Attractor Bifurcation and Final Patterns of the N-Dimensional and Generalized Swift-Hohenberg Equations

In this paper I will investigate the bifurcation and asymptotic behavior of solutions of the Swift-Hohenberg equation and the generalized Swift-Hohenberg equation with the Dirichlet boundary condition on a one- dimensional domain $(0,L)$. I will also study the bifurcation and stability of patterns in the n-dimensional Swift-Hohenberg equation with the odd-periodic and periodic boundary conditions. It is shown that each equation bifurcates from the trivial solution to an attractor, when the control parameter crosses the principal eigenvalue of the linearized equation. The local behavior of solutions and their bifurcation to an invariant set near higher eigenvalues are analyzed as well.


1.
Introduction. Pattern formation is an interesting phenomenon which is often observed in physics and chemistry. A physical system when driven sufficiently far from equilibrium tends to form geometric patterns [5]. This phenomena is determined by nonlinear aspects of the system under study.
To study how those patterns form and evolve is the subject of "non-equilibrium physics". Non-equilibrium phenomena include Taylor-Couette flow, parametric waves, reaction-diffusion systems, propagation of electromagnetic waves in certain types of media, and convection. Convection is a widely-studied example of the dynamics that can occur in a system under the influence of a constant, homogeneous temperature gradient. The mechanism of convection is responsible for many phenomena of great interest, such as cloud formation, ocean currents, plate tectonics, and crystal growth.
In the study of spatial patterns an important role is played by model equations. A model equation, while simpler than full system of equations, captures most features which control the pattern formation phenomenon of the system. Recently attention has been drawn to the study of fourth-order model equations involving bistable dynamics. The fourth order pattern forming equation is of central importance [6,7]: where q measures the pattern forming tendency. An extensive study of the stationary equation, known as the Symmetric Bistable (SBS) equation, can be found in [15]. can be written as and the latter can be scaled to the dimensionless form of the Swift-Hohenberg equation which shall be the main focus of this paper. The Swift-Hohenberg (SH) equation was proposed in 1977 in connection with Rayleigh-Bénard's convection. Although the SH equation cannot be derived systematically from the Boussinesq equations, it captures much of the observed physical behavior and has now become a general tool used to investigate not only Rayleigh-Bénard convection, but also other patternforming systems [5].
In this paper we will study qualitatively the bifurcation problem of the 1-dimensional Swift-Hohenberg (SH) equation and generalized Swift-Hohenberg (GSH) equation with the Dirichlet boundary condition. Some numerical results, compatible with our results concerning the SH equation with the Dirichlet boundary condition, can be found in [14].
The main technical tool is the new bifurcation theory developed recently by Ma and Wang [12]. The theory is based on a new notion of bifurcation called attractor bifurcation. The main theorem associated with the attractor bifurcation states that under certain conditions and provided the critical state is asymptotically stable, when the first eigenvalue of the linearized equation crosses the imaginary axis, the system bifurcates from a trivial steady state solution to an attractor with dimension between m − 1 and m, where m is the algebraic multiplicity of the first eigenvalue. Another important ingredient of the analysis is the reduction of the equation to its central manifold. This involves lengthy calculations and careful examination of the nonlinear interaction of higher order. The key idea is to derive an higher order non-degenerate approximation which is sufficient for the bifurcation analysis.
The main results obtained in this article can be summarized as follows. First, we have shown that as the parameter λ crosses the first critical value λ c , the Swift-Hohenberg equation bifurcates from the trivial solution to an attractor A λ , with a dimension between m − 1 and m depending on boundary conditions, where m is the multiplicity of the first eigenvalue of the linearized problem. We will determine the critical number λ c precisely. Second, as an attractor, the bifurcated attractor A λ has asymptotic stability in the sense that it attracts all solutions with initial data in the phase space outside of the stable manifold, with codimension m, of the trivial solution. Third, the 1-d SH equation with the Dirichlet boundary condition or odd periodic boundary condition bifurcates to exactly two steady state solutions; and with the periodic boundary condition, to an attractor homeomorphic to S 1 . Finally, in n-dimensional (n ≤ 3) case with the odd periodic boundary condition, the number of steady state solutions contained in the bifurcated attractor A λ , for λ > λ c , has been given precisely; and with the periodic boundary condition, it is shown that the bifurcated attractor A λ contains a torus T n which consists of steady state solutions.
It is worth mentioning that although the problem has certain symmetry, the method used in the article is crucial for proving the bifurcated object is precisely the attractor and for proving the stability property of the bifurcated object.
The paper is organized as follows. In section 2, we recall the main results of the attractor bifurcation and center manifold reduction method. In Section 3, we state and prove our main theorems concerning the bifurcation of the Swift-Hohenberg equation and the generalized Swift-Hohenberg equation with the Dirichlet boundary condition. In section four, the main results concerning the n-dimensional Swift-Hohenberg equation with the odd-periodic and periodic boundary condition shall be stated.
2. Abstract Bifurcation Theory and Reduction Method.

Attractor Bifurcation Theorem.
Here we shall recall some results of dynamic bifurcation of abstract nonlinear evolution equations developed in [12]. Also we will refer the reader to [13] for a comprehensive study of the dynamic bifurcation theory developed by Ma and Wang.
Let H and H 1 be two Hilbert spaces, and H 1 ֒→ H be a dense and compact inclusion. Consider the following nonlinear evolution equation where u : [0, ∞) → H is the unknown function, λ ∈ R is the system parameter, and L λ : H 1 → H are parameterized linear completely continuous fields continuously depending on λ ∈ R, which satisfy We can see that L λ generates an analytic semi-group {e −tL λ } t≥0 and then we can define fractional power operators L α λ for any 0 We now assume that the nonlinear terms G(·, λ) : H α → H for some 0 ≤ α < 1 are a family of parameterized C r bounded operators (r ≥ 1) continuously depending on the parameter λ ∈ R, such that

M. YARI
For the linear operator A we assume that there exists a real eigenvalue sequence {ρ k } ⊂ R and an eigenvector sequence For the compact operator B λ : H 1 → H, we also assume that there is a constant 0 < θ < 1 such that (5) and (6) is a sectorial operator. It generates an analytic semigroup {S λ (t)} t≥0 . Then the solution of (1) and (2) can be expressed as (1) is said to be an attractor if Σ is compact, and there exists a neighborhood U ⊂ H of Σ such that for any ϕ ∈ U we have The largest open set U satisfying (7) is called the basin of attraction of Σ.
The following dynamic bifurcation theorem for (1) was proved in [12].

The equation (1) bifurcates from
For any u λ ∈ A λ , u λ can be expressed as 4. If the number of equilibrium points of (1) in A λ is finite, then we have the following index formula (1) has a global attractor for all λ near λ 0 , then ε can be chosen independently of U.
The following theorem is an immediate result of the attractor bifurcation theorem: Theorem 2 (Pitchfork bifurcation). If the first eigenvalue is simple, i.e. m = 1, then the bifurcated attractor A λ consists of exactly two points u 1 and u 2 . Moreover, for any bounded open set U ⊂ H with 0 ∈ U there is an ε > 0 such that as λ 0 < λ < λ 0 + ε, U can be decomposed into two open sets U λ 1 and U λ 2 satisfying such that is the solution of (1). The following theorem will be useful later.

Reduction Method.
A useful tool in the study of bifurcation problems is the reduction of the equation to its local center manifold. The idea is to project the equation to a finite dimensional space after a change of basis. An extensive study of the method of reduction to center manifold can be found in [13]. Consider the following non-linear evolution equation: with L λ = −A + B λ : H 1 −→ H being a symmetric linear continuous field, G(., λ) : H 1 −→ H being a C ∞ , which can be expressed as is an n-multiple linear mapping, and G n (u, λ) = G n (u, ..., u, λ). Let β i (λ) and e i (λ) be the eigenvalue and eigenvector of L λ respectively. Since L λ is symmetric, β i (λ)'s are real. Assume that e i (λ)'s form an orthogonal basis for the space H. Also, assume the following conditions hold true: β j (λ 0 ) > 0 for m < j ≤ m + n, β j (λ 0 ) < 0 for m + n < j.
then with a change of basis, the equation can be written in the new basis (one can think of this as Fourier series). After projection to the subspace generated by the first eigenvalue, the equation can be reduced to the central manifold as follows: where x = (x 1 , ..., x m ) t , J mλ is the Jordan matrix corresponding to the first m eigenvalues of L λ , and g(x, λ) = (g 1 (x, λ), ..., g m (x, λ)) t with g i (x, λ) =< G(x + φ(x, λ)), e i >. Here φ(x, λ) is the center manifold function near λ 0 . Finally the above equation can be rewritten as follows: This last equation is called the N th -order approximation of the (13) 3. Bifurcation of The SH and The GSH Equations.
3.1. The SH and The GSH Equations. The following nonlinear equation is known as the Swift-Hohenberg (SH) equation when µ = 0, and as the generalized Swift-Hohenberg (GSH) when µ > 0. The SH equation, proposed in 1977 by Swift and Hohenberg [17], has been shown to be a useful tool in the study of a variety of problems, such as Taylor-Couette flow [10,16], and in the study of lasers [11]. The GSH equation was proposed later in connection with the study of localized patterns. Extensive numerical and analytical studies have been done on both equations; for example see [4,5,2,15] for the SH equation ; and for the GSH equation see [1,3,9,8].
The existence of solutions of (15) is established in [15] using a topological shooting method. Some bifurcation analyses and some numerical simulations, when 0 < λ < 1, are conducted in [14].
where the operators L λ and G are defined as follows: with Hilbert spaces H and H 1 : Since H 1 is compactly imbedded in H, H 1 ֒→ H, it is clear that B : H 1 ֒→ H is a compact operator. With an easy calculation one can see the eigenvalues of A : H 1 ֒→ H are λ n = (1 − ( nπ L ) 2 ) 2 ; hence, assuming L = nπ for any n ∈ N, A is a homeomorphism. Therefore, L λ = −A + B is a completely continuous field. When L = nπ for some integer n, A + I will be a homeomorphism. In this case, L λ is still a completely continuous field, for L λ can be written as the sum of A + I and (λ − 1)I. Figure1), and its principal eigenvalue is λ c = min{ P ( nπ L ) | n ≥ 1 }. Therefore, depending on the value of L, we might get a different critical value and a different final profile associated with it.

2.
For λ > λ c The SH-d bifurcates from (0, λ c ) to an attractor bifurcation A λ which consists of exactly two steady states (Figure 2). 3. For λ > λ c , the bifurcated attractor A λ consists of exactly two steady states u λ 1 , u λ 2 given by where β c (λ) is the first eigenvalue of the linearized equation, φ c the normalized eigenvector associated with it , and α =< (φ c ) 3 , φ c > H . 4. The stable manifold Γ ⊂ H of u = 0 separates the phase space H into two open sets U 1 λ and U 2 λ , where λ c < λ < λ c + ǫ for some ǫ > 0, which are the basin of attraction of u λ 1 and u λ 2 respectively, i.e.
for any ϕ ∈ U λ i (i = 1, 2), where u(t, ϕ) is the solution of the SH equation. 5. For any integer n, the SH equation bifurcates from (u, λ) = (0, λ n ),on λ > λ n , to an attractor consisting of two steady state solutions of the SH equation.
For instance, when L ≤ π, the final patterns are given by Proof. For the proof of the first part see Theorem (6). In fact the existence of the global attractor for the SH equation is known [14]. Without loss of generality, we will assume that L < π. The eigenvectors and eigenvalues of L λ : H 1 ֒→ H are known to be: Moreover, the eigenvalues have the following properties: Hence this theorem is a direct result of the pitchfork bifurcation theorem (2). We only need to prove (20). This can be proven by the Lyapunov-Schmidt reduction method near λ c = λ 1 . Let u ∈ H and u = ∞ k=1 x k φ k (x). Then the steady state bifurcation equation of the SH equation can be expressed as so we will have 450 M. YARI x j x k x l ], x j x k x l ], Hence, by induction, we have: x n = c n x n 1 + o(|x 1 | n ), for n ≥ 4 , where c n is a constant. Therefore, we have the following bifurcation equation for the SH equation: This completes the proof. The proof of the last part of the theorem is similar.
For the generalized Swift-Hohenberg (GSH) equation, the stability of the trivial solution u = 0 may not be true at λ c ; nevertheless, we can prove the existence of a bifurcation near the critical value.
For instance when L < π, that is λ c = (1 − ( π L ) 2 ) 2 , we have α = 8 Proof. By reducing the equation to its center manifold near λ c , we get the following equation when µ > 0 : where α =< (φ c ) 3 , φ c > H is a constant. Then the theorem is an obvious result of this reduced equation.
where Ω = (0, L) n , 1 ≤ n ≤ 3. And the equation is supplemented with one of the following boundary conditions: 1. The odd-periodic boundary condition: 2. The periodic boundary condition:

Functional settings.
In an appropriate functional setting the SH equation (23) can be expressed in the following form where the operators L λ and G are defined as follows: with Hilbert spaces H and H 1 : The subscript "per" stands for "periodic" and the dot, . , means L 0 f dx = 0, for f in H 4 or L 2 . In any case H 1 is a dense and compact subspace of H, H 1 ֒→ H. Theorem 6. (An a priori estimate) Assume λ c be the first eigenvalue of (I + ∆) 2 : H 1 ֒→ H. Then the following a priori estimates hold for the SH equation: Lemma 1. Suppose ψ : R + → R + satisfies the following: with b > 0. Then the following assertions are true: proof of theorem 6. Using the energy method, from (23) we will have: It is known that the principal eigenvalue of A = (I + ∆) 2  also by Hölders inequality : Therefore 27 follows from the lemma(1). By the above theorem we know that u = 0 is globally asymptotically stable for λ ≤ λ c . In fact, the existence of the global attractor for the SH equation can be proven in the same fashion as in [14].

4.3.
Bifurcation of odd-periodic solutions. Assume λ c denotes the first eigenvalues of (I + ∆) 2 : Theorem 7. The following assertions are true for the SH equation (23) with the odd periodic boundary condition : 1. For λ ≤ λ c , u = 0 is globally asymptotically stable.
(32) for 1 ≤ i ≤ n. This together with theorem (3) show that A λ = S 1 in the topological sense, when n = 1.
Since the subspace of the odd functions is an invariant set of L λ + G in H, the Lyapunov-Schmidt reduction equations of the SH equation in this subspace are the same as 29. Therefore the problem has solutions given in 30. Since the equation is invariant under the spatial translation, steady state solutions associated with 30 generate an n-dimensional torus T n as follows where y = (y 1 , ..., y n ). This completes the proof.
Acknowledgements. I would like to thank the referees very much for their valuable comments and suggestions.