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Attractor bifurcation for FKPP type equation with periodic boundary condition

Qiang Zhang

Author Affiliations

College of Mathematics, Sichuan University, Chengdu, Sichuan 610064, P. R. China

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

Boundary Value Problems 2012, 2012:41  doi:10.1186/1687-2770-2012-41


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2012/1/41


Received:31 December 2011
Accepted:13 April 2012
Published:13 April 2012

© 2012 Zhang; 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 article, we make bifurcation analysis on the FKPP type equation under periodic boundary condition. And we show that the solutions bifurcate from the trivial solution u = 0 to an attractor ∑λ as parameter crosses certain critical value. Moreover, we prove that the attractor ∑λ consists of only one cycle of steady state solutions and is homeomorphic to S1. The analysis is based on a new theory of bifurcation, called attractor bifurcation, which was developed by Ma and Wang.

2000 Mathematics Subject Classification: 35B; 35Q; 37G; 37L.

Keywords:
FKPP type equation; periodic boundary condition; attractor bifurcation; center manifold

1 Introduction

We consider the following reaction diffusion equation of FKPP type:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/41/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/41/mathml/M1">View MathML</a>

(1.1)

supplemented with the following natural constraint:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/41/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/41/mathml/M2">View MathML</a>

(1.2)

where a > 0, d > 0, b are given and λ > 0 is system parameter.

In 1937, the FKPP equation was first proposed by Fisher as a model to describe the propagation of advantageous genes [1] and was studied mathematically by Kolmogorov et al. [2]. Moreover, it was also used as biological models for population dynamics [3-5].

The FKPP equation has been extensively studied during the last decades. Among all the topics of these, the existence of traveling waves (exact form of solutions) and the asymptotic behavior of solutions attract much attention. Many different kinds of methods for the existence of traveling waves (exact form of solutions) haven been developed, such as Painleve expansion method [6,7], bilinear method [8,9], symmetry methods [10]. On the other hand, many results on the asymptotic behavior of solutions are also obtained; see among others [11-14] and references therein.

However, there is few work on the attractor bifurcation of the FKPP equation. As a new notion of bifurcation, attractor bifurcation was developed by Ma and Wang [15-17] and attracted researchers [18,19]. Ma and Wang [15] first proposed this new notion and applied it to Rayleigh-Benard Convection. Afterwards, with this new theory, Park [18] analyzed the bifurcation of the complex Ginzburg-Landau equation (CGLE) and Zhang et al. [19] studied the attractor bifurcation of the Kuramoto-Sivashinsky equation.

In this article, we focus on the attractor bifurcation of FKPP type Equation (1.1). The bifurcation analysis near the first eigenvalue of (1.1) will be discussed. The topology structure of the bifurcated solutions will also be studied. As a result, we show the system bifurcates from the trivial solution to an attractor ∑λ as system parameter λ crosses the critical value a, the first eigenvalue of the eigenvalue problem of the linearized equation of (1.1). Furthermore, we prove that ∑λ is homeomorphic to S1 and consists of only one cycle of steady state solutions.

This article is organized as follows. The mathematical setting are given in Section 2. The main results are stated in Section 3. The preliminaries are put in section 4. And Section 5 devote to the proof of the main theorem.

2 Mathematical setting

Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/41/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/41/mathml/M3">View MathML</a>

and we define Lλ = -A + Bλ : H1 H and G : H1 H by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/41/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/41/mathml/M4">View MathML</a>

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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/41/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/41/mathml/M5">View MathML</a>

(2.1)

Thanks to the existence result of semi-linear evolution equations, see Temam [20], Pazy [21], we can define a semigroup

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/41/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/41/mathml/M6">View MathML</a>

which satisfies the semigroup property.

3 Main results

3.1 The definition of attractor bifurcation

In order to state the main theorem of this article, we start with the definition of attractor bifurcation which was first proposed by Ma and Wang in [15-17].

Let H and H1 be two Hilbert spaces, and H1 H be a dense and compact inclusion. We consider the following nonlinear evolution equations

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/41/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/41/mathml/M7">View MathML</a>

(3.1)

where u : [0, ∞) → H is the unknown function, λ R is the system parameter, and Lλ:H1: → H are parameterized linear completely continuous fields depending continuously on λ, which satisfy

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/41/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/41/mathml/M8">View MathML</a>

(3.2)

Since Lλ is a sectorial operator which generates an analytic semigroup <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/41/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/41/mathml/M9">View MathML</a> for any λ R, we can define fractional power operators (-Lλ)μ for 0 ≤ μ ≤ 1 with domain Hμ = D((-Lλ)μ) such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/41/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/41/mathml/M10">View MathML</a> if μ1 > μ2, and H0 = H. In addition, we assume that the nonlinear terms G(., λ) : Hα H for some 0 ≤ α < 1 are a family of parameterized Cr bounded operator (r ≥ 1) continuously depending on λ, such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/41/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/41/mathml/M11">View MathML</a>

(3.3)

Definition 3.1.1 A set Σ ⊂ H is called an invariant set of (3.1) if S(t) Σ = Σ for any t ≥ 0. An invariant set Σ ⊂ H of (3.1) is said to be an attractor if Σ is compact, and there exists a neighborhood of W H of Σ such that for any ψ0 W we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/41/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/41/mathml/M12">View MathML</a>

Definition 3.1.2 (1) We say that the solution to Equation (3.1) bifurcates from (ψ, λ) = (0, λ0) to an invariant set Ωλ, if there exists a sequence of invariant sets <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/41/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/41/mathml/M13">View MathML</a> of (3.1) such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/41/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/41/mathml/M14">View MathML</a>, and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/41/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/41/mathml/M15">View MathML</a>

(2) If the invariant sets Ωλ are attractors of (3.1), then the bifurcation is called attractor bifurcation.

3.2 Main theorem

In this article, based on attractor bifurcation theory we obtain the following results.

Theorem 3.2.1 For the problem (1.1) with (1.2), if 2b2 < 9ad, following assertions hold true:

(1) if λ a, the steady state u = 0 is locally asymptotically stable. Furthermore, if b = 0, the steady state u = 0 is globally asymptotically stable.

(2) if λ > a, the Equation (1.1) bifurcates from u = 0 to an attractor Σλ which is homeomorphic to S1.

(3) Σλ consists of exactly one cycle of steady solutions of (1.1).

(4) There exists a neighborhood U of u = 0, such that Σλ attracts U/Γ, where Γ is the stable manifold of u = 0 with co-dimension 2 in H.

4 Preliminaries

4.1 Attractor bifurcation theory

In the following, we proceed with the principle of exchange of stabilities (PES). Let the eigenvalues (counting multiplicity) of Lλ be given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/41/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/41/mathml/M16">View MathML</a>

Suppose that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/41/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/41/mathml/M17">View MathML</a>

(4.1)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/41/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/41/mathml/M18">View MathML</a>

(4.2)

Let the eigenspace of Lλ at λ = λ0 be

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/41/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/41/mathml/M19">View MathML</a>

It is known that dim E0 = m.

Theorem 4.1.1[17] Assume that the conditions (3.2), (3.3), (4.1), and (4.2) hold true, and u = 0 is locally asymptotically stable for (3.1) at λ = λ0. Then the following assertions hold true:

(1) (3.1) bifurcates from (u, λ) = (0, λ0) to attractors Ωλ, having the same homology as Sm-1, for λ > λ0, with m - 1 ≤ dim Ωλ m, which is connected as m > 1;

(2) for any uλ ∈ Ωλ, uλ can be expressed as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/41/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/41/mathml/M20">View MathML</a>

(3) There is an open set U H with 0 ∈ U such that the attractor Ωλ bifurcated from (0, λ0) attracts U/Γ in H, where Γ is the stable manifold of u = 0 with co-dimension m.

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

Let v be a two-dimensional Cr (r ≥ 1) vector field given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/41/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/41/mathml/M21">View MathML</a>

(4.3)

for x R2. Here

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/41/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/41/mathml/M22">View MathML</a>

(4.4)

where Fk is a k-multilinear field, which satisfies an inequality

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/41/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/41/mathml/M23">View MathML</a>

(4.5)

for some constants 0 < C1 < C2 and k = 2m + 1, m ≥ 1.

Theorem 4.1.2 ([[16], Theorem 5.10, p. 134]) Under conditions (4.4) and (4.5), the vector field (4.3) bifurcates from (x, λ) = (0, 0) to an attractor Πλ for λ > 0, which is homeomorphic to S1. Moreover, one and only one of the following is true.

(1) Πλ is a period orbit,

(2) Πλ consists of infinitely many singular points,

(3) Πλ contains at most 2(k + 1) = 4(m + 1) singular points, and has 4N + n (N + n ≥ 1) singular points, 2N of which are saddle points, 2N of which are stable node points (possibly degenerate), and n of which have index zero.

4.2 Center manifold reduction

Since the key point in the proof of Theorem 3.2.1 is the center manifold function, we introduce an approximation formula of the center manifold function derived in [16].

We assume that the spaces H1 and H can be decomposed into

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/41/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/41/mathml/M24">View MathML</a>

(4.6)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/41/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/41/mathml/M25">View MathML</a> are invariant spaces of Lλ, i.e., Lλ can be decomposed into <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/41/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/41/mathml/M26">View MathML</a> such that for any λ near λ0,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/41/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/41/mathml/M27">View MathML</a>

(4.7)

where all eigenvalues of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/41/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/41/mathml/M28">View MathML</a> possess negative real parts, all eigenvalues of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/41/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/41/mathml/M29">View MathML</a> possess nonnegative real parts at λ = λ0.

Thus, for λ near λ0, (3.1) can be rewritten as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/41/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/41/mathml/M30">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/41/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/41/mathml/M31">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/41/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/41/mathml/M32">View MathML</a> are canonical projections. Moreover, let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/41/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/41/mathml/M33">View MathML</a>

where α < 1 given by (3.3).

By the classical center manifold theorem (see among others [20,22]), there exists a neighborhood of λ0, a neighborhood <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/41/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/41/mathml/M34">View MathML</a> of x = 0, and a C1 center manifold function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/41/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/41/mathml/M35">View MathML</a>, depending continuously on λ. Then to investigate the dynamic bifurcation of (3.1) it suffices to consider the finite dimensional system as follows

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/41/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/41/mathml/M36">View MathML</a>

Assume the nonlinear operator G be in the following form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/41/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/41/mathml/M37">View MathML</a>

(4.8)

for some integer k ≥ 2. Here Gk is a k-multilinear operator

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/41/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/41/mathml/M38">View MathML</a>

Theorem 4.2.1 ([[16], Theorem 3.8, p. 63]) Under the conditions (3.2), (4.1), (4.2), (4.6), and (4.8), the center manifold function Φ(x, λ) can be expressed as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/41/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/41/mathml/M39">View MathML</a>

(4.9)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/41/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/41/mathml/M28">View MathML</a> is as in (4.7), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/41/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/41/mathml/M40">View MathML</a> the canonical projection, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/41/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/41/mathml/M41">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/41/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/41/mathml/M42">View MathML</a> the eigenvalues of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/41/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/41/mathml/M29">View MathML</a>.

5 Proof of main theorem

In this section, we shall prove Theorem 3.2.1 by four steps.

Step 1. In this step, we shall study the eigenvalue problem of the linearized equation of (2.1) and shall 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/2012/1/41/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/41/mathml/M43">View MathML</a>

(5.1)

Note that the eigenvalues <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/41/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/41/mathml/M44">View MathML</a> and eigenvectors <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/41/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/41/mathml/M45">View MathML</a> of Laplace operator, which satisfy

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/41/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/41/mathml/M46">View MathML</a>

are

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/41/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/41/mathml/M47">View MathML</a>

Then the eigenvalues and eigenvectors of (5.1) are

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/41/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/41/mathml/M48">View MathML</a>

(5.2)

Now, we get the PES:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/41/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/41/mathml/M49">View MathML</a>

As a result, conditions (4.1) and (4.2) are verified.

Step 2. We verify that for any λ R, operator Lλ + G satisfies conditions (3.2) and (3.3).

From the theory of elliptic equations, operator A : H1 H is a homeomorphism. Note that conclusion H1 H is compact, then operator Bλ : H1 H is linear compact operator. Thanks to the results of analytic semigroup in [20-22], from (5.2) we know that operator Lλ : H1 H is a sectorial operator which generates an analytical semigroup. Condition (3.2) is verified.

It is easy to get the following inequality

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/41/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/41/mathml/M50">View MathML</a>

which implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/41/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/41/mathml/M51">View MathML</a>, where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/41/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/41/mathml/M52">View MathML</a>

condition (3.3) is verified.

Step 3. In this part, we shall prove the existence of attractor bifurcation and analyze the topological structure of attractor Σλ.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/41/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/41/mathml/M53">View MathML</a>. Let Φ be the center manifold function, in the neighborhood of (u, λ) = (0, a), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/41/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/41/mathml/M54">View MathML</a>

where y = x1e1 + x2e2.

Then the reduction equations of (2.1) are as follows

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/41/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/41/mathml/M55">View MathML</a>

(5.3)

To get the exact form of the reduction equations, we need to obtain the expression of < G(u), e1 > and < G(u), e2 >.

Let G2 : H1 × H1 H and G3 : H1 × H1 × H1 H are the bilinear and trilinear operators of G, respectively, i.e.,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/41/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/41/mathml/M56">View MathML</a>

Since

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/41/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/41/mathml/M57">View MathML</a>

the first-order approximation of (5.3) doesn't work. Now, we shall find out the second-order approximation of (5.3). And the most important of all is to obtain the approximation expression of the center manifold function.

Since

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/41/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/41/mathml/M58">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/41/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/41/mathml/M59">View MathML</a>

(5.4)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/41/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/41/mathml/M60">View MathML</a>

(5.5)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/41/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/41/mathml/M61">View MathML</a>

(5.6)

we can obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/41/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/41/mathml/M62">View MathML</a>

(5.7)

By the formula (4.9) in Theorem 4.2.1 and (5.7), the center manifold function Φ, in the neighborhood of (u, λ) = (0, a), can be expressed as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/41/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/41/mathml/M63">View MathML</a>

In the following, we calculate < G(u), ej >, j = 1, 2.

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/41/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/41/mathml/M64">View MathML</a>

Next we only need to find out < G2(y, Φ(y)), ej >, < G2(Φ(y),y), ej > and G3(y, y, y), ej >.

By calculation, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/41/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/41/mathml/M65">View MathML</a>

Note that (5.4) hold true, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/41/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/41/mathml/M66">View MathML</a>

Since

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/41/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/41/mathml/M67">View MathML</a>

we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/41/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/41/mathml/M68">View MathML</a>

then we obtain the expression of < G(u), ej >, j = 1,2.

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/41/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/41/mathml/M69">View MathML</a>

(5.8)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/41/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/41/mathml/M70">View MathML</a>.

Put (5.8) into (5.3), we have the approximation of reduction equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/41/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/41/mathml/M71">View MathML</a>

(5.9)

For the case of λ < a, it is obviously that u = 0 is locally asymptotically stable. For the case of λ = a, if 9ad > 2b2, which implies that A < 0, then u = 0 is also locally asymptotically stable. In particular, if b = 0, λ a, by Poincaré inequality we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/41/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/41/mathml/M72">View MathML</a>

which implies that u = 0 is globally asymptotically stable. Assertion (1) of Theorem 3.2.1 be proved.

Since the following equality holds true

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/41/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/41/mathml/M73">View MathML</a>

according to Theorem 4.1.2, we can conclude that if λ > a, the equation (1.1) bifurcates from u = 0 to an attractor Σλ which is homeomorphic to S1. Assertion (2) and (4) of Theorem 3.2.1 are proved.

Step 4. In the last step, we shall show that the bifurcated attractor of (2.1) contains a singularity cycle.

By Krasnoselskii Theorem for potential operator, at least, Lλ + G bifurcates from (u, λ) = (0,a) to a steady solutions (Vλ, λ).

Since the solutions of (2.1) are translation invariant,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/41/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/41/mathml/M74">View MathML</a>

the set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/41/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/41/mathml/M75">View MathML</a>

represents S1 in H1, which implies that assertion (3) of Theorem 3.2.1 is proved.

Competing interests

The author declares that he has no competing interests.

Acknowledgements

The author was very grateful to the anonymous referees whose careful reading of the manuscript and valuable comments enhanced presentation of the manuscript. This research was supported by Scientific Research Foundation of Civil Aviation Flight University of China (J2011-30).

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