Bifurcation analysis of the damped Kuramoto-Sivashinsky equation with respect to the period

In this paper, we study bifurcation of the damped Kuramoto-Sivashinsky equation on an odd periodic interval 
of period $2\lambda$. 
We fix the control parameter $\alpha \in (0,1)$ 
and study how the equation bifurcates to attractors as $\lambda$ varies. 
Using the center manifold analysis, we prove that the bifurcated attractors are 
homeomorphic to $S^1$ and consist of four or eight singular points and their connecting orbits. 
We verify the structure of the bifurcated attractors by investigating the stability of each singular point.

1. Introduction. In this paper, we are interested in the damped Kuramoto-Sivashinsky equation (DKSE): (1.1) Here, u : R × [0, ∞) → R and α is a control parameter related to the driving force of the system. The damped Kuramoto-Sivashinsky equation has emerged as a fundamental tool for understanding the onset and evolution of secondary instabilities in many driven nonequilibrium systems [1,3,4,7,14,15,18]. For example, it provides a crude model of directional solidification [4,5]. In this case, u can be considered as the interfacial position and x is the distance along the interface. In case α = 1, (1.1) is reduced to the usual Kuramoto-Sivashinsky equation (KSE) which has been known as an important model in the pattern formation arising from different contexts of hydrodynamics and moving interfaces. In this paper, we study the dynamical bifurcation of the DKSE (1.1) under the odd periodic boundary condition: u is periodic on Ω = [−λ, λ], i.e. u(−λ, t) = u(λ, t) for all t ≥ 0, u(−x, t) = −u(x, t) for all x ∈ Ω and t ≥ 0. (1.2) We shall treat the DKSE as a dynamical system in an appropriate phase space. In this setting, both the half period λ (domain size) and the control parameter α are regarded as bifurcation parameters. The Swift-Hohenberg equation (SHE) also enjoys a similar bifurcation scenario since both the KSE and the SHE share the same linear part. For the SHE, there have been intensive studies on the bifurcation phenomena according to the variations of the domain size ( [6,8,9,16,17,19,20]). In [2], the authors studied the dynamical bifurcation of the DKSE when λ is fixed and α varies. It turns out that as α passes through a critical number the DKSE bifurcates from the trivial solution to an attractor. Moreover, the structure of the bifurcated attractor was analyzed in detail.
In this paper, we study the dynamical bifurcation of the DKSE by fixing α and varying λ. This setting suggests more abundant structure of bifurcation. The study of bifurcation analysis for varying domain size was initiated in [16,17] for the SHE, where the authors reduced the SHE on the center manifold for the bifurcation analysis. However, for the DKSE, due to the nonlinearity in (1.1), the bifurcation analysis of (1.1) is more complicated than that of the SHE in [16,17]. We take care of this nonlinear analysis by employing the so-called second approximation formula of the center manifold function introduced in [10,11]. By the study of the reduced bifurcation equations on the center manifold, the structure of the bifurcated solutions is clearly analyzed. For readers' convenience, we briefly review the second approximation formula of the center manifold function in the Appendix.
Then, β k (α, λ) ≤ 0 for any k ≥ 1 and λ ∈ J n (α). Following [16], we will refer to each component of the set J n (α) as a gap corresponding to n and α.
In the following, we provide a brief outline of what follows in the remaining part of this paper. Since α ∈ (0, 1) and q(n) ∈ (0, 1] is a decreasing function of n ∈ N ∪ {0}, there is N ≥ 1 such that Under the condition (1.6), we will prove that the trivial solution u = 0 of the DKSE (1.3) is globally asymptotically stable for all λ ∈ J N −1 (α). Since β n (α, λ) > 0 on I n (α) for n ≤ N − 1, one may expect that there is a bifurcation as λ passes through the end points of I n (α). On the other hand, if we set α 0 := q(N ) 2 , it holds that λ 0 := N λ 2 (α 0 ) = (N + 1)λ 1 (α 0 ) and β N (α 0 , λ 0 ) = β N +1 (α 0 , λ 0 ) = 0. If α passes through the number α 0 to the right, then N λ 2 (α) > (N + 1)λ 1 (α) such that I N (α) and I N +1 (α) are overlapped slightly. On this overlapped interval of λ, β N (α, λ) > 0 and β N +1 (α, λ) > 0. Thus, a secondary bifurcation may occur on this overlapped region. This kind of reasoning was first introduced in [16,17] for the Swift-Hohenberg equation, where by means of the center manifold analysis the authors studied the qualitative properties of the structures around this secondary bifurcation. They also suggested a problem of bifurcation analysis for other phase transition equations which have similar structures as the SHE. This motivates our study on the DKSE in this paper. We will study the bifurcation phenomena for the DKSE (1.3) in a similar scenario. Using the attractor bifurcation theory of [10,11], we will prove that there happen several kinds of bifurcation inside the overlapped interval from which attractors appear. We analyze the final pattern of solutions by studying the stability of these attractors on the center manifold.
It is interesting to compare our result for the DKSE with that for the SHE in [16,17]. We recall that by the relation (1.6), α is closely related to the number N and Λ N (α) := I N (α) ∩ I N +1 (α) = ∅. For each λ ∈ Λ N (α), the bifurcated attractor A N (α, λ) is homeomorphic to S 1 and consists of singular points and their connecting orbits. The singular points come from the combination of the eigenvectors φ N and φ N +1 and their stability determines the structure of A N (α, λ). The stability of each singular points has quite different structure in the DKSE and the SHE.
For the SHE, the interval Λ N (α) is divided into three parts: K 1 = (N +1)λ 1 , δ 1 , K 2 = δ 1 , δ 2 , and K 3 = δ 2 , N λ 2 , where δ i depends on α for i = 1, 2. On K 1 ∪K 3 , the singular points are the perturbations of φ N or φ N +1 and stable modes are φ N on K 1 and φ N +1 on K 3 . On K 2 , mixed states of φ N and φ N +1 also emerge, and they are unstable while the single states φ N and φ N +1 are stable. This phenomena reflects the transition on the overlapped interval Λ N (α) and the bifurcated attractor has similar structure with that illustrated in Figure 3 in Section 3. As a consequence one can say that the bifurcation structure on the interval Λ N (α) is the same regardless of different values of α ∈ (0, 1).
Unlike the SHE, the bifurcation phenomena of the DKSE are dependent on α ∈ (0, 1) as well as λ ∈ Λ N (α). It turns out that there are three different types of transitions on Λ N (α) according to the values of N : (i) N = 2, 3, 4, (ii) N = 5, and (iii) N ≥ 6. If N = 2, 3, 4, the interval Λ N is divided into two parts (N + 1)λ 1 , δ 0 and δ 0 , N λ 2 . In the first interval the only singular points are the perturbations of φ N or φ N +1 and the stable modes are the perturbations of φ N . In the second interval, there appear stable mixed states of φ N and φ N +1 . The single states φ N and φ N +1 become unstable. On the other hand, for N ≥ 6 the interval Λ N is divided into three parts as in the SHE and shares the same transition structure. In particular, on the interval (δ 1 , δ 2 ) the single states of φ N and φ N +1 are stable and their mixtures are unstable. The case N = 5 connects these two different cases N = 2, 3, 4 and N ≥ 6 in the sense that Λ N =5 is split into three parts but the stable singular points on (δ 1 , δ 2 ) are mixed states. As a consequence, when the singular points arise from mixed states of φ N and φ N +1 on Λ N (α), (i) if α is large in (0, 1), then the mixed states are stable; (ii) if α is small in (0, 1), then the single state φ N or φ N +1 is stable and the mixed states are unstable. The structures of transition in each case are depicted in Figure 1-3 in Section 3. These results are given in Theorem 3.1 and Theorem 3.5.
The organization of this paper is as the following. In Section 2, we show that under the condition (1.6) the solutions of DKSE bifurcates to an attractor when λ passes through nλ 1 to the right for 1 ≤ n ≤ N or nλ 2 to the left for 1 ≤ n ≤ N − 1. The main theorems of this section are Theorem 2.2 and Theorem 2.3. In Section 3, we study bifurcation when α is slightly bigger than α 0 . We prove that if λ = λ 0 is fixed, then the DKSE bifurcates to an attractor as α crosses α 0 to the right. Then, we fix α which is slightly bigger than α 0 and study bifurcation on the overlapped interval I N (α) ∩ I N +1 (α). In Theorem 3.5, we show that there happen bifurcations mixing up two bifurcation branches emanating from the end points of the overlapped interval. In Section 4, we give a brief conclusion. Finally, in Section 5 we provide a summary of attractor bifurcation theory used in this paper.
Hence, multiplying (1.4) by u and integrating the result over the interval [−1, 1] with respect to the variable x, we obtain We then deduce that as λ ∈ J N −1 (α) the only possible invariant sets of the equation (1.3) u lies in the span of {φ k (x)} as λ = kλ 1 or λ = kλ 2 for k = 1, 2, 3, · · · N − 1. However, it is easy to see that the equation (1.3) does not have a solution in form of u = y k (t) sin kπx. By Theorem 3.16 of [10], this implies that u = 0 is globally asymptotically stable in H.
Theorem 2.2. Let N ≥ 2 be fixed and assume (1.6). If 1 ≤ n ≤ N , we have the following.
To show the part (b), we define E 1 = span{φ n } and is a center manifold function and v = P 1 u, then the reduced bifurcation equation of (1.3) on the center manifold reads which is equivalent to According to the second approximation formula (5.10) on the center manifold, this equation can be written as from which we obtain from (1.5) that Thus, α + σ 2n − 2σ n > 0 for small ε n > 0. Moreover, (2.5) has two asymptotically stable equilibrium points: y n = ±γ n (λ), where (2.6) Thus, we have a pitchfork bifurcation and the proof is complete.
By the same argument as in the proof of Theorem 2.2, we obtain the following theorem.

Bifurcation near
In the previous section, by considering the condition √ α < q(N −1), we have proved that (1.3) bifurcates from the trivial solution to an attractor as λ passes through the boundary of On the other hand, it holds that for √ α > q(N ), In this case, there are two nontrivial branches of bifurcation in the interval Λ N : the branch emanating from (N + 1)λ 1 (α) to the right and the branch emanating from N λ 2 (α) to the left. These two branches are mixed up in the interval Λ N . It is very interesting to study this mixed structure of A N (α, λ). We will study the bifurcation phenomena of the DKSE when the pair (α, λ) is close enough to (α 0 , λ 0 ), in particular, √ α is slightly bigger than q(N ). The bifurcation analysis is given in two ways. First, we fix λ = λ 0 and show that an attractor bifurcates for α > α 0 . Second, we fix α which is sufficiently close to α 0 and show that there are several mixed bifurcation on the interval Λ N .
Then as α crosses α 0 to the right and stays near α 0 , the equation (1.3) bifurcates from the trivial solution to an attractor A N (α, λ) which is homeomorphic to S 1 .

For any bounded open set
Furthermore, the attractor A N (α, λ) has the following structure.
(a) For 2 ≤ N < 8, the attractor A N (α, λ) consists of 4 singular points and their connecting orbits. The singular points can be expressed as where u ± 1 are stable nodes and u ± 2 are saddles. Here, a 1 and a 2 are defined in consists of 8 singular points and their connecting orbits. The singular points can be expressed as We need the following lemma for the proof of Theorem 3.1.
Based on the above observation, we prove Theorem 3.1 in the following. The proof is divided into several steps.
Using these, we obtain  In the sequel, (3.12) becomes (3.13) Here,F Here, we used (3.6). This completes the step 2.
By the Poincare-Bendixon Theorem, the bifurcated attractor A N (α, λ) is homeomorphic to S 1 and consists of singular points and their connecting orbits. In the following, we investigate the stability of equilibrium points which describes a complete diagram of the attractor A N (α, λ). This analysis is heavily dependent on the range of λ as we shall see.
Step 3. Finding singular points. We note from (3.1) and (3.6) that Hence, for N ≥ 2 Similarly, we obtain On the other hand, we deduce from (3.1) and (3.6) that which imply that In the sequel, we obtain (3.16) Formally, the truncated system of (3.13) has the following singular points: (y 1 , y 2 ) = (±a 1 , 0), (0, ±a 2 ), (b 1 , ±b 2 ), (−b 1 , ±b 2 ), (3.17) where 19) The points b 1 and b 2 may not exist according to the signs inside the square roots which depends on the range of λ. We shall examine the possibilities of the existence of (b 1 , ±b 2 ) and (−b 1 , ±b 2 ) and study the stabilities of all the singular points. The next lemma provides a useful information to do it.
Proof. The assertions (a) and (b) comes from the following identity: (3.20) We note that which is equivalent to This is true for only N = 2, 3, 4, 5.
Step 4. Stability of singular points. Using the facts β N (α 0 , λ 0 ) = β N +1 (α 0 , λ 0 ) = 0, by Lemma 3.3, we have Similarly, we have To study the stabilities of the equilibrium points of the truncated equation of (3.13), let F = (F 1 , F 2 ) with Then we have the following: First, suppose that z 2 1 = a 2 1 and z 2 = 0. Then, DF has two eigenvalues Similarly, if z 1 = 0 and z 2 = a 2 2 , then DF has two eigenvalues Hence, we obtain the following properties.

Proof. A simple computation gives
Since (N + 1)λ 1 < λ < N λ 2 , we infer that for N ≥ 2, The second result of this section is the following.

Conclusion.
We have studied the dynamic bifurcation of the DKSE on an interval with odd periodic condition. The DKSE (1.1) has two bifurcation parameters: the control parameter α and the period λ (domain size). In this paper, we consider the dynamic bifurcation of DKSE when α is a fixed number satisfying (1.6) and λ varies. A similar scenario of bifurcation analysis has been carried out in [16,17] for the Swift-Hohenberg equation. In particular, it was suggested in [17] to see the bifurcation phenomena due to the variation of period (domain size) for different phase transition equations. This paper answers this question for the DKSE. Using the center manifold analysis based on the second approximation formula, we classified the attractor bifurcating from the trivial solution. First, we proved that there happens a pitchfork bifurcation at the end points of each gap of type (0, λ 1 ], (N − 1)λ 2 , N λ 1 , or kλ 2 , (k + 1)λ 1 for k = 1, · · · , N − 2. The bifurcation is supercritical at λ = kλ 1 for 1 ≤ k ≤ N . It is subcritical at kλ 2 for 1 ≤ k ≤ N − 1.
The situation is quite interesting if α ≥ α 0 = q(N ) 2 and α − q(N ) 2 1. Since λ 0 = (N + 1)λ 1 (α 0 ) = N λ 2 (α 0 ), the intervals I N and I N +1 are slightly overlapped as α > α 0 . Then, we have some mixed states in the overlapped interval Λ N = (N + 1)λ 1 , N λ 2 . The structure of the bifurcated attractor A(α, λ), which consists of singular points and their connecting orbits, is heavily dependent on the number N and the value of λ ∈ Λ N . The singular points are either stable nodes or saddles. The possible stability of singular points is the following: (A1) The stable singular points are perturbations of the eigenvector of low frequency N .
(A2) The stable singular points are perturbations of the eigenvector of high frequency N + 1.
(A3) The stable singular points are perturbations of the eigenvectors of both low and high frequencies.
(A4) The stable singular points are perturbations of superpositions of eigenvectors.
The structures of the bifurcated attractors are illustrated in Figures 1-3. An interesting point in those pictures is that the structures for N = 2, 3, 4 and N ≥ 6 are completely different. For N = 2, 3, 4, the interval Λ N is divided into two parts and the stable singular points are perturbations of superpositions of eigenvectors φ N and φ N +1 . On the other hand, for N ≥ 6 the interval Λ N is divided into three parts and the stable singular points are perturbations of eigenvectors. The case N = 5 connects these two cases in the sense that Λ N is split into three parts but the stable singular points are perturbations of superpositions of eigenvectors. Thus, one may say that a transition occurs at N = 5 if we regard the number N as another transition parameter. This phenomena does not happen for the SHE as shown in [16,17] and discussed in the Introduction.

5.
Review of attractor bifurcation theory. In this section, we briefly review the attractor bifurcation theory developed by Ma and Wang in [10,11].
If the invariant sets Ω λ are attractors and are homotopy equivalent to an mdimensional sphere S m , then the bifurcation is called an S m -attractor bifurcation.
The following dynamic bifurcation theorem for (5.1), which comes from theorem 6.1 of [10], is the main tool for the study of the Swift-Hohenberg equation in this paper.
(b) For any u λ ∈ A α , u λ can be expressed as (c) If u = 0 is globally asymptotically stable for (5.1) at λ = λ 0 , then for any bounded open set U ⊂ H with 0 ∈ U , there exists ε > 0 such that as λ 0 < λ < λ 0 + ε, A α attracts U \Γ in H, where Γ is the stable manifold of u = 0 with codimension m.
In particular, if (5.1) has a global attractor for all λ near λ 0 , then ε can be chosen independent of U .
The second approximation formula (5.10) is very useful for calculating the reduced equation (5.9) when the inner product of the nonlinear term with the elements in E 0 becomes zero. Earlier research provides a good example [13]. We use this formula when we consider the DKSE (1.1) on the center manifolds.