STOCHASTIC ATTRACTOR BIFURCATION FOR THE TWO-DIMENSIONAL SWIFT-HOHENBERG EQUATION WITH MULTIPLICATIVE NOISE

. This article concerns the dynamical transitions of the stochastic Swift-Hohenberg equation with multiplicative noise on a two-dimensional domain ( − L,L ) × ( − L,L ). With α and L regarded as parameters, we show that the approximate reduced system corresponding to the invariant manifold undergoes a stochastic pitchfork bifurcation near the critical points, and the impact of noise on stochastic bifurcation of the Swift-Hohenberg equation. We ﬁnd the approximation representation of the manifold and the corresponding reduced systems for stochastic Swift-Hohenberg equation when L 2 and √ 2 L 1 are close together.


Introduction
The Swift-Hohenberg equation was initially proposed by Swift and Hohenberg ([27]) in 1977 as a simple model for the Rayleigh-Bénard instability of roll waves, which takes the form ∂u ∂t This equation plays an important role in the study of various phenomena in pattern formation, see [4,9]. It has been studied a great deal, both analytically and numerically. These fields include the Rayleigh-Bénard problem of convection in a horizontal fluid layer in the gravitational field [28], Taylor-Couette flow [15], some chemical reactions [25] and large-scale flows and spiral core instabilities [1]. These are effects which relate to systems far from equilibrium. In optics, this equation has been considered in relation to spatial structures in large aspect lasers, and synchronously pumped optical parametric oscillators. In [34], attractor bifurcation and asymptotic behavior of the real Swift-Hohenberg equation and the generalized Swift-Hohenberg equation with Dirichlet boundary condition and periodic boundary condition are investigated, in which the techniques are based on the results given in [20,21].
In [22,23,24], the authors studied the asymptotic behavior of the solutions of the Cauchy-Dirichlet problem for the Swift-Hohenberg equation on the domain (0, L), ∂u ∂t = αu − (1 + ∂ 2 ∂x 2 ) 2 u − u 3 , for 0 < x < L, t > 0, u = 0, ∂ 2 u ∂x 2 = 0, at x = 0, L, u(x, 0) = u 0 (x), for 0 < x < L, (1.2) where the initial function u 0 is a smooth function, α and σ are positive numbers. With α and the length of the domain L regarded as bifurcation parameters, different types of structures in the bifurcation diagrams are presented when the bifurcation points are closer. We have studied the asymptotic behavior of the solutions of the Cauchy-Dirichlet problem for the Swift-Hohenberg equation with quintic nonlinearity [29,30]. In [32], we have considered bifurcation of a modified Swift-Hohenberg equation in two spatial dimension with periodic boundary condition. There has been some research in the optimal distributed control for the modified Swift-Hohenberg equation, see [26].
The dynamical behavior of solutions to stochastic differential equations and stochastic partial differential equations, such as long time behavior, ergodicity, and periodicity, has been studied in [8,10,16,33]. In the recent two decades, there has been some research in the impacts of noise on the stochastic dynamics, see [5,15]. The study of the asymptotic behavior of the following stochastic equation driven by multiplicative noise in Stratonovich sense du = (L α u + G(u))dt + σu • dW t has an extensive literature, see [5,6,7,11], and the references therein. Here L α is a linear operator parameterized by a parameter α ∈ R, G(u) represents the nonlinear terms, W t is a two-sided one-dimensional Winner process, and σ ∈ R gives a measure of the amplitude of the noise.
The dynamics of the stochastic Swift-Hohenberg equation has attracted much attention in recent years. In [14], the authors studied the dynamic transitions of the two-dimensional Swift-Hohenberg equation with multiplicative noise, and showed that the approximate reduced system corresponding to the invariant manifold undergoes a stochastic pitchfork bifurcation. Li ([17]) studied the dynamic transitions of the two-dimensional Swift-Hohenberg equation with multiplicative noise, the study is based on the stochastic parameterizing manifolds developed by Chekroun, Liu and Wang ( [6,7]). They both considered α as a parameter to study the dynamic transitions. Approximation representation of parameterizing manifold and non-Markovian reduced systems for a stochastic Swift-Hohenberg equation with additive noise has been investigated in [12]. There are some other results for approximation of manifolds for stochastic Swift-Hohenberg equation with multiplicative noise in Stratonovich sense [3,18,27].
In [31], we studied the dynamical transitions of the stochastic Swift-Hohenberg equation with multiplicative noise on a one-dimensional domain (0, L). With α and the length of the domain L regarded as parameters, we showed that the approximate reduced system corresponding to the invariant manifold undergoes a stochastic pitchfork bifurcation near the critical points, and the impact of noise on stochastic bifurcation of the Swift-Hohenberg equation.
In this paper, we consider the stochastic attractor bifurcation of two dimensional Swift-Hohenberg equation (SHE) on the domain Q = (−L, L) × (−L, L) with multiplicative noise in Stratonovich sense 5) and the SHE with multiplicative noise in Ito sense where the initial function u 0 is a smooth function, α and σ are positive numbers, in particular, W t is a two-sided one-dimensional Winner process. With α and the length of the domain L regarded as parameters, we study the dynamic transitions of the the stochastic Swift-Hohenberg equation. One main objective of this paper is to extend the work in [22,23,24,32] to the two-dimensional stochastic Swift-Hohenberg equation, we will consider the stochastic attractor bifurcation of the Swift-Hohenberg equation near the critical points, and the case when the bifurcation points nearly coincide. With α and the length of the domain L regarded as parameters, we will study the dynamical transitions of the stochastic Swift-Hohenberg equation and the impact of noise on the stochastic dynamics of the Swift-Hohenberg equation. One standard technique in the analysis of deterministic dynamical systems is the center manifold reduction. However, the study of the reduction problem of SPDE to its corresponding stochastic invariant manifolds is much less, one reason is the incompatibility with large excursions of SPDE solutions caused by white noise. The above-mentioned difficulty can be overcome by using stochastic parameterizing manifolds developed ( [6,7]). This approach is based on approximate parameterizations of the small scales by the large ones via the concept of stochastic parameterizing manifolds, where the latter are random manifolds aiming to improve the partial knowledge of the full SPDE¡ s solution in mean square error when compared with its projection onto the resolved modes. Approximate parameterizing manifolds can be obtained by representing the modes with high wave numbers as a pullback limit depending on the time-history of the nodes with low wave numbers for the corresponding backward-forward systems.
This article is organized as follows. In section 2, we recall some results of deterministic the Swift-Hohenberg equation, and introduce some mathematical settings. In sections 3, 4, and 5, we analyze stochastic attractor bifurcation of the Swift-Hohenberg equation near the points √ m 2 + n 2 L 1 and √ m 2 + n 2 L 2 under three cases. The approximation representation of manifold and the corresponding reduced systems for stochastic Swift-Hohenberg equation when L 2 and √ 2L 1 are close together are obtained in section 6.

Mathematical setting
We will recall in this section some results of deterministic the Swift-Hohenberg equation, and introduce some mathematical settings.
Let us introduce the following spaces.
and with the inner product Notice that H and H 1 are Hilbert spaces, and H 1 → H is a dense and compact inclusion. We consider the nonlinear evolution equations where u : [0, ∞) → H is the unknown function, α ∈ R is the system parameter, and L α : H 1 → H are parameterized linear completely continuous fields continuously depending on α ∈ R, which satisfy where A : H 1 → H is a linear homeomorphism, B α : H 1 → H are parameterized linear compact operators. Furthermore, the nonlinear term G(u) = −u 3 is a bounded operator such that Hence the stochastic Swift-Hohenberg equation can be written as Then the eigenvalues of the following eigenvalue problem on H 1 , with the corresponding eigenfunctions e k,l = √ 2 sin π(kx + ly) L .
In the following two sections, we consider stochastic attractor bifurcation of the Swift-Hohenberg equation near the points √ m 2 + n 2 L 1 and √ m 2 + n 2 L 2 . This is done in two cases: first we assume that K = m 2 + n 2 has only two solutions (m, n) = (k, k) and (k, −k), this case will be discussed in section 3. Then, in section 4, we consider the case when K = m 2 + n 2 has only two solutions (m, n) = (k, 0) and (0, k).
3. Analysis of the case (m, n) = (k, k) and (k, −k) In this section, we consider the attractor bifurcation near the points √ m 2 + n 2 L 1 and √ m 2 + n 2 L 2 , in the case the intervals I m,n do not overlap, and K = m 2 +n 2 has only two solutions (m, n) = (k, k) and (k, −k). That is, we consider the attractor bifurcation near the points √ 2kL 1 and √ 2kL 2 . In this case, the space H 1 and H can be decomposed into where H c 1 = span{e k,k , e k,−k }, and H s is the closure of H s 1 in H. We will present a stochastic reduction procedure based on parameterizing manifolds (PM) associated with (1.3). A stochastic parameterizing manifolds [6,7], as the graph of a random continuous function h α (ξ, ω) from H c 1 to H s , and for each realization ω, the function is defined for ξ ∈ H c 1 . The PM-based reduced equation for the resolved modes is where ξ ∈ H c 1 . However, it's more involved to give an explicit expression of h α (ξ, ω), the key idea ( [6,7])is to provide an approximation of h α (ξ, ω) via the pullback characterization Indeed it is too cumbersome to use the above pullback characterization to approximate the vector field P c G(ξ + h α (ξ, θ t ω)) as ξ varies in H c 1 . We adopt instead a "Lagragian approach" which consists of approximating "on the fly" this vector along a trajectory ξ(t, ω) of interest, as the time t flows.
So this is much more manageable and leads naturally to consider, instead of (3.1), we consider the reduced equation where the notation ξ t emphasized the t-dependence of the variable ξ t , φ = P c u 0 , and u (2) s can be used to approximate the stochastic inertial manifold and is obtained from the following backward-forward systems (3.3)-(3.5).
For a given t > 0 and T sufficiently large, let us consider the following 2-layer auxiliary backward-forward system.
s can be used to approximate the stochastic inertial manifold, L c α = P c L α , L s α = P s L α . In the systems above, the initial value of u (1) c and u (2) c are prescribed in fiber θ t ω, and the the initial value of u (2) s is prescribed in fiber θ t−T ω. The solution of system (3.3)-(3.5) is obtained by using a backward-forward integration procedure made possible because of the partial coupling between the equations constituting this system. Here u (1) c and u (2) c emanate backward from ξ in H c 1 force the evolution equation of u (2) s to depend naturally on ξ but not reciprocally. Here, θ t is an element of a metric dynamical system, and ω is a given realization, θ t−T ω is called a fiber, see Arnold [2].
Since u Then by projecting equation After simple calculation, we can obtain Notice that 2 ) 3 ( Then we have It is easy to show that y m,n = 0, (m, n) ∈ Z.

(3.39)
To extract information from the PM reduction, we show that the asymptotic behavior can be completely captured by a sufficiently small neighborhood of the origin. Using the Ito formula, similar as the proposition 1 in [17], we have the following result.
From computations, we observe that For L near √ 2kL 1 and L > √ 2kL 1 , this system has the following 8 nontrivial solutions: For the system, we have the following theorems, whose proof is essentially the same as in [6]. As in [17], let ψ denote the flow associated with the system, and (ξ 1 (t), ξ 2 (t)) = ψ(t, θ −t ω)(ξ 0 1 , ξ 0 2 ) be a solution of the SDE with initial condition (ξ 0 1 , ξ 0 2 ). Using standard arguments in ODE given by [17], we have The other cases can be obtained in a similar fashion. In a similar manner, we have the following results.
For L near √ 2kL 1 and L > √ 2kL 1 , this system has the following 8 nontrivial solutions:S In a similar manner, we have the following results.
, then the origin is globally asymptotically stable.
(c) If L > √ 2kL 2 , then the origin is globally asymptotically stable.
Remark 3.7. For the deterministic Swift-Hohenberg equation, we can conclude that the equation undergoes a supercritical bifurcation at L = √ 2kL 1 and subcritical bifurcation at L = √ 2kL 2 . However, for the stochastic Swift-Hohenberg equation with multiplicative noise in Ito sense, we have the above theorem, that is to say, the noise may destroy the bifurcation, the size of parameter interval may be shortened.
From Theorems 3.2, 3.3, 3.5, and 3.6, we see the impact of noise in Stratonovich sense and Ito sense on the stochastic dynamics respectively, the multiplicative noise may destroy or induce bifurcations for different stochastic systems. 4. Analysis of the case (m, n) = (k, 0) and (0, k) In this section, we consider the attractor bifurcation near the points √ m 2 + n 2 L 1 and √ m 2 + n 2 L 2 in the case the intervals I m,n do not overlap, and K = m 2 +n 2 has only two solutions (m, n) = (k, 0) and (0, k); that is when the attractor bifurcates near the points kL 1 and kL 2 .
Notice that the space H 1 and H can be decomposed into where ξ ∈ H c 1 . Instead of (4.1), we consider the reduced equation where the notation ξ t emphasized the t-dependence of the variable ξ t , φ = P c u 0 , and u (2) s can be used to approximate the stochastic inertial manifold and is obtained from the following backward-forward systems (4.3)-(4.5).
We now consider approximation representation for stochastic parameterizing manifold as pullback limits of backward-forward systems. For a given t > 0 and T sufficiently large, 2 (τ, ω)e 0,k , (4.7) ξ(τ, ω) = ξ 1 (τ, ω)e k,0 + ξ 1 (τ, ω)e 0,k . (4.8) As in section 3, we obtain the reduced model, up to the leading order, which has a good description of the dynamics of (1.3). By computations, the leading order reduced equation is given below so for the reduced equation (4.3)-(4.5), we obtain the leading order equations When L is near kL 1 and L > kL 1 , this system has the following 8 nontrivial solutions: For the system, we have the following theorems.
Theorem 4.1. The reduced system (4.9)-(4.10) undergoes a stochastic supercritical bifurcation at L = kL 1 in the pullback sense. More precisely, for L near kL 1 , (a) If L < kL 1 , the origin is globally asymptotically stable. Theorem 4.2. The reduced system (4.9)-(4.10) undergoes a stochastic subcritical bifurcation at L = kL 2 in the pullback sense. More precisely, for L near kL 2 , (a) If L < kL 2 , the random compact set A L (ω) is a random pullback attractor. (b) If L > kL 2 , the origin is globally asymptotically stable. Now we consider the stochastic Swift-Hohenberg equation with multiplicative noise in Ito sense. As the case for the stochastic Swift-Hohenberg equation with multiplicative noise in Stratonovich sense, we can get the reduced equation, up to the leading order, When L is near kL 1 and L > kL 1 , this system has the following 8 nontrivial solutions: We have the following results.  (a) If L < kL 2 , P ( kπ L ) > σ 2 2 , then the random compact set A L (ω) is a random pullback attractor. (b) If L < kL 2 , and P ( kπ L ) < σ 2 2 , then the origin is globally asymptotically stable.
(c) If L > kL 2 , then the origin is globally asymptotically stable.

Remark 4.5.
For the deterministic Swift-Hohenberg equation, we can conclude that the equation undergoes a supercritical bifurcation at L = kL 1 and subcritical bifurcation at L = kL 2 . However, for the stochastic Swift-Hohenberg equation with multiplicative noise in Ito sense, we have the above theorem; that is to say, the noise may destroy the bifurcation, the size of parameter interval may be shortened. We may expect that the noise may induce bifurcation for some systems. 5. Analysis of the case (m, ±n) and (n, ±m) In this section, we consider the attractor bifurcation near the points √ m 2 + n 2 L 1 and √ m 2 + n 2 L 2 in the case the intervals I m,n do not overlap, and for fixed K. K = m 2 + n 2 has only four solutions (m, ±n) and (n, ±m) in Z, e.g. 5 = m 2 + n 2 has four solutions (1, ±2) and (2, ±1) in Z. Here m = n, mn = 0, as these two cases have been discussed in the previous two sections.
In this case, the space H 1 and H can be decomposed into where v c = P c u, and v s = P s u is the unresolved variable, and P c , P s are respectively canonical projections from H to H c 3 and H s . To obtain a closed form of the above equation, the unresolved variables u s is parameterized in terms of the resolved variables v c through a random continuous function h α (ξ, ω) : H c 3 × Ω → H s . The PM-based reduced equation for the resolved modes is dξ = (L c α ξ + P c G(ξ + h α (ξ, θ t ω)))dt + σξ • dW t , where ξ ∈ H c 3 . Using an approximation of h α (ξ, ω) via the pullback characterization as in section 3, we instead consider the reduced equation where T is sufficiently large, ϕ = P c u 0 , v (4) s is used to approximate the stochastic inertial manifold and is obtained from the following backward-forward systems (5.2)-(5.6).
(b) For L > √ m 2 + n 2 L 2 , the origin is globally asymptotically stable.
For the stochastic Swift-Hohenberg equation with multiplicative noise in Ito sense, we have similar results to the previous cases; here we do not state them.
By computations we obtain the reduced model on the PM, up to the leading order, From the above, we obtain the approximation representation of manifold and the corresponding reduced systems for stochastic Swift-Hohenberg equation when L 2 and √ 2L 1 are close together. The performances achieved by the above reduced system can approximate dynamics on the H c 4 modes in modeling of the pathwise SPDE (1.3). The dynamical behavior of the above reduced system is not easily to analyze because of its complex structure. In fact, it is possible to achieve good modeling performances of solution from these results.