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.
Keywords:
extended FisherKolmogorov equation; periodic boundary condition; attractor bifurcation; center manifold1 Introduction
In this paper we work with the extended FisherKolmogorov type equation with periodic boundary condition, which reads
where
where
The extended FisherKolmogorov (EFK) equation has been proposed as a model for phase
transitions and other bistable phenomena [13]. 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
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
2. As λ crosses
3.
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
where
Since
In addition, we assume that the nonlinear terms
Definition 2.1[15]
A set
where
Definition 2.2[15]
(1) We say that the solution to equation (2.1) bifurcates from
(2) If the invariant sets
To prove the main result, we introduce an important theorem.
Let the eigenvalues (counting multiplicity) of
and the principle of exchange of stabilities holds true:
Let the eigenspace of
It is known that
The following attractor bifurcation theorem can be found in [15].
Theorem 2.1Let
(1) Equation (2.1) bifurcates from
(2) The attractor
(3) For any
(4) If
Remark 2.1 As
To get the structure of the bifurcated solutions, we introduce another theorem.
Let v be a twodimensional
for
where
for some constants
Theorem 2.2 (Theorem 5.10 in [15])
Under conditions (2.7), (2.8), the vector field (2.6) bifurcates from
(1)
(2)
(3)
3 Mathematical setting
Let
and
We define
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 4.1For problem (1.1), if
(1) If
(2) If
(3)
(4)
where
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
Thanks to the results in [9,18,19], we know that the operator
It is easy to get the following inequality:
which implies that
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
Let
where
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
Let
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
In the following, we calculate
By direct calculation, we have
then we obtain the expression of
where
Putting (4.4) into (4.3), we have the reduction equations
For the case of
Since the following equality holds true:
according to Theorems 2.1, 2.2 and Remark 2.1, we can conclude that if
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
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
Since the solutions of (2.1) are translation invariant,
the set
represents
Remark 4.1 Suppose that
We have
Then near
where
Remark 4.2 If
Remark 4.3 If the higher order terms
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
It is not difficult to verify that
we derive the following system:
According to Remark 4.3, if the condition
Theorem 5.1For problem (5.1), if
(1) If
(2) If
(3)
(4)
where
Figure 1. Bifurcation diagram for the model of the population density for singlespecies. (1) Bifurcation appears at
Furthermore, Theorem 5.1 and the equality
yield the following biological results.
Biological results For the model (5.1), if
(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)).
Figure 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 singlespecies. 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
Taking
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
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
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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