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Attractor bifurcation for the extended FisherKolmogorov equation with periodic boundary condition
Boundary Value Problemsvolume 2013, Article number: 169 (2013)
Abstract
In this paper, we study the bifurcation and stability of solutions of the extended FisherKolmogorov 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.
1 Introduction
In this paper we work with the extended FisherKolmogorov type equation with periodic boundary condition, which reads
where $u=u(x,t):\mathbb{R}\times [0,\mathrm{\infty})\to \mathbb{R}$ is an unknown function, $\mu >0$, $\alpha >0$ are constants, $\lambda \in {\mathbb{R}}^{+}$ is the system parameter. $g(s)$ is a polynomial on $s\in \mathbb{R}$, which is given by
where $2\le p\in \mathbb{N}$ and ${a}_{k}$ are given constants.
The extended FisherKolmogorov (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 steadystate 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 fourthorder 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 ${H}^{k}$ for all $k>0$ provided that ${a}_{p}<0$ 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 $\lambda \le \mu +\alpha $, the steady state $u=0$ is locally asymptotically stable.

2.
As λ crosses $\mu +\alpha $, i.e., there exists an $\u03f5>0$ such that for any $\mu +\alpha <\lambda <\lambda +\u03f5$, system (1.1) bifurcates from the trivial solution to an attractor ${\mathrm{\Sigma}}_{\lambda}$.

3.
${\mathrm{\Sigma}}_{\lambda}$ is homeomorphic to ${S}^{1}$ 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 singlespecies 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 singlespecies 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 ${H}_{1}$ be two Hilbert spaces, and let ${H}_{1}\hookrightarrow H$ be a dense and compact inclusion. We consider the following nonlinear evolution equations
where $u:[0,\mathrm{\infty})\to H$ is the unknown function, $\lambda \in \mathbb{R}$ is the system parameter, and ${L}_{\lambda}:{H}_{1}\to H$ are parameterized linear completely continuous fields depending continuously on λ, which satisfy
Since ${L}_{\lambda}$ is a sectorial operator which generates an analytic semigroup ${S}_{\lambda}(t)={\{{e}^{t{L}_{\lambda}}\}}_{t\ge 0}$ for any $\lambda \in \mathbb{R}$, we can define fractional power operators ${({L}_{\lambda})}^{\mu}$ for $0\le \mu \le 1$ with domain ${H}_{\mu}=D({({L}_{\lambda})}^{\mu})$ such that ${H}_{{\mu}_{1}}\subset {H}_{{\mu}_{2}}$ if ${\mu}_{1}>{\mu}_{2}$, and ${H}_{0}=H$ (see [18, 19]).
In addition, we assume that the nonlinear terms $G:{H}_{\alpha}\to H$ for some $0\le \alpha <1$ are a family of parameterized ${C}^{r}$ bounded operators ($r\ge 1$) such that
Definition 2.1 [15]
A set $\mathrm{\Sigma}\subset H$ is called an invariant set of (2.1) if $S(t)\mathrm{\Sigma}=\mathrm{\Sigma}$ for any $t\ge 0$. An invariant set $\mathrm{\Sigma}\subset H$ of (2.1) is said to be an attractor if Σ is compact, and there exists a neighborhood of $W\subset H$ of Σ such that for any $\phi \in W$ we have
where ${dist}_{H}(u(t,\phi ),\mathrm{\Sigma})={inf}_{v\in \mathrm{\Sigma}}{\parallel u(t,\phi )v\parallel}_{H}$, $\mathrm{\forall}t\ge 0$.
Definition 2.2 [15]

(1)
We say that the solution to equation (2.1) bifurcates from $(u,\lambda )=(0,{\lambda}_{0})$ to an invariant set ${\mathrm{\Sigma}}_{\lambda}$ if there exists a sequence of invariant sets $\{{\mathrm{\Sigma}}_{{\lambda}_{n}}\}$ of (2.1) such that $0\notin {\mathrm{\Sigma}}_{{\lambda}_{n}}$, and
$$\begin{array}{c}\underset{n\to \mathrm{\infty}}{lim}{\lambda}_{n}={\lambda}_{0},\hfill \\ \underset{n\to \mathrm{\infty}}{lim}\underset{v\in {\mathrm{\Sigma}}_{{\lambda}_{n}}}{max}{\parallel v\parallel}_{H}=0.\hfill \end{array}$$ 
(2)
If the invariant sets ${\mathrm{\Sigma}}_{\lambda}$ 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 ${L}_{\lambda}$ be given by
and the principle of exchange of stabilities holds true:
Let the eigenspace of ${L}_{\lambda}$ at $\lambda ={\lambda}_{0}$ be
It is known that $dim{E}_{0}=m$.
The following attractor bifurcation theorem can be found in [15].
Theorem 2.1 Let ${H}_{1}=H={\mathbb{R}}^{n}$, conditions (2.4), (2.5) hold true, and $u=0$ is a locally asymptotically stable equilibrium point of (2.1) at $\lambda ={\lambda}_{0}$. Then the following assertions hold true:

(1)
Equation (2.1) bifurcates from $(u,\lambda )=(0,{\lambda}_{0})$ to attractors ${\mathrm{\Sigma}}_{\lambda}$ for $\lambda >{\lambda}_{0}$, with dimension $m1\le dim{\mathrm{\Sigma}}_{\lambda}\le m$, which is connected as $m>1$.

(2)
The attractor ${\mathrm{\Sigma}}_{\lambda}$ is a limit of a sequence of mdimensional annuli ${A}_{k}$ with ${A}_{k+1}\subset {A}_{k}$; especially, if ${\mathrm{\Sigma}}_{\lambda}$ is a finite simplicial complex, then ${\mathrm{\Sigma}}_{\lambda}$ has the homology type of the $(m1)$dimensional sphere ${S}^{m1}$.

(3)
For any ${u}_{\lambda}\in {\mathrm{\Sigma}}_{\lambda}$, ${u}_{\lambda}$ can be expressed as
$${u}_{\lambda}={v}_{\lambda}+o\left({\parallel {v}_{\lambda}\parallel}_{{H}_{1}}\right),\phantom{\rule{1em}{0ex}}{v}_{\lambda}\in {E}_{0}.$$ 
(4)
If $u=0$ is globally asymptotically stable for (2.1) at $\lambda ={\lambda}_{0}$, then for any bounded open set $U\subset H$ with $0\in U$, there is an $\u03f5>0$ such that ${\lambda}_{0}<\lambda <{\lambda}_{0}+\u03f5$, the attractor ${\mathrm{\Sigma}}_{\lambda}$ attracts $U\mathrm{\setminus}\mathrm{\Gamma}$ in H, where Γ is the stable manifold of $u=0$ with codimension m. In particular, if (2.1) has a global attractor for all λ near ${\lambda}_{0}$, then $U=H$.
Remark 2.1 As ${H}_{1}$ and H are infinite dimensional Hilbert spaces, if (2.1) satisfies conditions (2.2)(2.5) and $u=0$ is a locally (global) asymptotically stable equilibrium point of (2.1) at $\lambda ={\lambda}_{0}$, 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 twodimensional ${C}^{r}$ ($r\ge 1$) vector field given by
for $x\in {\mathbb{R}}^{2}$. Here
where ${F}_{k}$ is a kmultilinear field, which satisfies the inequality
for some constants $0<{C}_{1}<{C}_{2}$ and $k=2m+1$, $m\ge 1$.
Theorem 2.2 (Theorem 5.10 in [15])
Under conditions (2.7), (2.8), the vector field (2.6) bifurcates from $(x,\lambda )=(0,0)$ to an attractor ${\mathrm{\Sigma}}_{\lambda}$ for $\lambda >0$, which is homeomorphic to ${S}^{1}$. Moreover, one and only one of the following conclusions is true:

(1)
${\mathrm{\Sigma}}_{\lambda}$ is a period orbit.

(2)
${\mathrm{\Sigma}}_{\lambda}$ consists of infinitely many singular points.

(3)
${\mathrm{\Sigma}}_{\lambda}$ contains at most $2(k+1)=4(m+1)$ singular points and has $4N+n$ ($N+n\ge 1$) singular points, 2N of which are saddle points, 2N of which are stable node points (possibly degenerate), and n of which have index zero.
3 Mathematical setting
Let
and
We define ${L}_{\lambda}=A+{B}_{\lambda}:{H}_{1}\to H$ and $G:{H}_{1}\to H$ by
Consequently, we have an operator equation which is equivalent to problem (1.1) as follows:
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 4.1 For problem (1.1), if $2{a}_{2}^{2}+45\mu {a}_{3}+9\alpha {a}_{3}<0$ is satisfied, then the following assertions hold true:

(1)
If $\lambda \le \mu +\alpha $, the steady state $u=0$ is locally asymptotically stable.

(2)
If $\lambda >\mu +\alpha $, system (1.1) bifurcates from the trivial solution $u=0$ to an attractor ${\mathrm{\Sigma}}_{\lambda}$.

(3)
${\mathrm{\Sigma}}_{\lambda}$ is homeomorphic to ${S}^{1}$ and consists of exactly one cycle of steady solutions of (1.1).

(4)
${\mathrm{\Sigma}}_{\lambda}$ can be expressed as
$${\mathrm{\Sigma}}_{\lambda}=\{\tilde{x}cos(x+\theta )+o(\tilde{x})\theta \in \mathbb{R}\},$$
where $\tilde{x}=\sqrt{\frac{4(16\mu +4\alpha \lambda )(\mu +\alpha \lambda )}{3{a}_{3}(16\mu +4\alpha \lambda )+2{a}_{2}^{2}}}$ (${a}_{2}\ne 0$), or $\tilde{x}=\sqrt{\frac{4(\mu +\alpha \lambda )}{3{a}_{3}}}$ (${a}_{2}=0$), and $\mu +\alpha <\lambda <\mu +\alpha +\u03f5$, ϵ is 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,
It is not difficult to find that the eigenvalues and the normalized eigenvectors of (4.1) are
under condition that we get the principle of exchange of stabilities
Step 2. We verify that for any $\lambda \in \mathbb{R}$, operator ${L}_{\lambda}+G$ satisfies conditions (2.2) and (2.3).
Thanks to the results in [9, 18, 19], we know that the operator ${L}_{\lambda}:{H}_{1}\to H$ is a sectorial operator which implies that condition (2.2) holds true.
It is easy to get the following inequality:
which implies that $G(u)=o({\parallel u\parallel}_{{H}_{\frac{1}{2}}})$, where
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 ${\mathrm{\Sigma}}_{\lambda}$.
Let ${E}_{1}^{\lambda}={E}_{0}=span\{{e}_{1},{e}_{2}\}$, ${E}_{2}^{\lambda}={E}_{0}^{\mathrm{\perp}}$. Let Φ be the center manifold function, in the neighborhood of $(u,\lambda )=(0,\mu +\alpha )$, we have
where $y={x}_{1}{e}_{1}+{x}_{2}{e}_{2}$.
Then the reduction equations of (3.2) are as follows:
To get the exact form of the reduction equations, we need to obtain the expression of $\u3008G(u),{e}_{1}\u3009$ and $\u3008G(u),{e}_{2}\u3009$.
Let ${G}_{2}:{H}_{1}\times {H}_{1}\to H$ and ${G}_{3}:{H}_{1}\times {H}_{1}\times {H}_{1}\to H$ be the bilinear and trilinear operators of G respectively, i.e.,
Since
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
According to the formula of Theorem 3.8 in [15] (or Remark 4.1), the center manifold function Φ, in the neighborhood of $(u,\lambda )=(0,\mu +\alpha )$, can be expressed as
In the following, we calculate $\u3008G(u),{e}_{j}\u3009$, $j=1,2$.
By direct calculation, we have
then we obtain the expression of $\u3008G(u),{e}_{j}\u3009$, $j=1,2$.
where $A={(\lambda 16\mu 4\alpha )}^{1}\frac{{a}_{2}^{2}}{2\pi}+\frac{3{a}_{3}}{4\pi}$.
Putting (4.4) into (4.3), we have the reduction equations
For the case of $\lambda <\mu +\alpha $, it is obvious that $u=0$ is locally asymptotically stable. For the case of $\lambda =\mu +\alpha $, if $2{a}_{2}^{2}+(45\mu +9\alpha ){a}_{3}<0$, which implies that $A<0$, then $u=0$ is also locally asymptotically stable. Assertion (1) of Theorem 4.1 is proved.
Since the following equality holds true:
according to Theorems 2.1, 2.2 and Remark 2.1, we can conclude that if $\lambda >\mu +\alpha $, equation (1.1) bifurcates from $u=0$ to an attractor ${\mathrm{\Sigma}}_{\lambda}$, which is homeomorphic to ${S}^{1}$.
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 ${L}_{\lambda}+G$ defined by (3.1), we shall consider the bifurcation problem in the even function space and prove that system (1.1) bifurcates from $(u,\lambda )=(0,\mu +\alpha )$ to two steady solutions. For any function v in the even function space can be expressed as follows:
by the LyapunovSchmidt reduction method used in Step 3, we can deduce that the reduction equation of (1.1) is as follows:
which implies that (1.1) bifurcates from $(u,\lambda )=(0,\mu +\alpha )$ to two steady solutions ${V}_{\lambda}^{\pm}(x,t)=\pm \sqrt{\frac{4(16\mu +4\alpha \lambda )(\mu +\alpha \lambda )}{3{a}_{3}(16\mu +4\alpha \lambda )+2{a}_{2}^{2}}}cosx+\text{h.o.t.}$ in the space of even functions.
Since the solutions of (2.1) are translation invariant,
the set
represents ${S}^{1}$ in ${H}_{1}$, which implies that ${\sum}_{\lambda}$ consists of exactly one circle of steady solutions of (1.1). This completes the proof of Theorem 4.1. □
Remark 4.1 Suppose that $\{{e}_{i}\}$, the generalized eigenvectors of ${L}_{\lambda}$, form a basis of H with the dual basis $\{{e}_{i}^{\ast}\}$ such that
We have
Then near $\lambda ={\lambda}_{0}$, the center manifold function $\varphi (x,\lambda )$ in Theorem 3.8 in [15] can be expressed as follows:
where
Remark 4.2 If $g(s)$ in (1.1) is not a polynomial but a ${C}^{\omega}$ with Taylor’s expansion in $s=0$ as $g(s)={\sum}_{k=2}^{\mathrm{\infty}}{a}_{k}{s}^{k}$; if $2{a}_{2}^{2}+45\mu {a}_{3}+9\alpha {a}_{3}<0$ is satisfied, then the conclusions of Theorem 4.1 also hold true.
Remark 4.3 If the higher order terms ${\sum}_{k=4}^{p}{a}_{k}{u}^{k}$ in $g(u)$ 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 singlespecies as follows:
where μ, α are the diffusion coefficients, v is the population density for singlespecies, and ${a}_{2}<0$, ${a}_{3}<0$, ${b}_{0}=\lambda \frac{{a}_{2}}{4{a}_{3}}+\frac{3}{64}\frac{{a}_{2}^{3}}{{a}_{3}^{2}}$, ${b}_{1}=\lambda \frac{5}{16}\frac{{a}_{2}^{2}}{{a}_{3}}$, ${b}_{2}=\frac{{a}_{2}}{4}$. It is easy to see that ${b}_{0}<0$, ${b}_{1}>0$ and ${b}_{2}<0$. Inspired by the work of Murray [20], ${b}_{1}$ represents the birth rate, ${b}_{2}{v}^{2}+{a}_{3}{v}^{3}$ describes the intra specific competition, and ${b}_{0}$ stands for the emigration which arises from disease.
It is not difficult to verify that ${v}_{0}=\frac{{a}_{2}}{4{a}_{3}}$ is a positive steady solution of system (5.1). From the translation
we derive the following system:
According to Remark 4.3, if the condition $2{a}_{2}^{2}+45\mu {a}_{3}+9\alpha {a}_{3}<0$ 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 5.1 For problem (5.1), if $2{a}_{2}^{2}+45\mu {a}_{3}+9\alpha {a}_{3}<0$ is satisfied, then the following assertions hold true:

(1)
If ${b}_{1}\le \mu +\alpha \frac{5}{16}\frac{{a}_{2}^{2}}{{a}_{3}}$, the steady state ${v}_{0}=\frac{{a}_{2}}{4{a}_{3}}$ is locally asymptotically stable (Figure 1).

(2)
If ${b}_{1}>\mu +\alpha \frac{5}{16}\frac{{a}_{2}^{2}}{{a}_{3}}$, system (5.1) bifurcates from the solution ${v}_{0}$ to an attractor ${\mathrm{\Sigma}}_{{b}_{1}}$. This implies that the stability will switch from the original state (i.e., ${v}_{0}$) to a new one (i.e., ${\mathrm{\Sigma}}_{{b}_{1}}$) (Figure 1).

(3)
${\mathrm{\Sigma}}_{{b}_{1}}$ is homeomorphic to ${S}^{1}$ and consists of exactly one cycle of steady solutions of (5.1) (Figure 1).

(4)
${\mathrm{\Sigma}}_{{b}_{1}}$ can be expressed as
$${\mathrm{\Sigma}}_{{b}_{1}}=\{{v}_{0}+\tilde{x}cos(x+\theta )+o(\tilde{x})\theta \in \mathbb{R}\},$$
where $\tilde{x}=\sqrt{\frac{4(16\mu +4\alpha \lambda )(\mu +\alpha \lambda )}{3{a}_{3}(16\mu +4\alpha \lambda )+2{a}_{2}^{2}}}$, and $\mu +\alpha <\lambda <\mu +\alpha +\u03f5$, ϵ is sufficiently small.
Furthermore, Theorem 5.1 and the equality
yield the following biological results.
Biological results For the model (5.1), if $2{a}_{2}^{2}+45\mu {a}_{3}+9\alpha {a}_{3}<0$ is satisfied, we have the following assertions:

(1)
The population of this singlespecies 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)).
6 Discussion
Taking $\alpha =1$, $\lambda =1$, $g(u)={u}^{3}$ in (1.1), Peletier and Troy [6] analyzed stationary antisymmetric singlebump periodic solutions. They found that the coefficient of the fourthorder derivative term μ played a role of system parameter. If $\mu \le \frac{1}{8}$, the family of periodic solutions is still very similar to that of the FisherKolmogorov equations. However, if $\mu >\frac{1}{8}$, different families of periodic solutions emerged.
Taking $\mu =1$, $\lambda =1$ in (1.1), and under hypothesis that $g(1)=1$, ${g}^{\prime}(1)<1$, ${g}^{\prime}(u)<0$ for $0<u<1$, 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 ${c}^{\ast}$. If $c\ge {c}^{\ast}$, 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 $\lambda \le \mu +\alpha $, the bifurcated attractor consists of the trivial solution; if $\lambda >\mu +\alpha $, the bifurcated attractor consists of only one cycle of steady state solutions and is homeomorphic to ${S}^{1}$. Secondly, if $\lambda \le \mu +\alpha $, the trivial solution is locally asymptotically stable. However, if $\lambda >\mu +\alpha $, 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 $\mu +\alpha $, 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 $\lambda \le \mu +\alpha $, the trivial solution $u=0$ is locally asymptotically stable. However, if $\lambda >\mu +\alpha $, the stability switches from $u=0$ to ${\mathrm{\Sigma}}_{\lambda}$. Third, the topological structure of the attractor is investigated. We prove that the attractor ${\mathrm{\Sigma}}_{\lambda}$ consists of only one cycle of steady state solutions and is homeomorphic to ${S}^{1}$. At last, the expression of bifurcated solution is also obtained.
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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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Keywords
 extended FisherKolmogorov equation
 periodic boundary condition
 attractor bifurcation
 center manifold