Abstract and Applied Analysis

Volume 2013 (2013), Article ID 365436, 11 pages

http://dx.doi.org/10.1155/2013/365436

Research Article

Attractor Bifurcation for Extended Fisher-Kolmogorov Equation

Honglian You,¹ Rong Yuan,² and Ziheng Zhang³

¹Sino-European Institute of Aviation Engineering, Civil Aviation University of China, Tianjin 300300, China

²School of Mathematical Sciences, Beijing Normal University, Beijing 100875, China

³Department of Mathematics, Tianjin Polytechnic University, Tianjin 300160, China

Received 7 July 2013; Accepted 7 August 2013

Academic Editor: Feliz Minhós

Copyright © 2013 Honglian You et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

We consider the asymptotic stability and attractor bifurcation of the extended Fisher-Kolmogorov equation on the one-dimensional domain with Dirichlet or periodic boundary conditions. The novelty of this paper is that, based on a new method called attractor bifurcation, we investigate the existence of an attractor bifurcated from the trivial solution and give an explicit description of the bifurcated attractor. Moreover, the stability of the bifurcated branches is discussed.

1. Introduction

Fourth-order parabolic differential equations of the form

where , is a nonlinear function, arise from many physics models such as the theory of phase transitions [1], nonlinear optics [2], and shallow water waves [3]. The prototypical example of the nonlinearity is ; that is,

When , (2) is usually named extended Fisher-Kolmogorov (EFK) equation [4], while for the name Swift-Hohenberg (SH) equation [5] is more appropriate. In this paper, we restrict our attention to the case . For this case, (2) is a natural extension of the classical Fisher-Kolmogorov (FK) or Allen-Cahn equation [6, 7]. It has been proposed as a model equation for phase transitions in the neighborhood of a Lifshitz point [8] and is frequently used as a model system for the study of pattern formation from an unstable spatially homogeneous state; see [1, 4].

Due to the significance of fourth-order differential equations, substantial attention has been paid to (1). These results are mainly on basic properties of solutions (e.g., the existence, uniqueness, boundedness, and periodicity) [9–13], traveling waves [14], kinks [15, 16], asymptotic properties [17, 18], and global attractors [19]. In addition, (1) or similar equations play an important role in a variety of physics systems. Nevertheless we do not attempt to give a detailed account; instead refer the reader to [20] consulting a host of another related research.

To understand the behavior of solutions of (1), it is essential to understand the stationary (time-independent) solutions (or steady states), that is, the solutions of (1) satisfying the fourth-order differential equations of the form

Typically, the stationary problem displays a multitude of periodic, homoclinic, and heteroclinic solutions. The steady-state equation of (1), that is, (3), has been analysed using shooting methods [21] and with the help of variational methods [22]. An extensive literature on this subject exists, and we refer to [3, 23–26] and the reference listed therein.

In the recent paper [27], based on a variational gluing method, the authors considered the set of bifurcation stationary solutions of (2) for all with the Neumann boundary condition; that is,

Without loss of generality, they set and described the transition at the bifurcation point and gave a precise description of how the attractor is changing for , (see Section  8 in [27]).

Normally, to study (3), , , the potential (), the interval-length , and the boundary conditions are viewed as parameters. Motivated by the above works, in this paper we introduce new parameters and study the dynamical behavior of (2). Explicitly, we focus our attention on the asymptotic behavior and attractor bifurcation (the structure and the local stability of the bifurcated steady states) of the following extended Fisher-Kolmogorov (EFK) equation with (this can be done via a variable substitution ):

where is a constant and is the bifurcated parameter, using a new bifurcation theory (see Section 2) called attractor bifurcation developed recently by Ma and Wang [28, 29]. We are mainly dedicated to considering (5) associated to Dirichlet boundary condition

or odd-periodic boundary condition

or periodic boundary condition

respectively. For the three cases, the existence and local stability of bifurcated branches are described completely (see the main results in Section 3). At the end of this paragraph, we should point out that this new bifurcation theory has been proved to be very practical. In fact, with the help of this new theory, many long-time standing bifurcation problems for a variety of PDEs arising from science and engineering, such as Boussinesq equations [30], Rayleigh-Bénard convections [31], Swift-Hohenberg (SH) equations [32], Ginzburg-Landau (GL) equations [33, 34], and Kuramoto-Sivashinsky equations [28, 35], have been extensively taken into account.

The paper is organized as follows. In Section 2, we recall some preliminaries in terms of the attractor bifurcation and the center manifold reduction process. In Section 3, the main body of this paper, we are dedicated to considering the attractor bifurcation of (5) associated to Dirichlet, odd-periodic, or periodic boundary conditions, respectively. Firstly, we calculate the corresponding eigenvalues of the linear part with different boundary conditions and determine all critical values. Then as the parameter crosses the first critical value, we show that (5) bifurcates from the trivial solution an attractor . Moreover, we demonstrate an explicit description of that, with Dirichlet or odd-periodic boundary conditions, it consists of two steady states, while with periodic boundary condition, it is homeomorphic to . Finally, we extend these results to other critical values.

2. Preliminary Results

2.1. Attractor Bifurcation Theory

We first recall the general theory on attractor bifurcation of abstract evolution equations; see [28] (also [29]).

Let and be two Hilbert spaces and a dense and compact inclusion. Consider the following abstract nonlinear evolution equation

where is the unknown function, is the system parameter, and are parameterized linear completely continuous fields continuously depending on , which satisfy

Then generates an analytic semigroup , and we can define fractional power operators for any with domain such that if and . Furthermore, we assume that the nonlinear terms for some are family of parameterized () bounded operators depending continuously on the parameter such that

Definition 1 (see [28, Definition  5.1]). (1) We say that (9) bifurcates from an invariant set , if there exists a sequence of invariant sets such that , and

(2) If the invariant sets are attractors of (9), then the bifurcation is called attractor bifurcation.

(3) If the invariant sets are attractors of (9), which are homotopic equivalent to an -dimensional sphere, then we say that (9) has an -attractor bifurcation at .

Theorem 2 (see [28, Theorem  6.1]). Assume that conditions (10) and (11) hold true. Let the eigenvalues (counting the multiplicity) of be given by , where is the complex space. Suppose that

Let the eigenspace of at be

and let be a locally asymptotically stable equilibrium point of (9) at . Then the following assertions hold true.

(1)Equation (9) bifurcates from an attractor for , with , which is connected as .(2)The attractor is a limit of a sequence of -dimensional annulus with ; particularly if is a finite simplicial complex, then has the homotopy type of the -dimensional sphere .(3)For any , can be expressed as

(4)If the number of the equilibrium points of (9) in is finite, then one has the index formula

(5)If is globally asymptotically stable for (9) at , then for any bounded open set with , there is an such that as , the attractor attracts in , where is the stable manifold of with codimension . In particular, if (9) has a global attractor for all near , then can be chosen independently of  .

From the previous theorem we see that the asymptotic stability of the equilibrium for is crucial. In the following, we state a theorem on the asymptotic stability of for (9), that is, the critical state.

Theorem 3 (see [28, Theorems  3.16 and 3.17]). Let be symmetric with eigenvalues satisfying

Suppose that fulfills

where denotes the inner product of . Then is globally asymptotically stable.

When the first eigenvalue is simple, that is, , the following result describes the structure of bifurcated attractor more precisely.

Theorem 4 (see [28, Theorem  5.5 and Remark  6.1]). Let (10)–(14) hold true with . Assume that is analytic at and is locally asymptotically stable for (9). Then there exists an open set with such that if , (9) bifurcates from exactly two equilibrium points and , and the open set is decomposed into two open sets and , satisfying the following properties:

(1), , (2), ,    , (3)for any ,    ,

where is the solution of (9) initiated with .

If in (13) and condition (14) is replaced by

the bifurcated singular points of (9) are saddle points, of which the Morse index is obtained explicitly. In the following, we first give the definition of Morse index for nondegenerate singular points.

Definition 5 (see [28, Definition  3.5]). Let be a nondegenerate singular point of

and all eigenvalues of

have nonzero real part. Then the Morse index of is defined by “the number of eigenvalues of (23) having positive real part.”

Theorem 6 (see [28, Theorem  6.6]). Assume that (13), (21), and

hold and

where is the eigenvector of corresponding to . Then the bifurcated singular points of (9) from have Morse index on and have Morse index on . Moreover, has Morse index on and has Morse index on .

If the multiplicity of the first eigenvalue is , then there exists a -dimensional center manifold, and we can reduce the infinite dimensional system to this center manifold to investigate the attractor bifurcation.

Theorem 7 (see [28, Theorem  5.10]). Let be a 2-dimensional () vector field given by

for . Here

where is a -multilinear field, which satisfies

for some constants , , and . Then bifurcates from on an attractor , which is homeomorphic to . Moreover, one and only one of the following is true:

(1) is a periodic orbit, (2) consists of only singular points, or (3) contains at most singular points and has    singular points, of which are saddle points, of which are stable node points (possibly degenerate), and of which have index zero.

2.2. Reduction Procedure

During dealing with the bifurcation problem, the reduction of the equation to its local center manifold plays a crucial role; see [28, Section  3.2] for more information. Consider (9) and assume that eigenvalues of satisfy

for some , where is an -multilinear mapping and .

Let be the eigenvectors corresponding to . Under the above assumptions there is a center manifold near with the center manifold function given by

where is a neighborhood of and

Then (9) can be reduced to the center manifold in the following form:

where , is the Jordan matrix corresponding to eigenvalues of , and

By (30), we get the first approximation of (33), which is given by

where

3. Attractor Bifurcation of the EFK Equations

3.1. Dirichlet Boundary Condition

Consider the extended Fisher-Kolmogorov equation (5) with the Dirichlet boundary condition

In order to apply the bifurcation theory introduced in Section 2 to discuss the attractor bifurcation of (37), firstly we choose the Hilbert spaces and as follows:

and denote by the norm on which is induced by the inner product

It is well known that the embedding of is compact and dense. Then (37) can be transformed into the following abstract evolution equation in :

where , for all , with and defined as follows:

and, for some , is of the form

An easy calculation shows that the eigenvalues of are

with eigenvectors . Hence, is a linear homeomorphism. On the other hand, since is compactly imbedded in , is a compact operator. Therefore, is a linear completely continuous filed.

Proposition 8. For any , is a sectorial operator.

Proof. It is obvious that the spectrum of consists of eigenvalues; that is,

Let . Then

which yields that

Therefore, by [36, page 12, Corollary  3.8],

where , , is the -semigroup generated by . On the other hand, for with and , we have

By [36, page 61, Theorem  5.2], is a sectorial operator. Furthermore, is a linear bounded operator. Therefore, is a sectorial operator.

Lemma 9. For any , , for all .

Proof. For any , by [37, Theorem  1.6.1], we have

where denotes the supremum norm on , is some constant, and satisfies , that is, .

From the hypotheses in Theorem 2, we know that the eigenvalues of are vital to give rise to the bifurcation. After a simple computation, the eigenvalues and the eigenvectors of are given as follows.

Proposition 10. The eigenvalues of are

and the corresponding eigenvectors are

Moreover, forms an orthogonal basis of , as well as .

From (50) we know that every eigenvalue is simple and the first eigenvalue is . Take the first critical value . Now we give the result on bifurcation for (37) when crosses .

Theorem 11. For (37), one has the following assertions.

(1)As , is globally asymptotically stable.(2)Equation (37) bifurcates from an attractor on the right side of , which consists of exactly two steady states and expressed as follows:

(3)For some , if , there exists an open set containing , which can be decomposed by the stable manifold of into two open sets and with the property that , , and

where is the solution of (37) initiated with .

Proof. For any and ,

By Theorem 3, is globally asymptotically stable as .

The existence of follows from Propositions 8 and 10, Lemma 9, and Theorem 2 directly. Moreover, Theorem 4 indicates that the bifurcated attractor consists of exactly two steady states and (3) is also true. Now we give the formula of two steady states by reducing (40) to its local center manifold.

Let . Then

which deduces that the steady-state equation is of the form

Since

where is a -multilinear mapping, we have

For in (56), we have

which yields that

Similarly, one deduces that

Hence, by induction, we have

where is a constant. Then we obtain the steady-state bifurcation equation of (37) as follows:

and after an easy computation, we have

By (3) in Theorem 2, the elements in the bifurcated attractor are

As crosses the th critical value , , the local stability of the bifurcated steady states does not hold true. Nevertheless, as a direct consequence of Theorem 6, we have the following result.

Theorem 12. Equation (37) bifurcates from , , two saddle points with Morse index on and Morse index on .

3.2. Periodic Boundary Condition

In this section we are concerned with two cases: the odd-periodic boundary condition or the periodic boundary condition. Now we deal with the first case

Here we choose the Hilbert spaces and as follows:

It is obvious that is compact and dense. Similar to that in Section 3.1, and are defined by

and, for some , is of the form

Denote , for all . Then (66) can be deformed into the following abstract evolution equation in :

in which is a linear completely continuous field and a sectorial operator and symmetric and , for all .

Proposition 13. The eigenvalues of in are

and the corresponding eigenvectors are

In addition, forms an orthogonal basis of , as well as .

Now we are in the position to state the bifurcation result for (66) at the first eigenvalue.

Theorem 14. For (66), the following assertions hold true.

(1)For , is globally asymptotically stable.(2)Equation (66) bifurcates from , on the right side of , an attractor , which consists of exactly two steady states and , expressed by

(3)For some , if , there exists an open set containing , which can be decomposed by the stable manifold of into two open sets and with the property that , , and

where is the solution of (66) initiated with .

Proof. For the proof of (1) and the existence of , it is just the same as in Theorem 11. In what follows, we are devoted to describing the structure of .

Let be expressed as . Then we obtain that

Proposition 13 shows that the eigenvalues here are simple. Therefore, with the help of Theorem 4, we know that the bifurcated attractor consists of exactly two steady states. Hence, we consider the following steady-state equation:

By (57), we get

Similar to Theorem 11, for , we get the bifurcation equation

The remaining part is the same as in Theorem 11. So we omit the details.

As far as the th critical value is concerned, we have the following result.

Theorem 15. Equation  (66) bifurcates from , , two saddle points with Morse index on and Morse index on .

In what follows, we turn our attention to the periodic boundary condition

Let

Define and as follows:

and, for , is defined by

Denote , for all . Then (79) can be transformed into the following abstract evolution equation in :

in which is a linear completely continuous field and a sectorial operator and symmetric, and (), for all .

Proposition 16. The eigenvalues of in are

and the corresponding eigenvectors are

Moreover, forms an orthogonal basis of   and .

From (84) we know that every eigenvalue has multiplicity 2. Consider the first critical value . Then (79) has a 2-dimensional center manifold when . Reducing (79) to its center manifold and using Theorem 7, we get the following bifurcation result of (79) at .

Theorem 17. For (79), one has the following conclusions.

(1)For , is globally asymptotically stable.(2)Equation (79) bifurcates from , on the right side of , an attractor .(3) consists of steady states, which can be expressed as follows:

Proof . For the proof of (1) and (2), we can see Theorems 11 or 14. Next we focus on proving (3). Since , reducing (79) to the 2-dimensional center manifold, we obtain that

By (35), we get the first approximation of (87)

where . On the other hand,

Thus, we deduce that

where . Let

Then satisfies Theorem 7 with and . Thus, if , (79) bifurcates an attractor .

Finally, we prove that the attractor consists of steady states. It is clear that , , and that is, the subset is invariant under . Moreover, the eigenvalues of in are , which are simple. By the arguments in odd-periodic boundary condition we know that as crosses the first critical value , bifurcates from two steady states

Because of the invariance of (79) for the translation

the functions

are steady states of (79). Moreover, the set

is a circle in . Therefore,

where

As crosses the th critical value , similar to the proof of Theorem 17, we can obtain the following.

Theorem 18. Equation (79) bifurcates from , on the right side of , an invariant set . Furthermore, consists of steady states of (79).

Acknowledgments

This work was supported by the National Science Foundation for Young Scientists of China (Grant no. 11101304), and the Tian Yuan Special Foundation (Grant no. 11226137).

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