.3)

To get the exact form of the reduction equations, we need to obtain the expression of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M155">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M156">View MathML</a>.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M157">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M158">View MathML</a> be the bilinear and trilinear operators of G respectively, i.e.,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M159">View MathML</a>

Since

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M160">View MathML</a>

the first order approximation of (4.3) does not work. Now, we shall find out the second order approximation of (4.3). And the most important of all is to obtain the approximation expression of the center manifold function.

By direct calculation, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M161">View MathML</a>

According to the formula of Theorem 3.8 in [15] (or Remark 4.1), the center manifold function Φ, in the neighborhood of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M162">View MathML</a>, can be expressed as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M163">View MathML</a>

In the following, we calculate <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M164">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M165">View MathML</a>.

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M166">View MathML</a>

By direct calculation, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M167">View MathML</a>

then we obtain the expression of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M164">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M165">View MathML</a>.

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M170">View MathML</a>

(4.4)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M171">View MathML</a>.

Putting (4.4) into (4.3), we have the reduction equations

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M172">View MathML</a>

(4.5)

For the case of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M173">View MathML</a>, it is obvious that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M15">View MathML</a> is locally asymptotically stable. For the case of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M175">View MathML</a>, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M176">View MathML</a>, which implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M177">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M15">View MathML</a> is also locally asymptotically stable. Assertion (1) of Theorem 4.1 is proved.

Since the following equality holds true:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M179">View MathML</a>

according to Theorems 2.1, 2.2 and Remark 2.1, we can conclude that if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M127">View MathML</a>, equation (1.1) bifurcates from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M15">View MathML</a> to an attractor <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M19">View MathML</a>, which is homeomorphic to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M21">View MathML</a>.

Step 4. In the last step, we show that the bifurcated attractor of (3.2) consists of a singularity cycle.

Since the even function space is an invariant subspace of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M143">View MathML</a> defined by (3.1), we shall consider the bifurcation problem in the even function space and prove that system (1.1) bifurcates from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M185">View MathML</a> to two steady solutions. For any function v in the even function space can be expressed as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M186">View MathML</a>

by the Lyapunov-Schmidt reduction method used in Step 3, we can deduce that the reduction equation of (1.1) is as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M187">View MathML</a>

(4.6)

which implies that (1.1) bifurcates from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M185">View MathML</a> to two steady solutions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M189">View MathML</a> in the space of even functions.

Since the solutions of (2.1) are translation invariant,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M190">View MathML</a>

the set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M191">View MathML</a>

represents <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M21">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M22">View MathML</a>, which implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M194">View MathML</a> consists of exactly one circle of steady solutions of (1.1). This completes the proof of Theorem 4.1. □

Remark 4.1 Suppose that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M195">View MathML</a>, the generalized eigenvectors of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M29">View MathML</a>, form a basis of H with the dual basis <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M197">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M198">View MathML</a>

We have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M199">View MathML</a>

Then near <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M63">View MathML</a>, the center manifold function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M201">View MathML</a> in Theorem 3.8 in [15] can be expressed as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M202">View MathML</a>

(4.7)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M203">View MathML</a>

Remark 4.2 If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M6">View MathML</a> in (1.1) is not a polynomial but a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M205">View MathML</a> with Taylor’s expansion in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M206">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M207">View MathML</a>; if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M124">View MathML</a> is satisfied, then the conclusions of Theorem 4.1 also hold true.

Remark 4.3 If the higher order terms <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M209">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M210">View MathML</a> are omitted, from the proof of Theorem 4.1, it is easy to see that the conclusions of Theorem 4.1 also hold true.

5 Applications

In this section, we apply Theorem 4.1 to a model of the population density for single-species as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M211">View MathML</a>

(5.1)

where μ, α are the diffusion coefficients, v is the population density for single-species, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M212">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M213">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M214">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M215">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M216">View MathML</a>. It is easy to see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M217">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M218">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M219">View MathML</a>. Inspired by the work of Murray [20], <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M220">View MathML</a> represents the birth rate, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M221">View MathML</a> describes the intra specific competition, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M222">View MathML</a> stands for the emigration which arises from disease.

It is not difficult to verify that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M223">View MathML</a> is a positive steady solution of system (5.1). From the translation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M224">View MathML</a>

(5.2)

we derive the following system:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M225">View MathML</a>

(5.3)

According to Remark 4.3, if the condition <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M124">View MathML</a> is satisfied, the conclusions of Theorem 4.1 for system (5.3) also hold true. Consequently, from the translation (5.2), we have the following results for (5.1).

Theorem 5.1For problem (5.1), if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M124">View MathML</a>is satisfied, then the following assertions hold true:

(1) If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M228">View MathML</a>, the steady state<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M223">View MathML</a>is locally asymptotically stable (Figure 1).

(2) If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M230">View MathML</a>, system (5.1) bifurcates from the solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M231">View MathML</a>to an attractor<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M232">View MathML</a>. This implies that the stability will switch from the original state (i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M231">View MathML</a>) to a new one (i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M232">View MathML</a>) (Figure 1).

(3) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M232">View MathML</a>is homeomorphic to<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M21">View MathML</a>and consists of exactly one cycle of steady solutions of (5.1) (Figure 1).

(4) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M232">View MathML</a>can be expressed as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M238">View MathML</a>

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M134">View MathML</a>, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M138">View MathML</a>, ϵis sufficiently small.

thumbnailFigure 1. Bifurcation diagram for the model of the population density for single-species. (1) Bifurcation appears at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M241">View MathML</a>. (2) Bifurcated attractor <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M232">View MathML</a> is the boundary of the shaded region. (3) The first horizontal solid line from above denotes that the solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M243">View MathML</a> is stable, and the horizontal dotted line means this solution is unstable.

Furthermore, Theorem 5.1 and the equality

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M244">View MathML</a>

yield the following biological results.

Biological results For the model (5.1), if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M124">View MathML</a> is satisfied, we have the following assertions:

(1) The population of this single-species is a conservative quantity.

(2) If the birth rate is low, then the population density will keep a uniform spatial distribution (Figure 2(A)).

(3) If the birth rate becomes high enough, then the spatial distribution of the population density will not keep uniform but change periodically with space (Figure 2(B)).

thumbnailFigure 2. The spatial distribution of the population density. (1) Figure 2(A) shows that the population density keeps a uniform spatial distribution when the birth rate is low. (2) Figure 2(B) shows that the population density changes periodically with space when the birth rate becomes high enough. (3) The area of the shaded regions stands for the population of this single-species. And the area of the shaded region in Figure 2(A) is equal to the area of the shaded region in Figure 2(B).

6 Discussion

Taking <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M246">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M247">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M248">View MathML</a> in (1.1), Peletier and Troy [6] analyzed stationary antisymmetric single-bump periodic solutions. They found that the coefficient of the fourth-order derivative term μ played a role of system parameter. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M249">View MathML</a>, the family of periodic solutions is still very similar to that of the Fisher-Kolmogorov equations. However, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M250">View MathML</a>, different families of periodic solutions emerged.

Taking <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M251">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M247">View MathML</a> in (1.1), and under hypothesis that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M253">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M254">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M255">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M256','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M256">View MathML</a>, Rottschäer and Wayne [8] showed that for every positive wavespeed, there exists a traveling wave. And they also found that there exists a critical wavespeed <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M257">View MathML</a>. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M258">View MathML</a>, the solution is monotonic; otherwise, the solution is oscillatory.

Unlike the work mentioned above, which focuses on the structure of solutions varying with the system parameter (μ or c), the manuscript presented here investigates the topological structure and the stability of solutions varying with the system parameter, i.e.λ. Firstly, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M14">View MathML</a>, the bifurcated attractor consists of the trivial solution; if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M127">View MathML</a>, the bifurcated attractor consists of only one cycle of steady state solutions and is homeomorphic to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M21">View MathML</a>. Secondly, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M14">View MathML</a>, the trivial solution is locally asymptotically stable. However, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M127">View MathML</a>, the stability switches from the trivial solution to the bifurcated attractor.

Since the increment of dimension of spatial domain may lead to much richer bifurcated behavior, further investigation on higher dimension of spatial domain is necessary in the future.

7 Conclusions

In this article, we first prove the existence of attractor bifurcation when the system parameter crosses critical number <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M16">View MathML</a>, which is the first eigenvalue of the eigenvalue problem of the linearized equation of (1.1). Second, we show that the stability of solutions varies with the system parameter λ. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M14">View MathML</a>, the trivial solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M15">View MathML</a> is locally asymptotically stable. However, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M267','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M267">View MathML</a>, the stability switches from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M15">View MathML</a> to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M19">View MathML</a>. Third, the topological structure of the attractor is investigated. We prove that the attractor <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M19">View MathML</a> consists of only one cycle of steady state solutions and is homeomorphic to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/169/mathml/M21">View MathML</a>. At last, the expression of bifurcated solution is also obtained.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

Both authors read and approved the final manuscript.

Acknowledgements

The authors are grateful to the anonymous referees whose careful reading of the manuscript and valuable comments were very helpful for revising and improving our work.

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