BIFURCATION AND FINAL PATTERNS OF A MODIFIED SWIFT-HOHENBERG EQUATION

. In this paper, we study the dynamical bifurcation and ﬁnal patterns of a modiﬁed Swift-Hohenberg equation(MSHE). We prove that the MSHE bifurcates from the trivial solution to an S 1 -attractor as the control parameter α passes through a critical number ˆ α . Using the center manifold analysis, we study the bifurcated attractor in detail by showing that it consists of ﬁnite number of singular points and their connecting orbits. We investigate the stability of those points. We also provide some numerical results supporting our analysis.

1. Introduction. Pattern formation arises from many natural circumstances and has been an important subject in nonequilibrium physics. This happens when the underlying system undergoes phase transitions. Many examples of pattern formation occur in situation that a system is changing from one phase to another, for instance, from a liquid to a geometrically patterned solid, or from a uniform mixture of chemical constituents to a phase-separated pattern of precipitates( [12]). It also occurs in a variety of hydrodynamic systems like convection in pure fluids and mixtures, rotating fluids, and chemically reacting fluids.
The concept of instability plays an important role in the understanding of pattern formation. Spatial or temporal patterns emerge when relatively simple systems are 2544 YUNCHERL CHOI, TAEYOUNG HA, JONGMIN HAN AND DOO SEOK LEE driven into unstable states during the phase transition. The stable simple(or trivial) system will deform by large amount in response to small perturbation. Such an instability is closely related to a control parameter of the system. As the control parameter moves, solutions can appear or disappear, and change their stability. In particular, the instability usually happens when a control parameter of the system passes through a critical number such that the trivial state loses its stability and turns into a new state leading to a final pattern.
To study spatial or temporal patterns in a system, we do not need the full solutions of realistic equations describing the system. If we are interested in patternforming properties of the system, it is sufficient to consider a model equation which shares the long-range effects with the original system( [7]). 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 hydrodynamic instabilities. A canonical form of such fourth order equations is . The SHE is a widely accepted model in the study of the formation of patterns( [2,15]). It was derived in [22] as an approximate model for the Rayleigh-Bénard convection describing the pattern formation in layer fluids between horizontal plates. It has attracted a lot of interest in various areas of application regarding pattern formations such as Taylor-Couette flow and lasers( [7]). The GSHE also provides a useful tool to understand the evolution of hexagonal structure and patterns periodic in one direction( [15]). The DKSE has emerged as a fundamental tool for understanding the onset and evolution of secondary instabilities in many driven nonequilibrium systems( [10]). For example, it provides a crude model of directional solidification. Regarding the stability problem and pattern formation issues for (1.1), one of the basic approaches is to deal with it in the theory of nonlinear dynamical system( [12]). One may consider the model equation as an ODE in a phase space with a control parameter. A general form of this ODE is written as where L α is a linear operator in a phase space H, G is a nonlinear term, and α is a control parameter of the underlying system. We assume that some of eigenvalues of L α change from negative signs to positive signs as α passes through a critical number. Then, the trivial state of the system loses its stability and bifurcates to some nontrivial attractor A α . The bifurcated attractor is responsible for the long time behavior of the system and determines the final patterns. Indeed, if E 1 is corresponding eigenspace in the phase space H, the center manifold theory says that there exists a finite dimensional manifold which is locally represented by the graph of a function Φ : E 1 → E ⊥ 1 . This manifold, called the center manifold, is locally invariant and tangential to the eigenspace E 1 . Moreover, it attracts all flows in a neighborhood U of the trivial solution in H\Γ, where Γ is the stable manifold of the trivial solution. The bifurcated attractor is contained in the center manifold so that the long time dynamics of solutions in U is completely determined by the reduced equation of (1.2) on the center manifold. Thus, it is important to verify the structure of the bifurcated attractor, which leads us to the problem of finding the center manifold. Generally, it has been known that it is not easy to obtain the center manifold function since it is defined implicitly. Recently, Ma and Wang derive a useful formula for the approximation of the center manifold functions. This formula provides us the leading terms in certain order and helps us obtain the projected equation of (1.2) onto the center manifold. It will be one of main tools in proving the main theorem. See Theorem 6.1 of [16] for the precise statement.
There have been much efforts on the bifurcation analysis in the above framework as a way of understanding pattern formations for the equation (1.1). See [13,14,18,19,23,25] for the SHE, [6,11] for the GSHE, and [4,5] for the DKSE. The final patterns for these equations show different behaviors which will be mentioned in detail in Section 2. In this paper, as a variation of the Swift-Hohenberg equation, we are interested in a one dimensional modified Swift-Hohenberg equation(MSHE) which is given by (1.1) with f = µu 2 x − u 3 and µ ∈ R. This nonlinear term is reminiscent of the KSE and breaks the symmetry u → −u. It is needed to obtain stable hexagonal patterns( [8]). The MSHE arises in the study of various pattern formation phenomena involving some kind of phase turbulence or phase transition. If µ = 0, then (2.1) corresponds to the usual SHE. We will consider the MSHE on a periodic interval with period 2λ and study the bifurcation and pattern-forming phenomena in the above framework of dynamical system. We will prove that as the control parameter α moves through a critical valueα, the trivial solution will bifurcate to an attractor which determines final patterns. The critical numberα is the minimum eigenvalues of the linear part of the MSHE. We describe the structure of the bifurcated attractor in detail and provide some numerical results supporting our analysis.
Here is the organization of this paper. In Section 2, we state the main theorem of this paper. We also compare our result with those for SHE, GSHE and DKSE. In Section 3, we prove Theorem 2.1 via mathematically rigorous argument. We employ the attractor bifurcation theorem to show that an attractor bifurcates from the trivial solution as α passesα. Then, we study the structure of the bifurcated attractor by employing the center manifold analysis. The key ingredient is to find a representation of the center manifold function by using Theorem 3.8 of [16]. We provide some bifurcation diagrams. In Section 4, we prove Lemma 3.1 which says that the trivial solution u ≡ 0 is locally asymptotically stable for α ≤α. This is a necessary condition for the attractor bifurcation. In Section 5, we provide a numerical study for (1.2) which supports Theorem 2.1.
2. Statement of main Theorem. Let us rewrite the one dimensional MSHE as by setting L α u = −Au + B α u, and We also define the nonlinear operator G(u, µ) = G 2 (u, u, µ) + G 3 (u, u, u), where In what follows, when u = v or u = v = w, we simply write It is easy to check that A, B α , G : H 1 → H are well defined. The global wellposedness was established in [20]. Moreover, it was proved in [20,21] that a global attractor exists in the class H k per for any k ≥ 2. In particular, a bifurcation analysis with respect to α was given in [24] for two dimensional MSHE, where the authors characterized the bifurcation by using Lyapunov-Schmidt reduction method on (0, 2π) × (0, 2π). In this paper, we carry out the bifurcation analysis of one dimensional problem (2.1) in detail by using center manifold reduction on an interval (−λ, λ). Our method has an advantage by providing the stability of bifurcated singular points. Also, it is reported from various results (for example, [5,18,19]) that the period 2λ of the domain as well as the control parameter α is another factor causing bifurcation in the equation (1.1). In the following, we will see how the period 2λ is responsible for the bifurcation of the one dimensional MSHE in detail.
Let us investigate the eigenvalues of the operator L α on H. By a simple computation, one can find that L α has an eigenvalue sequence with the corresponding eigenvectors for n ≥ 1. We also define ψ n (x) = sin(nπx/λ) for n ≥ 1. We note that the eigenvectors are orthogonal to each other and φ n 2 H = λ for all n ≥ 0. Since α n is a quadratic function of (nπ/λ) 2 , there exists N ∈ N 0 := N ∪ {0} such that either α n > α N ∀ n = N, (2.4) or α n > α N = α N +1 ∀ n = N, N + 1. (2.5) In both cases, we denoteα = α N = inf{α n |n ∈ N 0 }.
We note that (2.5) occurs when λ = λ(N ) for some N ∈ N 0 , where The main result of this paper is to verify the dynamic bifurcation of the MSHE defined in H. For the case (2.4), the MSHE has one dimensional center manifold when α is slightly bigger than α N . In this case, the bifurcation phenomena of the MSHE for the case (2.4) was well established in [3]. It turns out that the MSHE has a pitchfork bifurcation. Moreover, the bifurcation also depends on the period 2λ but there is no dependence on µ.
In this paper, we deal with the second case (2.5). Our result has a big difference with the case (2.4) in that (i) the center manifold has two dimension, (ii) the MSHE bifurcates to an S 1 -attractor, and (iii) the structure of the bifurcated attractor depends on the number N and µ. We state the main theorems of this paper as follows.
Theorem 2.1. Suppose that (2.5) holds true for some N ∈ N 0 . Then, as α passeŝ α, the MSHE (2.1) defined in H bifurcates from the trivial solution to an attractor A α,N which is homeomorphic to S 1 . The bifurcated attractor A α,N consists singular points and their connecting orbits. The singular points and their stabilities are described in the following. (i) If N = 0 and µ > 0, there are four singular points: The singular points u ± 0 are asymptotically stable, while u ± 1 are saddle points. If N = 0 and µ < 0, there are four singular points u ± 0 and In this case, u ± 0 are asymptotically stable and u ± 1 are saddle points. (ii) If N = 1, there are two singular points If µ > 0, u − is asymptotically stable and u + is saddle. If µ < 0, u − is saddle and u + is a asymptotically stable.
(iii) Let 2 ≤ N ≤ 7. Then, there exists a number µ N > 1 satisfying the following. If |µ| < µ N and µ = 0, then there are eight singular points where c i is defined by (3.28) for i = 1, 2, 3, 4. Moreover, u ± N 1 and u ± N 2 are asymptotically stable, while u ± N 3 and u ± N 4 are saddle points. If |µ| > µ N , then there are four singular points u ± N 1 and u ± N 2 . The points u ± N 1 are asymptotically stable and u ± N 2 are saddle points. The value µ N is explicitly given in (3.32) for each 2 ≤ N ≤ 7. (iv) If N ≥ 8 and µ = 0, then there are eight singular points in (2.7). The points u ± N 1 and u ± N 2 are asymptotically stable. The points u ± N 3 and u ± N 4 are saddle. We will prove Theorem 2.1 in the next section. We give some remarks on our result. Theorem 2.1 shows the final patterns of the solutions of the MSHE if α is slightly bigger thanα = α N = α N +1 . Although the bifurcated attractor is homeomorphic to S 1 (which we call S 1 -attractor) with the scale O( √ α −α) for each N and µ = 0, its structure relies on both N and µ. If N ≥ 8, the structure is the same for any µ = 0: there are eight singular points. The stable singular points are perturbations of each single eigenvector φ N and φ N +1 . Whereas the saddle points are perturbations of superpositions of φ N and φ N +1 . If 2 ≤ N ≤ 7, there is no dependence on the sign of µ but the size of it. For small |µ|, we have the same structure as the case N ≥ 8. For large |µ|, we have only four singular points such that two perturbations of φ N are stable and two perturbations of φ N +1 are saddle. The cases N = 0, 1 are dependent on the signs of µ. Furthermore, in the case N = 0, although the bifurcated attractor is homeomorphic to S 1 , it almost looks like the perturbation of φ 0 . In fact, the coefficients a 0 and a 1 have a scale (α − α 0 ) 1/2 but a 2 has a scale (α − α 0 ) 3/4 so that a 2 a 0 , a 1 for all α close enough to α 0 .
It is quite interesting to compare our result with the dynamical bifurcation of other types of (1.1). Although the linear stability analysis is the same, the final patterns can be different due to the nonlinear effect f (u, u x , µ). We compare our result with the dynamical bifurcation phenomena for the SHE(f = −u 3 ), the GSHE(f = µu 2 − u 3 ), and the DKSE(f = −uu x ) under the condition (2.5) as the control parameter α →α.
First, it is known from [14] that the SHE bifurcates from the trivial solution to an S 1 -attractor A SHE α,N in the space of odd periodic functions. The attractor A SHE α,N has the same structure for all N ≥ 1. Indeed, it consists of eight static solutions and their connecting orbits for each N ≥ 1. Four of the static solutions are stable points coming from single eigenvectors φ N and φ N +1 . The others are saddle points which are superpositions of φ N and φ N +1 . In the sequel, A SHE α,N has the structure exhibited in Theorem 2.1 (iv) for all N ≥ 1.
Second, for the GSHE, since there is no invariance of odd or even periodic conditions, we need to treat it under the full periodic condition. It was shown in [6,11] that the GSHE bifurcates to an S 3 -attractor A GSHE α,N for the condition (2.5) for each N ≥ 1. The attractor A GSHE α,N contains two invariant circles of static solutions and two dimensional torus consisting of static solutions. Moreover, the direction of the bifurcation depends on the size of |µ| but not on the sign of µ. Indeed, there is a number H(N, λ, µ) such that if H(N, λ, µ) > 0, then the bifurcation is subcritical, i.e, the GSHE bifurcates as α passes throughα to the right. If H(N, λ, µ) < 0, then the bifurcation is supercritical, i.e, the GSHE bifurcates as α passes throughα to the left. It turns out that the µ-factor in the expression of H(N, λ, µ) contains only µ 2 . Consequently, the sign of µ is not important in the direction of bifurcation but the size of µ plays a crucial role. In particular, if |µ| is small enough, then we have a supercritical bifurcation. Third, regarding the DKSE, it was proved in [4] that the DKSE bifurcates from the trivial solution to an S 1 -attractor A DKSE α,N in the space of odd periodic functions. However, the structure of the bifurcated attractor depends on the number N ≥ 1. If N = 1, there are two singular points such that one is a stable node and the other is a saddle point. If 2 ≤ N ≤ 7, there are four singular points such that two of them are stable nodes and the others are saddle points. Stable nodes come from the eigenvector φ N and saddle points come from φ N +1 . If N ≥ 8, there are eight singular points such that A DKSE α,N has the same structure as A SHE α,N . Finally, we observe that the GSHE, the DKSE, and the MSHE have quadratic nonlinear terms coupled with a parameter µ. These terms give different structures of the bifurcated attractors when the critical frequency N is low, precisely N ≤ 7. However, if N ≥ 8, then the bifurcated attractors have the same structure as in the case of the SHE: it consists of the eight singular points and their connecting orbits. There are two single modes φ N and φ N +1 which provide the four stable points. The superposition of φ N and φ N +1 become saddle points. For low numbers N ≤ 7, the structures are different from each other. For example, in the MSHE, if 2 ≤ N ≤ 7, then single modes are stable and the superpositions are saddle(Theorem 2.1 (iii)). However, in the DKSE, if 2 ≤ N ≤ 7, then there are only single modes such that two of them are stable nodes and the others are saddle points.
3. Proof of Theorem 2.1. This section is devoted to the proof of Theorem 2.1. We will apply the attractor bifurcation Theorem of [16,17]. The main point is the center manifold analysis. We will use some formula to calculate the center manifold function derived in [16]. We assume that α is slightly bigger thanα = α N = α N +1 . We set We note that and Reβ n (α) < 0, ∀n = N, N + 1. For the proof this lemma, we need the center manifold analysis for the case α =α given in this section. So, We postpone the proof of this lemma in Section 4. Lemma 3.1 together with (3.1) satisfies the assumptions of the attractor bifurcation Theorem 6.1 of [16]. Hence, (2.1) bifurcates from the trivial solution to an attractor A α,N as α passes throughα to the right. Furthermore, by the Poincare-Bendixon Theorem, the bifurcated attractor A α,N is homeomorphic to S 1 and consists of singular points and their connecting orbits. In the following, we study the stability of singular points, which gives us an information of the final patterns of A α,N . This depends on the behavior of solutions on the center manifold since the center manifold at the trivial solution is locally attractive.
Let E 1 = span{φ N , φ N +1 } and E 2 = E ⊥ 1 in H. For j = 1, 2, let P j : H → E j be the canonical projections and L α j = L α | Ej . If Φ(·, α, µ) : E 1 → E 2 is a center manifold function and v = P 1 u = y N φ N + y N +1 φ N +1 , then the reduced equation of (2.1) on the center manifold is By taking the inner product of (3.4) with φ N and φ N +1 , we obtain In the following, we compute F 1 and F 2 . We deduce from Theorem 3.8 of [16] that the center manifold function Φ can be expressed as (3.6) as α α, where the last equality comes from (3.1). The computation is split into three cases according to the value N .
Using this expression for the center manifold function, we can compute F 1 and F 2 . First, we note that As a consequence, we are led to On the other hand, since Φ(v, α, µ) = O(y 2 0 + y 2 1 ), we obtain
where we used the following formula If we set h = (h 1 , h 2 ), then
3.3. The Case: N ≥ 2. In this case, by (2.6),  For simplicity, let z 1 = y N and z 2 = y N +1 , and set z = (z 1 , z 2 ). Then for Hence, we derive from (3.6) that So, we obtain that Applying the formula (3.7), we deduce that On the other hand, we get Using (3.8), we infer that Gathering the above results, we are led that where We note from (3.25) that Similarly, Using the above identities and (3.24), we have a precise expression for d ij : Let us find nontrivial singular points of the truncated system of (3.5): . (3.28) Since d 11 , d 22 > 0, c 1 and c 2 are well-defined. We need to show that c 3 and c 4 are also well-defined. By direct calculation, we have We note that d 11 − d 21 < 0 for all N ≥ 2 and µ = 0. This means that d 11 d 22 − d 12 d 21 < 0 for the existence of c 4 , which in turn implies that d 22 − d 12 < 0 for the existence of c 3 . We note that Hence, for any µ = 0, In the following, we study the stability of singular points for each N ≥ 2 in detail. Before proceeding further, we notice that the singular points (±c 1 , 0) are always asymptotically stable. Indeed, if we set h = (h 1 , h 2 ), then It is not difficult to see that Dh(±c 1 , 0) have two eigenvalues −2β, (d 11 − d 21 )β/d 11 which are negative by (3.29). Hence, (±c 1 , 0) are asymptotically stable. For existence and stability of other singular points depends on N and µ. In the following, we study these subjects by dividing the value N into two cases. First suppose that N ≥ 8. In this case, we have eight singular points. We note that which are both negative by (3.29). So, (0, ±c 2 ) are asymptotically stable. Next, we see that which implies that Dh ± (c 3 , ±c 4 ) have a positive eigenvalue and a negative eigenvalue. Therefore, ±(c 3 , ±c 4 ) are all saddle points. See Figure 5. Second, we consider the case 2 ≤ N ≤ 7. By direct computation, we deduce that    Suppose that u 0 2 < δ, where δ ∈ (0, 1) is to be determined below. We expand the solution of (2.2) as u(x, t) = ∞ n=0 a n (t)φ n (x).
Since u ∈ C([0, ∞), H), there exists T > 0 such that for all t ∈ [0, T ]. We note that λ|a n (t)| 2 < δ for all t ∈ [0, T ] and n ≥ 0. Multiplying (2.2) by u, we obtain where C ε,µ is a positive constant depending only on ε and µ. By the one dimensional Gagliardo-Nirenberg inequality, we have where D ε,µ relies only on ε and µ. In the sequel, We notice that by the choice of ε As a consequence, Thus, if we choose δ > 0 so small such that then u(·, t) 2 is decreasing for all t ∈ [0, T ]. Applying continuation argument, we can conclude that the trivial solution is asymptotically stable in the ball B δ (0) ⊂ H.
Case 2. α =α In this case, since β N (α) = β N +1 (α) = 0, we employ the center manifold reduction. Since the center manifold is locally attractive, it suffices to consider the locally asymptotic stability of the trivial solution on the center manifold. We rewrite the truncated form of the reduction of (3.5) as whereF i is the truncation of F i . In the following, we show that (y N , y N +1 ) = (0, 0) is locally asymptotically stable for this truncated system. First, suppose that N = 0. By (3.10), we have (2y 2 0 y 1 + y 3 1 ).
So, u ≡ 0 is locally asymptotically stable. This completes the proof of Lemma 3.1.
5. Numerical study. In this section, we will validate Theorem 2.1 using the numerical scheme. Let us consider the one dimensional MSHE (2.1) for 0 ≤ x ≤ λ(N ) and t > 0, where λ(N ) is defined by (2.6) for given N ∈ N 0 . For numerical simulation, we employ the Crank-Nicholson method. To this end, we assume the space is discretized by the equal mesh spacing ∆x = λ/M and x j = j∆x, j = 0.1.2. · · · , M . And T max is enough time so that the numerical solution is close to steady state solution and ∆t = T max /K and t n = n∆t, n = 0, 1, 2, · · · , K. Let u n j or u h (x, t) be an approximate solution to the solution u(x j , t n ) at the discrete grid point (x j , t n ). Then, by applying Crank-Nicholson method to (2.1), we obtain where 0 ≤ γ ≤ 1. In our numerical simulation, γ = 1/2 is chosen. Here, we use the same notations L α and G(·, µ) as in Section 2. Finally, the functions for initial condition are chosen by three types as shown in Figure 6: u 0 (x) = u(x, 0) = 1 + φ 1 (x), u 1 (x) = u(x, 0) = 1 + 2φ 1 (x) + φ 2 (x), u 2 (x) = 0.1φ 8 (x) + φ 9 (x).
Most numerical simulations of this section use u 0 (x) and u 1 (x). But, u 2 (x) is adopted to the test for the case N = 8, We also choose α = 1 + 10 −4 for all numerical tests.
Recalling the final patterns of solutions described in Theorem 2.1, we decompose u(·, t) as u(·, t) = v(·, t) + o(β 1/2 N ). For example, if N = 0 and µ > 0, then v = ±a 0 φ 0 as in Theorem 2.1 (i). To verify the results of Theorem 2.1 numerically, we show the status that the numerical solution goes to v(·, t) as time goes to infitnity. The following Figures illustrates that the dotted curves are the numerical u h (·, t) solutions and the red solid curve is the leading term v(·, t) of the steady state solution u(·, t) obtained in Theorem 2.1.   Let us consider the case 2 ≤ N ≤ 7. We recall from Theorem 2.1 that the bifurcated attractor shows different shapes according to whether µ > µ N or µ < µ N . To capture this phenomena, we choose N = 2 and set µ = 1/2 or 2. We note from (3.32) that µ 2 = 1.2 · · · . Then, Theorem 2.1 (iii) says that if µ = 1/2 < µ 2 , both of the single modes φ 2 and φ 3 are asymptotically stable. If µ = 2 > µ 2 , only the single mode φ 2 is asymptotically stable. Figures 8 illustrate these facts. Figure 8a and Figure 8b show that if µ = 1/2, u h (x, t) goes to c 1 φ 2 (x) for the initial condition u 0 (x), and u h (x, t) goes to c 2 φ 3 (x) for the initial condition u 1 (x) respectively. Figure 8c shows that if µ = 2, u h (x, t) goes to c 1 φ 2 (x) for the initial conditions u 1 (x). For the initial condition u 0 (x), we can obtain the same result in this case. Therefore structures shown at Figure 5a and 5b are verified. . Tests for N = 2 given by (a) the initial value u 0 (x) and µ = 1/2, (b) the initial value u 1 (x) and µ = 1/2 and (c) the initial value u 1 (x) and µ = 2 Last, for the test in the case N ≥ 8, we choose N = 8 and set µ = 1. By Theorem 2.1 (iv), the perturbations of the single modes φ 8 and φ 9 are asymptotically stable. Figure 9a shows that u h (x, t) goes to v(x) = c 1 φ 8 (x) for the initial conditions u 1 (x). For the initial condition u 0 (x), we can obtain the same result in this case. On the other hand, if we choose the initial condition u 2 (x) = 0.1φ 8 ((x) + φ 9 (x), u h (x, t) goes to v(x) = c 2 φ 9 (x) as shown in Figure 9b. This also confirms the structure shown at Figure 5a for the case N ≥ 8.