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

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國立臺灣大學 數學系 預印本 Department of Mathematics, National Taiwan University

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

Jongmin Han and Chun-Hsiung Hsia

November 12, 2012

SWIFT-HOHENBERG EQUATION WITH ODD PERIODIC CONDITION

JONGMIN HAN AND CHUN-HSIUNG HSIA

Abstract. 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.

1. 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 sys- tem, 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 exam- ple, 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 phenom- ena 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

1991 Mathematics Subject Classification. Primary: 35B32, 35B41.

Key words and phrases. Attractor bifurcation, Swift-Hohenberg equation.

1

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 + ∆)²u + λu− u³,

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:

(1.2)

� u(x1, x2) = u(x1+ 2l, x2, t) = u(x1, x2+ 2l, t), ∀ (x1, x2) ∈ R², t≥ 0, u(−x1,−x2, t) =−u(x1, x₂, t), ∀ (x1, x₂) ∈ Ω, t ≥ 0.

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-B´enard heat convection.

The Rayleigh-B´enard 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-B´enard 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 λ0. 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.

2. 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 H = {u ∈ L²(Ω; R) : u satisfies (1.2) and

�

Ω

u(x)dx = 0}, H₁ = H⁴(Ω; R) ∩ H.

Now, the problem (1.1) with condition (1.2) is expressed in the abstract form



 du

dt = L_λu + G(u), u(0) = u₀,

where L_λu =−Au + Bλu, G(u) =−u³,

Au = (∆ + I)² : H₁ → H, B_λu = λI : H₁ → H.

The eigenvalues and the corresponding eigenvectors of the operator L_λ on H are given by

β_K(λ) = λ − λK, λ_K=�

1 − |K|²π² l²

�2

≡ h(K), φ_K(x) = sin�k1π

l x₁+k₂π l x₂�

,

where x = (x1, x2) and K = (k1, k2) �= (0, 0) with ki∈ Z. Let

Z = {(k1, k₂) : k1 ∈ N, k2 ∈ Z} ∪ {(0, k2) : k2∈ N}.

We see that {βK(λ) : K ∈ Z} is a complete set of the eigenvalues of Lλ and {φK : K ∈ Z}

forms a basis of H.

The critical value of the first bifurcation is

(2.1) λ0= min{λK : K ∈ Z}.

Indeed, we will show that the primary instability of the system happens at λ = λ0 and the SHE bifurcates from the trivial solution (u, λ) = (0, λ₀) to an attractor as λ crosses to λ₀. Therefore, it is important to find the critical value λ0 and its multiplicity which gives the dimension of the center manifold near λ0. Since λ_K = h(K) is a quadratic function of

|K|²π²/l², there are two cases to be considered:

(P1) there exists K ∈ Z such that h(K) < h(N) for all N ∈ Z with |N|² �= |K|². (P2) there exist K1, K₂ ∈ Z with |K1|² �= |K2|² such that h(K1) = h(K2) < h(N) for

all N ∈ Z with |N|²�= |K1|²,|K2|². The case (P2) occurs when

(2.2) l² = π²

2

�|K1|²+ |K2|²�

In this paper, for simplicity, we deal with the case (P1). The case (P2) resembles the case (P₁). Suppose that

λ₀=�

1 −n₀π² l²

�2

for some n0 ∈ N. Let

Γ(n0) = {K ∈ Z : |K|² = n0}.

Then, λ0 = h(K) for all K ∈ Γ(n0) and h(K) > λ0 for K ∈ Z \ Γ(n0). Since the number of elements of Γ(n0) determines the dimension of the center manifold near the bifurcation point λ₀, we shall take a glimpse at the set Γ(n₀).

First, if n0 = k²₀ for some k0 ∈ N, then (k0, 0), (0, k₀) ∈ Γ(n0). Furthermore, if there exist two positive numbers k₁and k₂such that n₀ = k²₀ = k²₁+k₂², it is obvious that k₁ �= k2. Hence, (k1,±k2), (k2,±k1) ∈ Γ(n0). We therefore conclude the number of elements of the set Γ(n₀) = 4p + 2 for some nonnegative integer p.

Secondly, if n0 is not a perfect square, there are two cases. If n0/2 is not a perfect square, each pair of positive integers (k1, k2) that satisfies n0 = k²₁ + k²₂ generates four elements of Γ(n₀). Hence, the number of Γ(n₀) is 4p for some positive integer p. In the case that n0/2 is a perfect square, the number of Γ(n₀) is 4p + 2 for some positive integer p.

The above discussion gives the following classification of the set Γ(n0) for case (P1).

(Q1) At least one element K in Γ(n0) has zero component. We have two subcases.

(Q₁₁) Multiplicity is two and Γ(n₀) = {(k0, 0), (0, k₀) : k₀² = n₀}.

(Q12) Multiplicity is 4p + 2 for some positive integer p, and there exist ki1, k_i2 ∈ N with k_i1r �= ki2s for (i₁, r)�= (i2, s), i, i₁, i₂ = 1, 2, · · · , p and r, s = 1, 2, such that

Γ(n0) = {(k0, 0), (0, k₀), (ki1,±ki2), (ki2,±ki1) : k₀² = k_i1² + k_i2² = n0}.

(Q2) Every element K in Γ(n0) has nonzero component. We have two subcases.

(Q21) Multiplicity is 4p for some integer p ≥ 1. For 1 ≤ i ≤ p, there exist ki1, ki2∈ N such that k_i1r �= ki2s for (i₁, r)�= (i2, s) and

Γ(n0) = {(ki1,±ki2), (ki2,±ki1) : k_i1² + k_i2² = n0}.

(Q22) Multiplicity is 4p + 2 for some integer p ≥ 0. For 0 ≤ i ≤ p, there exist k₀, k_i1, k_i2∈ N such that k0 �= kij for i ≥ 1, kij �= krs for (i, j) �= (r, s), and Γ(n0) = {(k0,±k0), (ki1,±ki2), (ki2,±ki1) : 2k₀²= k²_i1+ k²_i2= n0}.

Remark 2.1. We single out the simplest case (Q11) to demonstrate the detailed analysis in Theorem 3.1. The general situation for the case (Q₁) is stated in Theorem 3.2.

Examples for each case are the following:

case n₀ = |K|² K = (k₁, k₂) (Q₁₁) 1 (1,0), (0,1) (Q₁₁) 49 (7,0), (0,7)

(Q₁₂) 25 (4, ±3), (3, ±4), (5, 0), (0, 5) (Q₁₂) 169 (5, ±12), (12, ±5), (13, 0), (0, 13) (Q₂₁) 5 (2, ±1), (1, ±2)

(Q₂₁) 65 (7, ±4), (4, ±7), (8, ±1), (1, ±8) (Q₂₂) 2 (1, ±1)

(Q₂₂) 50 (7, ±1), (1, ±7), (5, ±5)

(Q₂₂) 1250 (35, ±5), (5, ±35), (31, ±7), (7, ±31), (25, ±25)

In the next section, we prove that for each case above (Q₁₁− Q22), the SHE bifurcates from the trivial solution (0, λ) to an attractor as λ crosses λ0. For the case (Q11) under assumption (P1), part of the bifurcation result is treated for the two dimensional SHE in [13]. Theorem 3.1 extends this result by demonstrating the whole structure of singular points on the bifurcated attractor. Moreover, the general cases ((Q12), (Q21), (Q22)) under assumption (P₁) are also treated in the Section 3. We classify the structure of the singular points on the bifurcated attractors for each case.

It is worthwhile to recall that the critical value λ₀ is a function of l. The scale of the bifurcation varies in response to the change of l. In fact, for given n0 ∈ N, one can find the critical number l0 such that n0π²/l²₀ is closest to 1 among all l > 0. Since Γ(n0) determines the dimension of the center manifold near λ0, 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 (P2) 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

(2.3) u_t+ ∆(∆u + λu) + uux= 0,

which has the form (5.1) with L_λu =−Au + Bλu and

Au =−∆²u, Bλu =−λ∆u, G(u) = −uux. The eigenvalues of L_λ are

αK(λ) =�|K|π L

�2�

λ−�|K|π L

�2�

=: τK(λ − τK),

with the corresponding eigenvectors φK in H. Since αK(λ) is a linear function of λ, the case (P₂) does not happen for KSE. Even when we consider the case (P₁) for the SHE, the primary instability can arise at λ1 as well as at λK for any K according to the length l. However, the primary instability for the KSE arises when λ crosses τ₁ = π²/L² 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.

3. Main Results

In this section, we show that the SHE bifurcates from the trivial solution to an at- tractor as λ crosses λ0. Under assumption P1, we study all the possible cases, i.e., (Q11), (Q12), (Q21) and (Q22). First, we consider the case (Q11).

Theorem 3.1. Suppose that (P1) and (Q₁₁) hold true. Assume K₁ = (k₀, 0), K₂ = (0, k0) ∈ Γ(n0) as defined in (Q11). Then, we have the following.

(a) For λ > λ₀, (1.1) bifurcates from (u, λ) = (0, λ₀) to an attractor Aλ which is homeomorphic to S¹.

(b) For any bounded open set U ⊂ H with 0 ∈ U, there exists ε > 0 such that as λ₀ < λ < λ₀+ ε, Aλ attracts U\Γ in H, where Γ is the stable manifold of u = 0 with codimension 2.

(c) The bifurcated attractor Aλ consists of eight singular points and their connecting orbits. The singular points can be expressed as

u^±₁(x) = ± αλ

�sink₀πx₁

l + sink₀πx₂ l

� + o(�

λ− λ0), u^±₂(x) = ± αλ

�sink₀πx₁

l − sink₀πx₂ l

� + o(�

λ− λ0), u^±₃(x) = ± γλ sink₀πx₁

l + o(�

λ− λ0), u^±₄(x) = ± γλ sink₀πx₂

l + o(�

λ− λ0), where

(3.1) α_λ = 2

3

�λ− λ0, γ_λ=

�4(λ − λ0)

3 .

Moreover, under the conditions (P1) and (Q11), u^±₁ and u^±₂ are saddle points, while u^±₃ and u^±₄ are stable nodes.

Proof. We note that

β_K1(λ) = β_K2(λ)





< 0, if λ < λ0,

= 0, if λ = λ₀,

> 0, if λ > λ0,

and Reβn(λK) < 0 for all K �= K1, K₂. Then, we may deduce from Lemma 4.3 and Theorem 5.1 that (1.1) bifurcates from (u, λ) = (0, λ0) to an attractor Aλ satisfying the statement (b).

It remains to show (c) is valid and Aλis homeomorphic to S¹. Let E1 = span{φK1, φK2} and E2 = E₁^⊥ in H. Let Pj : H → Ej be the canonical projections and L^λj = Lλ|Ej, for j = 1, 2. For u∈ H, we can write u =�

K∈Zy_Kφ_K. Let us write yi = yKi, φi = φKi for i = 1, 2, and β(λ) = βK1(λ) for simplicity. Let Φ(·, λ) : E1 → E2 be a center manifold function, then the reduced equation of (5.1) on the center manifold is

(3.2)





 dy₁

dt = β(λ)y1+ 1 2l²

�G�

y₁φ₁+ y2φ₂+ Φ(y1, y₂, λ)� , φ₁�

, dy2

dt = β(λ)y₂+ 1 2l²

� G�

y₁φ₁+ y₂φ₂+ Φ(y₁, y₂, λ)� , φ₂�

.

Since Φ(y1, y2, λ) = o(|y|) with y = (y1, y2), we have

�G�

y₁φ₁+ y2φ₂+ Φ(y1, y₂, λ)� , φ₁�

= −�

Ω

�y³₁φ⁴₁+ 3y₁²y₂φ³₁φ₂+ 3y1y₂²φ²₁φ²₂+ y³₂φ₁φ³₂�

dx + o(|y|³),

� G�

y1φ1+ y2φ2+ Φ(y1, y2, λ)� , φ2

�

= −�

Ω

�

y³₁φ³₁φ₂+ 3y₁²y₂φ²₁φ²₂+ 3y₁y₂²φ₁φ³₂+ y₂³φ⁴₂�

dx + o(|y|³).

Performing explicit calculation by use of (4.3) in Appendix 1, we see that the bifurcation equation (3.2) becomes

(3.3) dy

dt = β(λ)y − F (y) + o(|y|³), where

F (y) = 3

4(y₁³+ 2y1y₂², 2y²₁y₂+ y³₂).

We note that

3

4|y|⁴≤ < F (y), y >_R² ≤ 3 2|y|⁴.

Hence, by Theorem 5.2 to show that the bifurcated attractor Aλ is homeomorphic to S¹. Finally, we show that Aλ 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) dy

dt = β(λ)y − F (y) =: vλ(y)

have the same dynamic behavior near (u, λ) = (0, λ₀). For λ > λ₀ near λ₀, (3.4) admits eight steady state solutions

(3.5) ± (y1, y₂) = (αλ, α_λ), (αλ,−αλ), (γλ, 0), (0, γ_λ), where α_λ and γ_λ are given by the formula (3.1). Then,

Dv_λ(λ) =





β(λ)− 3

4(3y₁²+ 2y₂²) −3y1y₂

−3y1y₂ β(λ)−3

4(2y²₁+ 3y²₂)





If y²₁ = y₂² = α²_λ, then

Dv_λ =





−2

3β(λ) −3y1y₂

−3y1y2 −2 3β(λ)





has eigenvalues ²₃β(λ),−2β(λ). Hence, they are regular solutions of (3.4) and correspond to saddle points of (3.3).

If y²₁ = γ_λ² and y2 = 0, then

Dv_λ =

� −2β(λ) 0

0 −β(λ)

�

has eigenvalues −β, −2β. Thus, (y1, y2) = ±(γλ, 0) are regular solutions of (3.4) and correspond to stable nodes of (3.3). Similarly, (y₁, y₂) = ±(0, γλ) are regular solutions of (3.4) and correspond to stable nodes of (3.3). Now, we conclude that there are exactly eight regular steady state solutions for (3.3) and the corresponding bifurcated solutions

u = y1φ1+ y2φ2+ �

K�=K1,K2

yKφK

of SHE are regular by Theorem 3.10 of [7]. �

Figure 1 illustrates the bifurcated attractor on the center manifold for the case (Q11) when λ > λ0 near λ0.

u⁻₁ u⁻₃

u⁻₂

u⁺₄

u⁺₁

u⁺₃

u⁺₂ u⁻₄

Figure 1.

Throughout this paper, we denote C(r, s) =

� r s

�

= r!

s!(r− s)!.

The second main result is the dynamic bifurcation of the SHE under the condition (P1) and (Q₁₂).

Theorem 3.2. Suppose that (P1) and (Q12) hold true. For 1 ≤ i ≤ p, we put K₁= (k₀, 0), K₂= (0, k₀),

K_4i−3 = (ki1, k_i2), K_4i−2 = (ki2, k_i1), K_4i−1 = (ki1,−ki2), K4i= (ki2,−ki1) such that corresponding eigenvectors are

φ_K1 = sin�k0π l x₁�

, φ_K2 = sin�k0π l x₂�

, φ_K4i−3 = sin�ki1π

l x₁+k_i2π l x₂�

, φ_K4i−2 = sin�ki2π

l x₁+k_i1π l x₂�

, φ_K4i−1 = sin�ki1π

l x₁−k_i2π l x₂�

, φ_K4i = sin�ki2π

l x₁−k_i1π l x₂�

. Then we have the followings.

(a) For λ > λ₀, (1.1) bifurcates from (u, λ) = (0, λ₀) to an attractor Aλ which is homeomorphic to S^4p+1.

(b) For any bounded open set U ⊂ H with 0 ∈ U, there exists ε > 0 such that as λ₀ < λ < λ₀+ ε, Aλ attracts U\Γ in H, where Γ is the stable manifold of u = 0 with

codimension 4p + 2.

(c) For any u_λ∈ Aλ, u_λ can be expressed as u_λ =

4p+2�

i=1

y_iφ_Ki + o(|y|), y = (y1,· · · , y4p+2).

(d) The bifurcated attractor Aλ contains exactly 3^4p+2− 1 singular points. More pre- cisely, each singular point u ∈ Aλ can be written as u = w + o�√

λ− λ0�

, where w belongs to one of Γ_q defined by

Γq =�

yi1φK_i1 + · · · + yiqφK_iq

��

� 1 ≤ i¹ <· · · < iq≤ 4p + 2, yi1,· · · , yiq = ±α(λ, q)� . Here, q = 1, · · · , 4p + 2 and

α(λ, q) =

�4(λ − λ0) 3(2q − 1).

For each q = 1, · · · , 4p + 2, Γq has 2^q· C(4p + 2, q) elements. If w ∈ Γ1, then u is a stable node on the center manifold. If w ∈ Γq 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,

β_Ki(λ)





< 0, if λ < λ₀

= 0, if λ = λ0

> 0, if λ > λ0

and Reβn(λK) < 0 for all K �= Ki. 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 E1 = span{φKi|i = 1, · · · , 6} and E2 = E₁^⊥ in H. Let Pj : H → Ej be the canonical projections and L^λj = L_λ|Ej, for j = 1, 2. For u∈ H, we can write u =�_∞

K∈ZyKφK. Let us write yi = yK_i, φi = φK1 and β = βK1 for simplicity. The reduced equation of (5.1) on the center manifold is the following system:

for i = 1, · · · , 6, (3.6) dy_i

dt = βyi+ 1 2l²

�G�

y₁φ₁+ y2φ₂+ y3φ₃+ y4φ₄+ y5φ₆+ y6φ₆� , φ_i�

+ o(|y|³), where y = (y₁, y₂, y₃, y₄, y₅, y₆). After some calculation by use of (4.3), we obtain the truncated system for (3.6)

(3.7) dy_i

dt = βy_i−3 4

�

y³_i + 2y_i�

|y|²− y²i

��=: βy_i− Fi(y), i = 1, · · · , 6.

If we set F = (F1,· · · , F6), then we have 3

4|y|⁴≤ < F (y), y >_R⁶ ≤ 3 2|y|⁴,

which implies by Theorem 5.2 that the bifurcated attractor Aλ is homeomorphic to S⁵. Next, let us consider nontrivial steady state solutions of (3.7). If yi �= 0 for i = 1, · · · , q and y_i = 0 for i = q + 1, · · · 6, then it follows from (3.7) that

4

3β = 2|ˆy|²− yi², i = 1,· · · , q,

where ˆy = (y1,· · · , yq, 0,· · · , 0). Summing up these equations, we have 4qβ/3 = (2q − 1)|ˆy|², and thus

(3.8) y_i²= 2|ˆy|²−4

3β = 4

3(2q − 1)β≡ α(λ, q)², i = 1,· · · , q.

Hence, there are 2^q· C(6, q) singular points y = (y1, y₂,· · · , y6) with exactly q nonzero yi

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

�6 q=1

2^q· C(6, q) = 3⁶− 1.

Next, we show that these singular points are nondegenerate. Let v_λ be a vector field defined by

dy

dt = v_λ(y) Then, the Jacobian matrix of Dv_λ of v_λ is

















β− h1(y) −3y1y2 −3y1y3 −3y1y4 −3y1y5 −3y1y6

−3y1y₂ β− h2(y) −3y2y₃ −3y2y₄ −3y2y₅ −3y2y₆

−3y1y3 −3y2y3 β− h3(y) −3y3y4 −3y3y5 −3y3y6

−3y1y₄ −3y2y₄ −3y3y₄ β− h4(y) −3y4y₅ −3y4y₆

−3y1y5 −3y2y5 −3y3y5 −3y4y5 β− h5(y) −3y5y6

−3y1y₆ −3y2y₆ −3y3y₆ −3y4y₆ −3y5y₆ β− h6(y)















 ,

where hi’s are defined by hi(y) = 3

4(3y²_i + 2|ˆyi|²), ˆyi = (y1,· · · , yi−1, 0, yi+1,· · · , y6).

Let q be the number of nonzero components of singular points y obtained above. By virtue of symmetry, we may assume that yi �= 0 for 1 ≤ i ≤ q and yi = 0 for q + 1 ≤ i ≤ 6.

Then, we have

y_i = ±� 4β 3(2q − 1)

�¹₂

for 1 ≤ i ≤ q, h_i =2q + 1

2q − 1 β for 1 ≤ i ≤ q, hi = 2q

2q − 1 β for q + 1 ≤ i ≤ 6.

Suppose that q = 1. Then,

Dv_λ(y) = diag�

β− h1(y), · · · , β − h6(y)� has two eigen values

(3.9) σ₁= −2β, σ₂ = −β

The multiplicity of σ2 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 yi’s are positive for 1 ≤ i ≤ q. This follows from a simple observation that

det�

Dv_λ(y1,· · · , yi,· · · , yq, 0,· · · , 0) − ρI�

= det�

Dv_λ(y1,· · · , −yi,· · · , yq, 0,· · · , 0) − ρI� . Keeping these information in mind, we derive that

Dv_λ=







E_q� −2

2q − 1β, −4 2q − 1β�

0

0 −1

2q − 1βI_6−q





 ,

where Eq is a q × q matrix defined by (4.5) and Ik is the k × k identity matrix. Hence, by Lemma 4.2 in Appendix 1 the eigenvalues of Dv_λ are

(3.10) ρ₁= −1

2q − 1β < 0, ρ₂= −4q + 2

2q − 1 β < 0, ρ₃ = 2

2q − 1β > 0.

The multiplicities of ρ1, ρ₂, ρ₃ are 6 − q, 1, q − 1 by Lemma 4.2, respectively. As a conse- quence, 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 Q21 and Q22. Essentially, these cases share the similar bifurcation structure with Theorem 3.2. Indeed, Lemma 4.1 is valid for all cases Q11−Q22. Hence, we expect the reduced equations on the center manifold to have the same structure.

Theorem 3.3. Suppose that (P1) and (Q21) hold true. For 1 ≤ i ≤ p, we put K_4i−3= (k_i1, k_i2), K_4i−2= (k_i2, k_i1), K_4i−1 = (k_i1,−ki2), K_4i= (k_i2,−ki1), such that corresponding eigenvectors are

φK_4i−3 = sin�ki1π

l x1+ki2π l x2

�

, φK_4i−2 = sin�ki2π

l x1+ki1π l x2

� , φK_4i−1 = sin�ki1π

l x1−ki2π l x2

�

, φK_4i = sin�ki2π

l x1−ki1π l x2

� . Then, we have the following.

(a) For λ > λ0, (1.1) bifurcates from (u, λ) = (0, λ0) to an attractor Aλ which is homeomorphic to S^4p−1.

(b) For any bounded open set U ⊂ H with 0 ∈ U, there exists ε > 0 such that as λ0 < λ < λ0+ ε, Aλ attracts U\Γ in H, where Γ is the stable manifold of u = 0 with codimension 4p.

(c) For any uλ∈ Aλ, uλ can be expressed as u_λ =

�4p i=1

yiφ_Ki+ o(|y|), y = (y1,· · · , y4p).

(d) The bifurcated attractor Aλ contains exactly 3^4p− 1 singular points. More precisely, each singular point u ∈ Aλ can be written as u = w + o�√

λ− λ0

�, where w belongs to one of Γq defined by

Γq =�

y_i1φ_Ki1 + · · · + yiqφ_Kiq ��� 1 ≤ i¹ <· · · < iq≤ 4p, yi1,· · · , yiq = ±α(λ, q)� .

Here, q = 1, · · · , 4p and

α(λ, q) =

�4(λ − λ0) 3(2q − 1).

For each q = 1, · · · , 4p, Γq has 2^q· C(4p, q) elements. If w ∈ Γ1, then u is a stable node on the center manifold. If w ∈ Γq 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 3.2. Let E1 = span{φKi|i = 1, · · · , 4p} and E2 = E₁^⊥ in H. Let Pj : H → Ej

be the canonical projections and L^λ_j = L_λ|Ej, for j = 1, 2. For u ∈ H, we can write u =

�4p

i=1y_iφ_Ki+ Φ(y, λ), where y = (y₁,· · · , y4p) 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 = βyi+ 1 2l²

�G��^4p

j=1

y_jφ_Kj� , φ_Ki�

+ o(|y|³).

Using (4.3), we can calculate the nonlinear terms to obtain the truncated system for (3.11)

(3.12) dy_i

dt = βyi−3 4

�

y³_i + 2yi�

|y|²− y²i

��, i = 1,· · · , 4p.

By Theorem 5.2, the bifurcated attractor Aλ 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 yi �= 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) y²_i = 4

3(2q − 1)β, i = 1,· · · , q.

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

�4p q=1

2^q· C(4p, q) = 3^4p− 1.

We show that these singular points are nondegenerate. Let v_λ be a vector field defined by

dy

dt = v_λ(y) Then, the Jacobian matrix of Dvλ of vλ is











β− h1(y) −3y1y₂ · · · −3y1y_4p

−3y1y₂ β− h2(y) · · · −3y2y_4p

... ... ... ...

−3y1y_4p −3y2y_4p · · · β − h4p(y)









 ,

where hi’s are defined by h_i(y) = 3

4(3y_i²+ 2|ˆyi|²), ˆy_i= (y₁,· · · , yi−1, 0, y_i+1,· · · , y4p).

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 σ1 and σ2 are 1 and 4p − 1, respectively. If 2 ≤ q ≤ 4p, then the multiplicities of ρ₁, ρ₂, ρ₃ 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 (P1) and (Q22) hold true. For 1 ≤ i ≤ p, we put K₁= (k0, k₀), K₂= (k0,−k0),

K4i−3 = (ki1, ki2), K4i−2 = (ki2, ki1), K_4i−1 = (k_i1,−ki2), K_4i= (k_i2,−ki1) such that corresponding eigenvectors are

φ_K1 = sin�k0π

l x₁+k₀π l x₂�

, φ_K2 = sin�k0π

l x₁−k₀π l x₂�

, φ_K4i−3 = sin�ki1π

l x₁+k_i2π l x₂�

, φ_K4i−2 = sin�ki2π

l x₁+k_i1π l x₂�

, φ_K4i−1 = sin�ki1π

l x₁−k_i2π l x₂�

, φ_K4i = sin�ki2π

l x₁−k_i1π l x₂�

. Then, we have the following.

(a) For λ > λ0, (1.1) bifurcates from (u, λ) = (0, λ0) to an attractor Aλ which is homeomorphic to S^4p+1.

(b) For any bounded open set U ⊂ H with 0 ∈ U, there exists ε > 0 such that as λ₀ < λ < λ₀+ ε, Aλ attracts U\Γ in H, where Γ is the stable manifold of u = 0 with codimension 4p + 2.

(c) For any u_λ∈ Aλ, u_λ can be expressed as

u_λ =

4p+2�

i=1

y_iφ_Ki + o(|y|), y = (y1,· · · , y4p+2).

(d) The bifurcated attractor Aλ contains exactly 3^4p+2− 1 singular points. More pre- cisely, each singular point u ∈ Aλ can be written as u = w + o�√

λ− λ0

�, where w belongs to one of Γq defined by

Γq =�

y_i1φ_Ki1 + · · · + yiqφ_Kiq ��� 1 ≤ i¹ <· · · < iq≤ 4p + 2, yi1,· · · , yiq = ±α(λ, q)� . Here, q = 1, · · · , 4p + 2 and

α(λ, q) =

�4(λ − λ0) 3(2q − 1).

For each q = 1, · · · , 4p + 2, Γq has 2^q· C(4p + 2, q) elements. If w ∈ Γ1, then u is a stable node on the center manifold. If w ∈ Γq 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. �

4. 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.

Lemma 4.1. Let k1,· · · , k8 be integers and

Ki= (k2i−1, k2i) �= (0, 0), φi(x1, x2) = sin�k2i−1π

l x1+k2iπ l x2

�

, i = 1, 2, 3, 4.

Suppose that

K_i �= Kj and K_i�= −Kj for i �= j, (4.1)

|Ki|²= n0 for all i = 1, 2, 3, 4.

(4.2)

Then, we have the following identities: for distinct i, j, r,

(4.3)

(i) �

Ω

φ⁴_i dx₁dx₂= 3

2l², (ii) �

Ω

φ²_iφ²_j dx₁dx₂= l², (iii) �

Ω

φ³_iφ_j dx₁dx₂ = 0, (iv) �

Ω

φ₁φ₂φ₃φ₄ dx₁dx₂= 0,

(v) �

Ω

φ²_iφ_jφ_r dx₁dx₂= 0.

Proof. The assertions (i) and (ii) are easily derived from the half-angle identity of trigono- metric functions. For other assertions, we recall the identity

(4.4) sin α · sin β = 1

2

�cos(α − β) − cos(α + β)� , cos α · cos β = 1

2

�cos(α − β) + cos(α + β)� .

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

(iii) = 3 4

�

Ω

sin� k2i−1π

l x₁+k_2iπ l x₂�

· sin�k_2j−1π

l x₁+k_2jπ l x₂�

dx₁dx₂

−1 4

�

Ω

sin�3k_2i−1π

l x₁+3k_2iπ l x₂�

· sin�k_2j−1π

l x₁+k_2jπ l x₂�

dx₁dx₂. The first integral vanishes by (4.1) and (4.4). If the second term is nonzero, then we deduce from (4.4) that either 3Ki + Kj = 0 or 3Ki− Kj = 0. In both cases, we obtain

|Ki|²= 9|Kj|², 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

K₁+ (−1)^aK₂+ (−1)^bK₃+ (−1)^cK₄ = 0, for some a, b, c ∈ N.

Without loss of generality, we may assume

K₁+ K2+ K3+ K4 = 0,

and Ki’s are in counter-clockwise orientation on the circle of radius √n0 centered at the origin. Since K₁+ K₂= −(K3+ K₄), it follows from (4.2) that K₁· K2= K₃· K4. Again

by (4.2), we obtain |K1− K2|² = |K3− K4|². Similarly, we have

|K1− K3|² = |K2− K4|², |K1− K4|² = |K2− K3|²,

which implies that K1, K₂, K₃, K₄ are vertices of a right rectangle inscribed in a circle.

Hence, we have K₁ = −K3 and K₂ = −K4, 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 2Ki+ Kj + Kr= 0. Hence,

4n0= 4|Ki|²= |Kj|²+ |Kr|²+ 2Kj · Kr= 2n0+ 2Kj· Kr.

Moreover, this implies that n0 = Kj· Kr ≤ |Kj| |Kr| = n0. Thus the equality holds true in this inequality, which concludes that Kj = Kr. 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 En(a, b) be a n × n symmetric real matrix defined by

(4.5) E_n(a, b) =









a b · · · b b a · · · b ... ... ... ...

b b · · · a







 ,

where a and b are nonzero real numbers. Then, En(a, b) admits two distinct eigenvalues ρ1 = a + b(n − 1) and ρ2= a − b such that the multiplicities of ρ1 and ρ2 are 1 and n − 1, respectively.

Proof. A direct computation shows that ρ1is an eigenvalue with eigenvector e1 = (1, 1, · · · , 1)^t and ρ2 is an eigenvalue with linearly independent eigenvectors

e2 = (1, −1, 0, 0, · · · , 0), e3 = (1, 0, −1, 0, · · · , 0),

...

en = (1, 0, 0, · · · , 0, −1).

� 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.

Lemma 4.3. If λ ≤ λ0, then u = 0 is globally asymptotically stable equilibrium point of (1.1).

Proof. See [13]. �

5. 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 H1 and H be two Hilbert spaces with a dense inclusion H1 �→ H. Let us consider the nonlinear evolution equation

(5.1)



 du

dt = L_λu + G(u, λ), u(0) = u₀,

where u : [0, ∞) → H is the unknown function and λ ∈ R is the system parameter. The parameterized operator Lλ : H1 → H are linear completely continuous fields depending continuously on λ and satisfy

(5.2)





L_λ = −A + Bλ, a sectorial operator, A : H1 → H, a linear homeomorphism,

B_λ : H₁→ H, parameterized linear compact operator.

Then, Lλ generates an analytic semigroup {Sλ(t) = e^−tLλ}t≥0 and we can define fractional power operators L^α_λ for any 0 ≤ α ≤ 1 with domain Hα = D(L^α_λ). Moreover, H₀ = H and if α1 > α₂, then Hα1 ⊂ Hα2. We assume that G(·, λ) : Hα → H are parameterized 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 {ρk} ⊂ C and eigenvector sequence {ek, h_k} ⊂ H1 of A satisfying

(5.4)

















Az_k = ρkz_k, z_k= ek+ ihk, {ek, h_k} is a basis of H,

Reρk → ∞ as k → ∞,

��

� Imρk

b + Reρ_k

��

� ≤ C,

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 Hα = D(A^α) for any 0 ≤ α ≤ 1. For the compact operator Bλ : H1 → H, we assume that there exists a constant 0 ≤ θ < 1 such that

(5.5) B_λ : H_θ → H is bounded for all λ ∈ R.

It is known that L_λ = −A + Bλ is a sectorial operator if (5.4) and (5.5) hold.

Let β1(λ), · · · , βk(λ), · · · ∈ C be the eigenvalues of Lλ counting multiplicities. Suppose that

(5.6) Reβ_j(λ) =





< 0, if λ < λ0

= 0, if λ = λ0

> 0, if λ > λ₀ (1 ≤ j ≤ m) and

(5.7) Reβ_j(λ₀) < 0, ∀ j ≥ m + 1.

We define the eigenspace of L_λ at λ0 by E₀ =

�m j=1

�∞ k=1

{u ∈ H1 : (Lλ0 − βj(λ0))^ku = 0}.

Then, it is known that dimE0 = m.

We now introduce the notion of attractor bifurcation. We say that (5.1) bifurcates from (u, λ) = (0, λ0) to an attractor Ωλ if there exists a sequence of attractors {Ωλn} of (5.1) with 0 /∈ Ωλn such that

nlim→∞ max

u∈Ωλn|u| = 0, lim

n→∞λ_n= λ0.

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 dy- namic 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.

Theorem 5.1 (Attractor Bifurcation theorem). Suppose that (5.2)-(5.7) hold true, and u = 0 is a locally asymptotically stable equilibrium point of (5.1) at λ = λ0. Then, we have the following:

(a) The equation (5.1) bifurcates from (u, λ) = (0, λ0) to an attractor Aλ for λ > λ0, with m− 1 ≤ dimAλ ≤ m, which is connected if m > 1.

(b) For any uλ ∈ Aλ, uλ can be expressed as

u_λ = v_λ+ o(�vλ�H1), v_λ ∈ E0.

(c) If u = 0 is globally asymptotically stable for (5.1) at λ = λ₀, then for any bounded open set U ⊂ H with 0 ∈ U, there exists ε > 0 such that as λ0 < λ < λ₀+ ε, Aλ 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 λ0, then ε can be chosen independent of U.

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

(5.8) dy

dt = β(λ)y − G(y, λ), y ∈ Rⁿ.

Here, β(λ) is a continuous function of λ and G(y, λ) = Gk(y, λ) + o(|y|^k) is a two dimen- sional vector field for some integer k = 2m+1 ≥ 3, where Gk(y1,· · · , yk, λ) is a k–multiple linear function and G_k(y, λ) = G_k(y, · · · , y, λ). The following theorem from [7] gives a criterion when the system (5.8) bifurcates to an S^m-attractor.

Theorem 5.2. Suppose that

(5.9) β(λ)







< 0 if λ < λ₀,

= 0 if λ = λ₀,

> 0 if λ > λ₀, and G_k(y, λ) satisfies

(5.10) C₁|y|^k+1 ≤ < Gk(y, λ), y >_Rⁿ ≤ C2|y|^k+1

for some constants C1, C2 > 0. Then, for λ > λ0, the system (5.8) bifurcates from (y, λ) = (0, λ₀) to an attractor Aλ which is homeomorphic to Sⁿ⁻¹.