Attractor bifurcation for FKPP type equation with periodic boundary condition
© Zhang; licensee Springer. 2012
Received: 31 December 2011
Accepted: 13 April 2012
Published: 13 April 2012
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.
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-Be nard 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
which satisfies the semigroup property.
3 Main results
3.1 The definition of attractor bifurcation
If the invariant sets Ω λ 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.
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.
if λ > a, the Equation (1.1) bifurcates from u = 0 to an attractor Σ λ which is homeomorphic to S 1.
Σ λ consists of exactly one cycle of steady solutions of (1.1).
There exists a neighborhood U of u = 0, such that Σ λ attracts U/Γ, where Γ is the stable manifold of u = 0 with co-dimension 2 in H.
4.1 Attractor bifurcation theory
It is known that dim E0 = m.
(3.1) bifurcates from (u, λ) = (0, λ 0) to attractors Ω λ , having the same homology as S m -1, for λ > λ 0, with m - 1 ≤ dim Ω λ ≤ m, which is connected as m > 1;
- (2)for any u λ ∈ Ω λ , u λ can be expressed as
There is an open set U ⊂ H with 0 ∈ U such that the attractor Ω λ bifurcated from (0, λ 0) 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.
for some constants 0 < C1 < C2 and k = 2m + 1, m ≥ 1.
Π λ is a period orbit,
Π λ consists of infinitely many singular points,
Π λ 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 .
where all eigenvalues of possess negative real parts, all eigenvalues of possess nonnegative real parts at λ = λ0.
where α < 1 given by (3.3).
where is as in (4.7), the canonical projection, , and the eigenvalues of .
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.
As a result, conditions (4.1) and (4.2) are verified.
Step 2. We verify that for any λ ∈ R, operator L λ + 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 B λ : H1 → H is linear compact operator. Thanks to the results of analytic semigroup in [20–22], from (5.2) we know that operator L λ : H1 → H is a sectorial operator which generates an analytical semigroup. Condition (3.2) is verified.
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 Σ λ .
where y = x1e1 + x2e2.
To get the exact form of the reduction equations, we need to obtain the expression of < G(u), e1 > and < G(u), e2 >.
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.
Next we only need to find out < G2(y, Φ(y)), e j >, < G2(Φ(y),y), e j > and G3(y, y, y), e j >.
which implies that u = 0 is globally asymptotically stable. Assertion (1) of Theorem 3.2.1 be proved.
according to Theorem 4.1.2, we can conclude that if λ > a, the equation (1.1) bifurcates from u = 0 to an attractor Σ λ 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, L λ + G bifurcates from (u, λ) = (0,a) to a steady solutions (V λ , λ).
represents S1 in H1, which implies that assertion (3) of Theorem 3.2.1 is proved.
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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