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
We consider the following reaction diffusion equation of FKPP type:
supplemented with the following natural constraint:
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  and was studied mathematically by Kolmogorov et al. . 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 . 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  first proposed this new notion and applied it to Rayleigh-Benard Convection. Afterwards, with this new theory, Park  analyzed the bifurcation of the complex Ginzburg-Landau equation (CGLE) and Zhang et al.  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
and we define Lλ = -A + Bλ : H1 → H and G : H1 → H by
Consequently, we have an operator equation which is equivalent to the problem (1.1):
which satisfies the semigroup property.
3 Main results
3.1 The definition of attractor bifurcation
Let H and H1 be two Hilbert spaces, and H1 ↪ H be a dense and compact inclusion. We consider the following nonlinear evolution equations
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
Since Lλ is a sectorial operator which generates an analytic semigroup for any λ ∈ R, we can define fractional power operators (-Lλ)μ for 0 ≤ μ ≤ 1 with domain Hμ = D((-Lλ)μ) such that 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
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
(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.
4.1 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
Let the eigenspace of Lλ at λ = λ0 be
It is known that dim E0 = m.
Theorem 4.1.1 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
(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
for x ∈ R2. Here
where Fk is a k-multilinear field, which satisfies an inequality
for some constants 0 < C1 < C2 and k = 2m + 1, m ≥ 1.
Theorem 4.1.2 ([, 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 .
We assume that the spaces H1 and H can be decomposed into
Thus, for λ near λ0, (3.1) can be rewritten as
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 of x = 0, and a C1 center manifold function , depending continuously on λ. Then to investigate the dynamic bifurcation of (3.1) it suffices to consider the finite dimensional system as follows
Assume the nonlinear operator G be in the following form
for some integer k ≥ 2. Here Gk is a k-multilinear operator
Theorem 4.2.1 ([, 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
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,
Then the eigenvalues and eigenvectors of (5.1) are
Now, we get the PES:
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
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.
where y = x1e1 + x2e2.
Then the reduction equations of (2.1) are as follows
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.,
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.
we can obtain
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
In the following, we calculate < G(u), ej >, j = 1, 2.
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
Note that (5.4) hold true, we have
then we obtain the expression of < G(u), ej >, j = 1,2.
Put (5.8) into (5.3), we have the approximation of reduction equation
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
which implies that u = 0 is globally asymptotically stable. Assertion (1) of Theorem 3.2.1 be proved.
Since the following equality holds true
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,
represents S1 in H1, which implies that assertion (3) of Theorem 3.2.1 is proved.
The author declares that he has no competing interests.
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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