Dynamical Bifurcation of the Two Dimensional Swift-Hohenberg Equation with Odd Periodic Condition

In this article, we study the stability and dynamic bifurcation for the two dimensional Swift-Hohenberg equation with an odd periodic condition. It is shown that an attractor bifurcates from the trivial solution as the control parameter crosses the critical value. The bifurcated attractor consists of finite number of singular points and their connecting orbits. Using the center manifold theory, we verify the nondegeneracy and the stability of the singular points.


Introduction
Pattern formation arises when a system undergoes phase transitions.It is an interesting phenomenon that appears in a lot of natural circumstances and has been an important subject in nonequilibrium physics.To study the pattern-forming properties for a system, instead of extracting information from the full solutions of the realistic equations modeling the system, it is sufficient to investigate a relatively simple system of model equations which shares the same large-range effects with original system [2].For example, the complex Ginzburg-Landau equation is accepted as a model equation describing a variety of phenomena from the nonlinear waves to second-order phase transitions [1].Recently, it is noticed that fourth-order model equations are responsible for lots of phenomena from bistable dynamics.For examples, see the Kuramoto-Sivashinsky equation, the Swift-Hohenberg equation, the Cahn-Hilliard equation, the extended Fisher-Kolmogorov equation, and the suspension bridge equations, etc. [10].
The concept of instability plays an important role in the understanding of pattern formation [4].Spatial or temporal patterns emerge when relatively simple systems are driven into unstable states during the phase transition.The basic state of the system will deform by large amount in response to small perturbation.For the stability issue of such problems, one shall deal with it in the point of view of dynamical system.Namely, the model equations are considered as ordinary differential equations in a phase space with a control parameter which constitute a dynamical system.The instability usually accompanies with a bifurcation from a basic state to another state while the control parameter varies.In such a process, as the control parameter crosses the critical value, the basic state loses its stability and bifurcates to some nontrivial attractor.Therefore, the structure of the bifurcated attractor illustrates the properties of the long time behavior of the system.The study of the bifurcated attractor is based on the center manifold theorem saying that after the primary instability, the unstable trajectories move away from the basic state to a low dimensional subspace (called a center manifold) of the phase space.As a consequence, bifurcated attractor is contained in the center manifold which transforms the problem to the finite dimensional dynamical system.In this study, we employ the bifurcation theory regarding this approach established by Ma and Wang [7,8].For the convenience of quotation, a brief account of this theory is summarized in Section 4.
In this paper, we consider the dynamical bifurcation for the Swift-Hohenberg equation (SHE) among fourth-order model equations.More precisely, we consider the dynamical bifurcation for the following two dimensional SHE (1.1) ∂u ∂t = −(I + ∆) 2 u + λu − u 3 , defined on the periodic spatial domain Ω = (−l, l) × (−l, l).Here, λ is a positive real number that serves as a control parameter for the system.In this study, we impose the following odd periodic boundary condition on the domain Ω: The issue of the existence of solutions can be achieved by standard Galerkin's method.
Hereafter, we focus on the long time behavior of the solutions.
The SHE was proposed in [12] to describe the onset of Rayleigh-Bénard heat convection.The Rayleigh-Bénard convection describes a fluid placed between at horizontal plates such that the lower plate is maintained at a temperature beyond that at the upper plate.As the Rayleigh number is close to the critical Rayleigh number at which the onset of the convection occurs, the Rayleigh-Bénard convection model may be approximated by the SHE.
It is natural to take λ to be the system parameter for SHE.However, as shown in the next section, the dimension of the center manifold may vary according to the size of the spatial domain Ω, namely, the side length l.Thus, we encounter two control parameters for the bifurcation analysis which was detected in many articles [3,5,9,11].This phenomenon is not common for general fourth-order equations.For instance, the Kuramoto-Sivashinsky equations (KSE) does not share this phenomenon.See the comment in the end of Section 2 for a comparison between SHE and KSE.
The organization of this article is as follows.Some preliminaries including the eigenvalue analysis of the linear operator associated with the SHE are given in Section 2. The discussion of the eigenspaces is divided into four cases according to the spatial period 2l.We also compare the SHE and the KSE regarding the dependence of the eigenspaces on the number l.The main results and proofs are addressed in Section 3.For each case considered in Section 2, we show that the SHE bifurcates from the trivial solution to an attractor as the control parameter λ crosses the critical value λ 0 .As shown in Section 3, the number of the singular points on the bifurcated attractor is 3 m − 1, where m is the dimension of the center manifold.The classifications of the singular points are given in Theorem 3.1 -Theorem 3.4 for different cases.It is worth mentioning that the dimension of the center manifold, hence the number of the singular points on the bifurcated attractors, could be very large.In section 4, we state and prove some lemmas which are used in the proof of main theorems.Finally, in section 5, we summarize the attractor bifurcation theory developed by Ma and Wang [7,8] which is employed in this article.In particular, we list the characterization of the S m -attractor bifurcation.

Preliminaries
In this section, we address the linearized eigenvalue problem associated with the SHE.The multiplicity of the first eigenvalue determines the dimension of the center manifold and has a variety range according to the periodicity 2l.This phenomenon is not common for general fourth-order equations.To see this, we demonstrate the difference between the SHE and the KSE in the end of this section.
We start with the functional setting of the bifurcation problem for SHE.Let where The eigenvalues and the corresponding eigenvectors of the operator L λ on H are given by We see that {β K (λ) : K ∈ Z} is a complete set of the eigenvalues of L λ and {φ K : K ∈ Z} forms a basis of H.
The critical value of the first bifurcation is (2.1) λ 0 = min{λ K : K ∈ Z}.Indeed, we will show that the primary instability of the system happens at λ = λ 0 and the SHE bifurcates from the trivial solution (u, λ) = (0, λ 0 ) to an attractor as λ crosses to λ 0 .Therefore, it is important to find the critical value λ 0 and its multiplicity which gives the dimension of the center manifold near λ 0 .Since π 2 /l 2 , there are two cases to be considered: In this paper, for simplicity, we deal with the case (P 1 ).The case (P 2 ) resembles the case (P 1 ).Suppose that Then, λ 0 = h(K) for all K ∈ Γ(n 0 ) and h(K) > λ 0 for K ∈ Z \ Γ(n 0 ).Since the number of elements of Γ(n 0 ) determines the dimension of the center manifold near the bifurcation point λ 0 , we shall take a glimpse at the set Γ(n 0 ).First, if n 0 = k 2 0 for some k 0 ∈ N, then (k 0 , 0), (0, k 0 ) ∈ Γ(n 0 ).Furthermore, if there exist two positive numbers k 1 and k 2 such that Hence, (k 1 , ±k 2 ), (k 2 , ±k 1 ) ∈ Γ(n 0 ).We therefore conclude the number of elements of the set Γ(n 0 ) = 4p + 2 for some nonnegative integer p.
Secondly, if n 0 is not a perfect square, there are two cases.If n 0 /2 is not a perfect square, each pair of positive integers (k 1 , k 2 ) that satisfies n 0 = k 2 1 + k 2 2 generates four elements of Γ(n 0 ).Hence, the number of Γ(n 0 ) is 4p for some positive integer p.In the case that n 0 /2 is a perfect square, the number of Γ(n 0 ) is 4p + 2 for some positive integer p.
The above discussion gives the following classification of the set Γ(n 0 ) for case (P 1 ).
It is worthwhile to recall that the critical value λ 0 is a function of l.The scale of the bifurcation varies in response to the change of l.In fact, for given n 0 ∈ N, one can find the critical number l 0 such that n 0 π 2 /l 2 0 is closest to 1 among all l > 0. Since Γ(n 0 ) determines the dimension of the center manifold near λ 0 , this means that the complexity of the bifurcated attractor strongly depends on the number l.Such an observation was observed in many literatures, for instance, [3,5,9,11].In particular, the dynamic bifurcation of the one dimensional SHE under the assumption (P 2 ) was studied in [5].In [9,11], for fixed λ ∈ (0, 1), the authors study the dynamic bifurcation of the one dimensional SHE when l serves as the control parameter.
We close this section by giving a comparison of the SHE with the Kuramoto-Sivashinsky equation (KSE).The two dimensional KSE is given by which has the form (5.1) with L λ u = −Au + B λ u and The eigenvalues of L λ are with the corresponding eigenvectors φ K in H. Since α K (λ) is a linear function of λ, the case (P 2 ) does not happen for KSE.Even when we consider the case (P 1 ) for the SHE, the primary instability can arise at λ 1 as well as at λ K for any K according to the length l.However, the primary instability for the KSE arises when λ crosses τ 1 = π 2 /L 2 such that the multiplicity is always two with Γ(1) = {(1, 0), (0, 1)} for any l.Thus the SHE allows more phase transition phenomena than KSE as discussed in the previous section.One can refer to Chapter 9 of [7] for the dynamic bifurcation for the KSE.
(a) For λ > λ 0 , (1.1) bifurcates from (u, λ) = (0, λ 0 ) to an attractor A λ which is homeomorphic to S 1 .(b) For any bounded open set U ⊂ H with 0 ∈ U , there exists ε > 0 such that as (c) The bifurcated attractor A λ consists of eight singular points and their connecting orbits.The singular points can be expressed as where Moreover, under the conditions (P 1 ) and (Q 11 ), u ± 1 and u ± 2 are saddle points, while u ± 3 and u ± 4 are stable nodes.

It remains to show (c) is valid and
a center manifold function, then the reduced equation of (5.1) on the center manifold is Performing explicit calculation by use of (4.3) in Appendix 1, we see that the bifurcation equation (3.2) becomes where Hence, by Theorem 5.2 to show that the bifurcated attractor A λ is homeomorphic to S 1 .Finally, we show that A λ consists of four minimal attractor, four saddle points, and their connecting orbits.It is known that under nondegenerate conditions the bifurcated equation (3.3) and its truncation (3.4) have the same dynamic behavior near (u, λ) = (0, λ 0 ).For λ > λ 0 near λ 0 , (3.4) admits eight steady state solutions where α λ and γ λ are given by the formula (3.1).Then, has eigenvalues 2 3 β(λ), −2β(λ).Hence, they are regular solutions of (3.4) and correspond to saddle points of (3.3).
Figure 1 illustrates the bifurcated attractor on the center manifold for the case (Q 11 ) when λ > λ 0 near λ 0 .
Throughout this paper, we denote The second main result is the dynamic bifurcation of the SHE under the condition (P 1 ) and (Q 12 ).Theorem 3.2.Suppose that (P 1 ) and (Q 12 ) hold true.For 1 ≤ i ≤ p, we put such that corresponding eigenvectors are Then we have the followings.
(c) For any u λ ∈ A λ , u λ can be expressed as (d) The bifurcated attractor A λ contains exactly 3 4p+2 − 1 singular points.More precisely, each singular point u ∈ A λ can be written as , where w belongs to one of Γ q defined by Here, q = 1, .
For each q = 1, • • • , 4p + 2, Γ q has 2 q • C(4p + 2, q) elements.If w ∈ Γ 1 , then u is a stable node on the center manifold.If w ∈ Γ q with 2 ≤ q ≤ 4p + 2, then u is a saddle point such that the dimension of the unstable manifold of it on the center manifold is given by q − 1.
Proof.To demonstrate the proof concretely, without loss of generality, we assume p = 1.
The proof for the general case is exactly the same as this case.To obtain bifurcated attractors, we utilize Theorem 5.1.We note that for i = 1, • • • , 6, > 0, if λ > λ 0 and Reβ n (λ K ) < 0 for all K = K i .Then, it is easy to check that every hypothesis of theorem 5.1 is fulfilled.Thus, the assertions (a)-(c) follows.
It remains to show (d) is valid.Let E 1 = span{φ K i |i = 1, • • • , 6} and E 2 = E ⊥ 1 in H. Let P j : H → E j be the canonical projections and L λ j = L λ | E j , for j = 1, 2. For u ∈ H, we can write u = ∞ K∈Z y K φ K .Let us write y i = y K i , φ i = φ K 1 and β = β K 1 for simplicity.The reduced equation of (5.1) on the center manifold is the following system: where y = (y 1 , y 2 , y 3 , y 4 , y 5 , y 6 ).After some calculation by use of (4.3), we obtain the truncated system for (3.6) which implies by Theorem 5.2 that the bifurcated attractor A λ is homeomorphic to S 5 .Next, let us consider nontrivial steady state solutions of (3.7).If y i = 0 for i = 1, • • • , q and y i = 0 for i = q + 1, • • • 6, then it follows from (3.7) that where ŷ = (y 1 , • • • , y q , 0, • • • , 0).Summing up these equations, we have 4qβ/3 = (2q − 1)|ŷ| 2 , and thus (3.8) Hence, there are 2 q • C(6, q) singular points y = (y 1 , y 2 , • • • , y 6 ) with exactly q nonzero y i components.Moreover, the nonzero components take the values ± 4β/3(2q − 1).In the sequel, the number of nonzero singular points of (3.7) is Next, we show that these singular points are nondegenerate.Let v λ be a vector field defined by Then, the Jacobian matrix of , where h i 's are defined by Let q be the number of nonzero components of singular points y obtained above.By virtue of symmetry, we may assume that y i = 0 for 1 ≤ i ≤ q and y i = 0 for q + 1 ≤ i ≤ 6.Then, we have Suppose that q = 1.Then, has two eigen values (3.9) The multiplicity of σ 2 is five.Hence, the singular points are regular and correspond to stable nodes.
Next, suppose that q ≥ 2. We may assume that the signs of all y i 's are positive for 1 ≤ i ≤ q.This follows from a simple observation that Keeping these information in mind, we derive that where E q is a q × q matrix defined by (4.5) and I k is the k × k identity matrix.Hence, by Lemma 4.2 in Appendix 1 the eigenvalues of Dv λ are (3.10) The multiplicities of ρ 1 , ρ 2 , ρ 3 are 6 − q, 1, q − 1 by Lemma 4.2, respectively.As a consequence, all the singular points of (3.7) are nondegenerate.Moreover, the singular points are saddle points such that the dimension of the unstable manifolds is q−1.This completes the proof.Now we turn to the cases Q 21 and Q 22 .Essentially, these cases share the similar bifurcation structure with Theorem 3.2.Indeed, Lemma 4.1 is valid for all cases Q 11 −Q 22 .Hence, we expect the reduced equations on the center manifold to have the same structure.Theorem 3.3.Suppose that (P 1 ) and (Q 21 ) hold true.For 1 ≤ i ≤ p, we put such that corresponding eigenvectors are Then, we have the following.
For each q = 1, • • • , 4p, Γ q has 2 q • C(4p, q) elements.If w ∈ Γ 1 , then u is a stable node on the center manifold.If w ∈ Γ q with 2 ≤ q ≤ 4p, then u is a saddle point such that the dimension of the unstable manifold of it on the center manifold is given by q − 1.
Proof.The assertions (a)-(c) can be proved as in the previous theorems.So, we pay attention to the proof of (d).The basic idea of the proof is the same as the proof of Theorem 3.
For u ∈ H, we can write u = 4p i=1 y i φ K i + Φ(y, λ), where y = (y 1 , • • • , y 4p ) and Φ is the center manifold function..The reduced equation of (5.1) on the center manifold is the following system: for i = 1, • • • , 4p, (3.11) dy i dt Using (4.3), we can calculate the nonlinear terms to obtain the truncated system for (3.11) (3.12) By Theorem 5.2, the bifurcated attractor A λ is homeomorphic to S 4p−1 .The proof is exactly the same as that of Theorem 3.2.
To find nontrivial singular points of (3.12), we suppose that y i = 0 for i = 1, • • • , q and y i = 0 for i = q + 1, • • • , 4p.Proceeding as in the proof of Theorem 3.2, we obtain (3.13) Hence, for each q = 1, • • • , 4p, there are 2 q • C(4p, q) singular points with exactly q nonzero components.As a consequence, the number of nonzero singular points of (3.12) is We show that these singular points are nondegenerate.Let v λ be a vector field defined by Then, the Jacobian matrix of , where h i 's are defined by Now, by the similar argument as in the proof of Theorem 3.2, we can show that the eigenvalues of Dv λ are given by (3.9) for q = 1 and (3.10) for 2 ≤ q ≤ 4p.If q = 1, the multiplicities of σ 1 and σ 2 are 1 and 4p − 1, respectively.If 2 ≤ q ≤ 4p, then the multiplicities of ρ 1 , ρ 2 , ρ 3 are 4p − q, 1, q − 1, respectively.Hence, all the singular points are nondegenerate.If q = 1, then the singular points are stable.If 1 < q ≤ 4p, then the singular points are saddle points such that the dimension of the unstable manifolds is q − 1.This finishes the proof.
Theorem 3.4.Suppose that (P 1 ) and (Q 22 ) hold true.For 1 ≤ i ≤ p, we put such that corresponding eigenvectors are Then, we have the following.
For each q = 1, • • • , 4p + 2, Γ q has 2 q • C(4p + 2, q) elements.If w ∈ Γ 1 , then u is a stable node on the center manifold.If w ∈ Γ q with 2 ≤ q ≤ 4p + 2, then u is a saddle point such that the dimension of the unstable manifold of it on the center manifold is given by q − 1.
Proof.The proof is the same as the proof of Theorem 3.3 and we shall omit the details.

Appendix 1: Auxilliary Lemmas
In this section, we state and prove the lemmas used in the proof of Theorem 3.1-3.4.We begin with following lemma which is useful for computing the reduced equation of the SHE on the center manifold.
Suppose that Then, we have the following identities: for distinct i, j, r, Proof.The assertions (i) and (ii) are easily derived from the half-angle identity of trigonometric functions.For other assertions, we recall the identity By these formulae, the integrands in (iii), (iv) and (v) are expressed as a sum of cosine functions.Thus if we show that the angles does not vanish for each term, then the integrals should be zero.First, we consider The first integral vanishes by (4.1) and (4.4).If the second term is nonzero, then we deduce from (4.4) that either 3K i + K j = 0 or 3K i − K j = 0.In both cases, we obtain , which contradicts to (4.2).This proves (iii).Now we consider the case (iv).The integral (iv) is a sum of integrals of cosine functions.If one of these integrals is not zero, then by (4.4) we are led to for some a, b, c ∈ N. Without loss of generality, we may assume and K i 's are in counter-clockwise orientation on the circle of radius √ n 0 centered at the origin.Since which implies that K 1 , K 2 , K 3 , K 4 are vertices of a right rectangle inscribed in a circle.Hence, we have K 1 = −K 3 and K 2 = −K 4 , which contradicts to (4.1).Hence, (iv) is proved.
Finally, let us consider the case (v).As in the case of (iv), if the integral is nonvanishing, then we may assume that 2K i + K j + K r = 0. Hence, Thus the equality holds true in this inequality, which concludes that K j = K r .This is a contradiction to (4.1).The proof of the lemma is complete.
The next lemma is important when we study the stability of singular points on the center manifold.Lemma 4.2.For n ≥ 2, let E n (a, b) be a n × n symmetric real matrix defined by Proof.A direct computation shows that ρ 1 is an eigenvalue with eigenvector e 1 = (1, 1, • • • , 1) t and ρ 2 is an eigenvalue with linearly independent eigenvectors The next lemma states that the trivial solution of the SHE is asymptotically globally stable.This fact will be useful for showing by Theorem 5.1 that the bifurcated attractor attracts all solutions with initial data in the phase space outside of the stable manifold of the trivial solution.

Appendix 2: Abstract Attractor Bifurcation Theory
In this section, we briefly review the attractor bifurcation theory developed by Ma and Wang in [7,8].
Let H 1 and H be two Hilbert spaces with a dense inclusion H 1 → H. Let us consider the nonlinear evolution equation Then, L λ generates an analytic semigroup {S λ (t) = e −tL λ } t≥0 and we can define fractional power operators L α λ for any 0 ≤ α ≤ 1 with domain C r bounded operators for some 0 ≤ α < 1 and r ≥ 0, and depend continuously on λ such that (5.3) G(u, λ) = o(u Hα ), ∀λ ∈ R. In this paper, we are interested in the case that there exists an eigenvalue sequence {ρ k } ⊂ C and eigenvector sequence {e k , h k } ⊂ H 1 of A satisfying (5.4) for some constants b, C > 0. The condition (5.4) implies that A is a sectorial operator.Hence, we can define fractional power operators A α with domain H α = D(A α ) for any 0 ≤ α ≤ 1.For the compact operator B λ : H 1 → H, we assume that there exists a constant 0 ≤ θ < 1 such that We define the eigenspace of L λ at λ 0 by Then, it is known that dimE 0 = m.We now introduce the notion of attractor bifurcation.We say that (5.1) bifurcates from (u, λ) = (0, λ 0 ) to an attractor Ω λ if there exists a sequence of attractors {Ω λn } of (5.If the invariant sets Ω λ are attractors and are homotopy equivalent to an m-dimensional 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 [7], is the main tool for the study of the Swift-Hohenberg equation in this paper.
Finally, we introduce a useful theorem when we study the structure of the bifurcated attractor for the two dimensional case.We consider a two dimensional system The following theorem from [7] gives a criterion when the system (5.8)bifurcates to an S m -attractor.
a and b are nonzero real numbers.Then, E n (a, b) admits two distinct eigenvalues ρ 1 = a + b(n − 1) and ρ 2 = a − b such that the multiplicities of ρ 1 and ρ 2 are 1 and n − 1, respectively.
u + G(u, λ), u(0) = u 0 , where u : [0, ∞) → H is the unknown function and λ ∈ R is the system parameter.The parameterized operator L λ : H 1 → H are linear completely continuous fields depending continuously on λ and satisfy(5.2)