Electronic Journal of Differential Equations, Vol. 2023 (2023), No. 20, pp. 1–22.

ISSN: 1072-6691. URL: https://ejde.math.txstate.edu or https://ejde.math.unt.edu

STOCHASTIC ATTRACTOR BIFURCATION FOR THE

TWO-DIMENSIONAL SWIFT-HOHENBERG EQUATION

WITH MULTIPLICATIVE NOISE

QINGKUN XIAO, HONGJUN GAO

Abstract. This article concerns the dynamical transitions of the stochastic
Swift-Hohenberg equation with multiplicative noise on a two-dimensional do-

main (−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 L2 and

√
2L1

are close together.

1. 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
= αu− (1 +

∂2

∂x2
)2u− u3. (1.1)

This equation plays an important role in the study of various phenomena in pat-
tern 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 bound-
ary condition are investigated, in which the techniques are based on the results
given in [20, 21].

2020 Mathematics Subject Classification. 35B40, 35B41, 37H20, 37L55.
Key words and phrases. Swift-Hohenberg equation; stochastic bifurcation;

dynamical transition; parameterizing manifold.
©2023. This work is licensed under a CC BY 4.0 license.

Submitted December 30, 2022. Published February 27, 2023.

1



2 Q. XIAO, H. GAO EJDE-2023/20

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

∂x2
)2u− u3, for 0 < x < L, t > 0,

u = 0,
∂2u

∂x2
= 0, at x = 0, L,

u(x, 0) = u0(x), for 0 < x < L,

(1.2)

where the initial function u0 is a smooth function, α and σ are positive num-
bers. 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 con-
dition. 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 sto-
chastic 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 = (Lαu+G(u))dt+ σu ◦ dWt

has an extensive literature, see [5, 6, 7, 11], and the references therein. Here Lα
is a linear operator parameterized by a parameter α ∈ R, G(u) represents the
nonlinear terms, Wt 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 man-
ifold 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 approx-
imation 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.



EJDE-2023/20 STOCHASTIC ATTRACTOR BIFURCATION OF 2D S-H EQUATIONS 3

In this paper, we consider the stochastic attractor bifurcation of two dimen-
sional Swift-Hohenberg equation (SHE) on the domain Q = (−L,L)×(−L,L) with
multiplicative noise in Stratonovich sense

du = (αu− (1 + ∆)2u− u3)dt+ σu ◦ dWt, (x, y) ∈ Q, t > 0, (1.3)

with boundary conditions

u(x, y, t) = u(x, y + 2L, t) = u(x+ 2L, y, t), (x, y) ∈ Q, t ≥ 0, (1.4)

u(x, y, t) = −u(−x,−y, t), (x, y) ∈ Q, t ≥ 0, (1.5)

and the SHE with multiplicative noise in Ito sense

du = (αu− (1 + ∆)2u− u3)dt+ σudWt, (x, y) ∈ Q, t > 0,

u(x, y, t) = u(x, y + 2L, t) = u(x+ 2L, y, t), (x, y) ∈ Q, t ≥ 0,

u(x, y, t) = −u(−x,−y, t), (x, y) ∈ Q, t ≥ 0,

(1.6)

where the initial function u0 is a smooth function, α and σ are positive numbers,
in particular, Wt 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 bifur-
cation 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 pa-
rameterizing 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 im-
prove the partial knowledge of the full SPDE¡¯s solution in mean square error
when compared with its projection onto the resolved modes. Approximate pa-
rameterizing 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 deter-
ministic 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

√
m2 + n2L1 and

√
m2 + n2L2 under three

cases. The approximation representation of manifold and the corresponding re-
duced systems for stochastic Swift-Hohenberg equation when L2 and

√
2L1 are

close together are obtained in section 6.



4 Q. XIAO, H. GAO EJDE-2023/20

2. 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.

H = {u ∈ L2(Q) : u satisfies (1.4)-(1.5) and

∫
Q

u dx dy = 0},

H1 = H4(Q) ∩H,

and with the inner product

〈u, v〉 =
1

4L2

∫ L

0

∫ L

0

uv dx dy.

Notice that H and H1 are Hilbert spaces, and H1 ↪→ H is a dense and compact
inclusion. We consider the nonlinear evolution equations

du

dt
= Lαu+G(u), (2.1)

u(0) = u0, (2.2)

where u : [0,∞)→ H is the unknown function, α ∈ R is the system parameter, and
Lα : H1 ↪→ H are parameterized linear completely continuous fields continuously
depending on α ∈ R, which satisfy

Lα = A+Bα is a sectorial operator, (2.3)

where A : H1 ↪→ H is a linear homeomorphism, Bα : H1 ↪→ H are parameter-
ized linear compact operators. Furthermore, the nonlinear term G(u) = −u3 is a
bounded operator such that

G(u, α) = o(‖u‖H1
), ∀α ∈ R.

Hence the stochastic Swift-Hohenberg equation can be written as

du = (Lαu+G(u))dt+ σu ◦ dWt. (2.4)

Let

Z = {(m,n) : m ∈ N, n ∈ Z} ∪ {(0, n) : n ∈ N}. (2.5)

Then the eigenvalues of the following eigenvalue problem on H1,

Lαϕ = λϕ,

are

λm,n = P (

√
m2 + n2π

L
), where P (ξ) = α− (ξ2 − 1)2,

with the corresponding eigenfunctions

ek,l =
√

2 sin
π(kx+ ly)

L
. (2.6)

for all (k, l) which satisfies k2 + l2 = m2 +n2, and (k, l) 6= (0, 0), (k, l) ∈ Z. We can
see that {em,n, (m,n) ∈ Z} forms a basis of H, and 〈em,n, em,n〉 = 1.

Throughout this article, we consider α positive and small enough, and we let
α ∈ (0, 1), then P (ξ) has two positive zeros:

ξ− = (1−
√
α)1/2, ξ+ = (1 +

√
α)1/2.



EJDE-2023/20 STOCHASTIC ATTRACTOR BIFURCATION OF 2D S-H EQUATIONS 5

This implies that λm,n > 0, when L ∈ (
√
m2 + n2L1,

√
m2 + n2L2), where L1 =

π/ξ+, L2 = π/ξ−. If L ∈ (0, L1), then
√
m2 + n2π

L
>

√
m2 + n2π

L1
≥ π

L1

for all (m,n) ∈ Z. This implies that

λm,n = P (

√
m2 + n2π

L
) > P (ξ+) = 0, for all (m,n) ∈ Z.

Thus, if L < L1, the trivial solution is asymptotically stable.
When L is larger than L1, we denote

Im,n =
{
L > 0 : P

(√m2 + n2π

L

)
≥ 0
}
,

then Im,n = [
√
m2 + n2L1,

√
m2 + n2L2] for every (m,n) ∈ Z.

For α small,

L1(α) ∼ π − π

2

√
α, L2(α) ∼ π +

π

2

√
α, as α→ 0+. (2.7)

If α is small enough, the intervals Im,n will not overlap, but be separated by intervals
in which λm,n > 0.

Suppose there is a gap between the intervals Im,n. Then we define

Πm,n = (0,
√
m2 + n2L2)− ∪1≤i≤m,1≤j≤nIi,j .

From the Fourier series of the solution, as discussed in our previous work [29, 30],
we have the following theorem.

Theorem 2.1. Suppose there is a gap between the intervals Im,n, and let u(t) be
the solution of Problem (1.2). Then for all L ∈ Πm,n, we have u(t)→ 0 as t→∞.

Remark 2.2. For the deterministic case of Problem (1.3), we can conclude that the

equation undergoes a supercritical bifurcation at L =
√
m2 + n2L1 and a subcritical

bifurcation at L =
√
m2 + n2L2.

Note that for fixed K, the number of solutions (m,n) to K = m2 + n2 may be
lager than two, such as m2 +n2 = 25 has 6 different solutions (m,n) = (5, 0), (0, 5),
(3, 4), (3,−4), (4, 3), and (4,−3). Here we consider only m ≥ 0.

In the following two sections, we consider stochastic attractor bifurcation of the
Swift-Hohenberg equation near the points

√
m2 + n2L1 and

√
m2 + n2L2. This

is done in two cases: first we assume that K = m2 + n2 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 = m2 + n2 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
√
m2 + n2L1

and
√
m2 + n2L2, in the case the intervals Im,n do not overlap, and K = m2+n2 has

only two solutions (m,n) = (k, k) and (k,−k). That is, we consider the attractor

bifurcation near the points
√

2kL1 and
√

2kL2.
In this case, the space H1 and H can be decomposed into

H1 = Hc
1 ⊕Hs

1 , H = Hc
1 ⊕ H̃s,



6 Q. XIAO, H. GAO EJDE-2023/20

where

Hc
1 = span{ek,k, ek,−k},

and H̃s is the closure of Hs
1 in H.

We will present a stochastic reduction procedure based on parameterizing man-
ifolds (PM) associated with (1.3). A stochastic parameterizing manifolds [6, 7], as

the graph of a random continuous function hα(ξ, ω) from Hc
1 to H̃s, and for each

realization ω, the function is defined for ξ ∈ Hc
1 .

The PM-based reduced equation for the resolved modes is

duc = (Lcαuc + PcG(uc + us))dt+ σuc ◦ dWt, (3.1)

where ξ ∈ Hc
1 . However, it’s more involved to give an explicit expression of hα(ξ, ω),

the key idea ([6, 7])is to provide an approximation of hα(ξ, ω) via the pullback
characterization

hα(ξ, ω) := lim
T→+∞

u(2)s (ξ)(T, θ−Tω; 0)).

Indeed it is too cumbersome to use the above pullback characterization to ap-
proximate the vector field PcG(ξ+ hα(ξ, θtω)) as ξ varies in Hc

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

dξt = (Lcαξt + PcG(ξt + u(2)s [ξ(t, ω)](t+ T, θ−Tω; 0)))dt+ σξt ◦ dWt,

ξ(0, ω) = φ, t > 0,
(3.2)

where the notation ξt emphasized the t-dependence of the variable ξt, φ = Pcu0,

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.

du(1)c = Lcαu
(1)
c dτ + σu(1)c ◦ dWτ , τ ∈ [t− T, t], (3.3)

du(2)c = (Lcαu
(2)
c + PcG(u(1)c )dτ + σu(2)c ◦ dWτ , τ ∈ [t− T, t], (3.4)

du(2)s = (Lsαu
(2)
s + PsG(u(2)c (τ − T, ω)))dτ + σu(2)s ◦ dWτ−T , τ ∈ [t, t+ T ], (3.5)

with

u(1)c (τ, ω)
∣∣
τ=t

= ξ(t, ω),

u(2)c (τ, ω)
∣∣
τ=t

= ξ(t, ω),

u(2)s (τ, θ−Tω)
∣∣
τ=t

= 0,

where u
(2)
s can be used to approximate the stochastic inertial manifold, Lcα = PcLα,

Lsα = PsLα. In the systems above, the initial value of u
(1)
c and u

(2)
c are prescribed

in fiber θtω, and the the initial value of u
(2)
s is prescribed in fiber θt−Tω. 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 Hc

1 force

the evolution equation of u
(2)
s to depend naturally on ξ but not reciprocally. Here,



EJDE-2023/20 STOCHASTIC ATTRACTOR BIFURCATION OF 2D S-H EQUATIONS 7

θt is an element of a metric dynamical system, and ω is a given realization, θt−Tω
is called a fiber, see Arnold [2].

Since u
(1)
c ∈ Hc

1 , we write

u(1)c (τ, ω) = x
(1)
1 (τ, ω)ek,k + x

(1)
2 (τ, ω)ek,−k, (3.6)

ξ(τ, ω) = ξ1(τ, ω)ek,k + ξ2(τ, ω)ek,−k. (3.7)

Then by projecting equation (3.2) onto Hc
1 , we obtain

dx
(1)
1 = P (

√
2kπ

L
)x

(1)
1 dτ + σx

(1)
1 ◦ dWτ , τ ∈ [t− T, t], (3.8)

dx
(1)
2 = P (

√
2kπ

L
)x

(1)
2 dτ + σx

(1)
2 ◦ dWτ , τ ∈ [t− T, t], (3.9)

with

x
(1)
1 (τ, ω)

∣∣
τ=t

= ξ1(t, ω), (3.10)

x
(1)
2 (τ, ω)

∣∣
τ=t

= ξ2(t, ω). (3.11)

After simple calculation, we can obtain

〈PcG(u(1)c ), ek,k〉 = −3

2
(x

(1)
1 )3 − 3x

(1)
1 (x

(1)
2 )2, (3.12)

〈PcG(u(1)c ), ek,−k〉 = −3

2
(x

(1)
2 )3 − 3(x

(1)
1 )2x

(1)
2 . (3.13)

We write

u(2)c (τ, ω) = x
(2)
1 (τ, ω)ek,k + x

(2)
2 (τ, ω)ek,−k.

Then we have

dx
(2)
1 = (P (

√
2kπ

L
)x

(2)
1 −

3

2
(x

(1)
1 )3 − 3x

(1)
1 (x

(1)
2 )2)dτ

+ σx
(2)
1 ◦ dWτ , τ ∈ [t− T, t],

(3.14)

dx
(2)
2 = (P (

√
2kπ

L
)x

(2)
2 −

3

2
(x

(1)
2 )3 − 3(x

(1)
1 )2x

(1)
2 )dτ

+ σx
(2)
2 ◦ dWτ , τ ∈ [t− T, t],

(3.15)

with

x
(2)
1 (s, ω)

∣∣
s=t

= ξ1(t, ω), (3.16)

x
(2)
2 (s, ω)

∣∣
s=t

= ξ2(t, ω). (3.17)

Notice that

G(u(2)c ) = −2
√

2[(x
(2)
1 )3(

3

4
sin

kπ(x+ y)

L
− 1

4
sin

3kπ(x+ y)

L
)

+
3

2
(x

(2)
1 )2x

(2)
2 (sin

kπ(x− y)

L
− 1

2
sin

kπ(3x+ y)

L

+
1

2
sin

kπ(x+ 3y)

L
) +

3

2
x
(2)
1 (x

(2)
2 )2(sin

kπ(x+ y)

L

+
1

2
sin

kπ(x− 3y)

L
− 1

2
sin

kπ(3x− y)

L
)

+ (x
(2)
2 )3(

3

4
sin

kπ(x− y)

L
− 1

4
sin

3kπ(x− y)

L
)].

(3.18)



8 Q. XIAO, H. GAO EJDE-2023/20

Then we have

〈PsG(u(2)c ), e3k,3k〉 =
1

2
(x

(2)
1 )3, (3.19)

〈PsG(u(2)c ), e3k,k〉 =
3

2
(x

(2)
1 )2x

(2)
2 , (3.20)

〈PsG(u(2)c ), ek,3k〉 = −3

2
(x

(2)
1 )2x

(2)
2 , (3.21)

〈PsG(u(2)c ), e3k,−k〉 =
3

2
x
(2)
1 (x

(2)
2 )2, (3.22)

〈PsG(u(2)c ), ek,−3k〉 = −3

2
x
(2)
1 (x

(2)
2 )2, (3.23)

〈PsG(u(2)c ), e3k,−3k〉 =
1

2
(x

(2)
2 )3, (3.24)

〈PsG(u(2)c ), em,n〉 = 0, (m,n) ∈ Z̃, (3.25)

where Z̃ = Z−{(k, k), (k,−k), (3k, 3k), (3k, k), (k, 3k), (3k,−k), (k,−3k), (3k,−3k)}.
Since u

(2)
s ∈ H̃s, we set

u(2)s = y1e3k,3k + y2e3k,k + y3ek,3k + y4e3k,−k + y5ek,−3k + y6e3k,−3k

+
∑

(m,n)∈Z̃

ym,nem,n. (3.26)

By projecting (3.5) onto em,n ((m,n) ∈ {(3k, 3k), (3k, k), (k, 3k), (3k,−k),
(k,−3k), (3k,−3k)}), and with (3.19)-(3.25), for τ ∈ [t, t+ T ], we obtain

dy1 = (P (
3
√

2kπ

L
)y1 +

1

2
(x

(2)
1 )3)dτ + σy1 ◦ dWτ−T , (3.27)

dy2 = (P (

√
10kπ

L
)y2 +

1

2
(x

(2)
1 )3)dτ + σy2 ◦ dWτ−T , (3.28)

dy3 = (P (

√
10kπ

L
)y3 +

1

2
(x

(2)
1 )3)dτ + σy3 ◦ dWτ−T , (3.29)

dy4 = (P (

√
10kπ

L
)y4 +

1

2
(x

(2)
1 )3)dτ + σy4 ◦ dWτ−T , (3.30)

dy5 = (P (

√
10kπ

L
)y5 +

1

2
(x

(2)
1 )3)dτ + σy5 ◦ dWτ−T , (3.31)

dy6 = (P (
3
√

2kπ

L
)y6 +

1

2
(x

(2)
1 )3)dτ + σy6 ◦ dWτ−T , (3.32)

dym,n = P (

√
m2 + n2π

L
)ym,ndτ + σym,n ◦ dWτ−T , (3.33)

where (m,n) ∈ {(3k, 3k), (3k, k), (k, 3k), (3k,−k), (k,−3k),(3k,−3k)}, and

u(2)m,n(s, θ−Tω)|s=t = 0, (m,n) ∈ Z̃. (3.34)

It is easy to show that

ym,n = 0, (m,n) ∈ Z̃. (3.35)



EJDE-2023/20 STOCHASTIC ATTRACTOR BIFURCATION OF 2D S-H EQUATIONS 9

Hence, from (3.26) and (3.35), we have

u(2)s [ξt](s, θ−Tω; 0)

= y1[ξt](s, θ−Tω; 0)e3k,3k + y2[ξt](s, θ−Tω; 0)e3k,k + y3[ξt](s, θ−Tω; 0)ek,3k

+ y4[ξt](s, θ−Tω; 0)e3k,−k + y5[ξt](s, θ−Tω; 0)ek,−3k

+ y6[ξt](s, θ−Tω; 0)e3k,−3k.

(3.36)

From the form of u
(2)
s , we now can consider equation (3.2), by writing

ξ(t, ω) = ξ1(t, ω)ek,k + ξ2(t, ω)ek,−k.

Notice that

G(ξ + u(2)s ) = −(ξ1ek,k + ξ2ek,−k + y1e3k,3k + y2e3k,k + y3ek,3k

+ y4e3k,−k + y5ek,−3k + y6e3k,−3k)3.
(3.37)

Hence by projecting (3.2) onto ek,k and ek,−k respectively, and with (3.37), we
have

dξ1

= (P (

√
2kπ

L
)ξ1 −

3

2
ξ31 − 3ξ1ξ

2
2 − 3ξ1y

2
1 − 3ξ1y

2
2 − 3ξ1y

2
3 − 3ξ1y

2
4 − 3ξ1y

2
5

− 3ξ1y
2
6 +

3

2
ξ21y1 +

3

2
ξ22y4 −

3

2
ξ22y5 + 3ξ1ξ2y2 − 3ξ1ξ2y3 + 3ξ1y4y5

− 3ξ2y1y2 + 3ξ2y1y3 − 3ξ2y2y4 − 3ξ2y3y5 − 3ξ2y4y6 + 3ξ2y5y6

− 3y1y2y3 − 3y1y4y5 − 3y2y3y4 + 3y2y3y5 − 3y2y5y6 − 3y3y4y6)dt

+ σξ1 ◦ dWt,

(3.38)

dξ2 = (P (

√
2kπ

L
)ξ2 −

3

2
ξ32 − 3ξ21ξ2 − 3ξ2y

2
1 − 3ξ2y

2
2 − 3ξ2y

2
3 − 3ξ2y

2
4

− 3ξ2y
2
5 − 3ξ2y

2
6 +

3

2
ξ21y2 −

3

2
ξ21y3 +

3

2
ξ22y6 + 3ξ1ξ2y4 − 3ξ1ξ2y5

− 3ξ1y1y2 + 3ξ1y1y3 − 3ξ1y2y4 − 3ξ1y3y5

− 3ξ1y4y6 + 3ξ1y5y6 + 3ξ2y2y3 − 3y1y2y5 − 3y1y3y4

− 3y2y3y6 − 3y2y4y5 + 3y3y4y5 − 3y4y5y6)dt+ σξ2 ◦ dWt.

(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.

Proposition 3.1. For L near
√

2kL1 and L >
√

2kL1 , there exists a random
closed ball BL(ω) such that for every bounded set B ⊂ H and almost every ω, there
exists a time TB(ω) > 0 so that

φ(t, θ−tω)B ⊂ BL(ω), ∀t > TB(ω).

It is known from Chekroun et al [6] that the center manifold reduction for the
system (1.3) holds in deterministic neighborhood of the origin. By considering only
those ω such that BL(ω) is small enough so that the center manifold reduction
holds, we expect that the reduced system (3.38)-(3.39) has a good description of



10 Q. XIAO, H. GAO EJDE-2023/20

the dynamics of (1.3), then we turn to the study of the reduced systems (3.38)-

(3.39). In the study of dynamic transition near
√

2kL1, higher-order terms can be
dropped, and we can consider the reduced model, up to the leading order.

Now we give the solutions of some above-mentioned equations. From (3.8)-(3.11),
we have

x
(1)
1 (τ, ω) = ξ1(t, ω)eP (

√
2kπ
L )(τ−t)+σ(Wτ (ω)−Wt(ω)), τ ∈ [t− T, t],

x
(1)
2 (τ, ω) = ξ2(t, ω)eP (

√
2kπ
L )(τ−t)+σ(Wτ (ω)−Wt(ω)), τ ∈ [t− T, t].

From (3.14)-(3.17), we have

x
(2)
1 (τ, ω)

= x
(1)
1 (τ, ω)−

∫ t

τ

eP (
√

2kπ
L )(τ−ρ)+σ(Wτ (ω)−Wρ(ω))[

3

2
(x

(1)
1 )3 + 3x

(1)
1 (x

(1)
2 )2]dρ,

x
(2)
2 (τ, ω)

= x
(1)
2 (τ, ω)−

∫ t

τ

eP (
√

2kπ
L )(τ−ρ)+σ(Wτ (ω)−Wρ(ω))[3(x

(1)
1 )2x

(1)
2 +

3

2
(x

(1)
2 )3]dρ.

From (3.27)-(3.34), we have

y1(τ, θ−Tω) =
1

2

∫ τ

t

eP ( 3
√

2kπ
L )(τ−ρ)+σ(Wτ−T (ω)−Wρ−T (ω))(x

(2)
1 (ρ− T, ω))3dρ,

y2(τ, θ−Tω)

=
3

2

∫ τ

t

eP (
√

10kπ
L )(τ−ρ)+σ(Wτ−T (ω)−Wρ−T (ω))(x

(2)
1 (ρ− T, ω))2x

(2)
2 (ρ− T, ω)dρ,

y3(τ, θ−Tω)

= −3

2

∫ τ

t

eP (
√

10kπ
L )(τ−ρ)+σ(Wτ−T (ω)−Wρ−T (ω))(x

(2)
1 (ρ− T, ω))2x

(2)
2 (ρ− T, ω)dρ,

y4(τ, θ−Tω)

=
3

2

∫ τ

t

eP (
√

10kπ
L )(τ−ρ)+σ(Wτ−T (ω)−Wρ−T (ω))x

(2)
1 (ρ− T, ω)(x

(2)
2 (ρ− T, ω))2dρ,

y5(τ, θ−Tω)

= −3

2

∫ τ

t

eP (
√

10kπ
L )(τ−ρ)+σ(Wτ−T (ω)−Wρ−T (ω))x

(2)
1 (ρ− T, ω)(x

(2)
2 (ρ− T, ω))2dρ,

y6(τ, θ−Tω) =
1

2

∫ τ

t

eP ( 3
√

2kπ
L )(τ−ρ)+σ(Wτ−T (ω)−Wρ−T (ω))(x

(2)
2 (ρ− T, ω))3dρ.

From computations, we observe that

u(2)s [ξt](s, θ−Tω) = O(‖ (ξ1, ξ2) ‖3), (3.40)

so for the reduced equation (3.38)-(3.39), we obtain the leading order equations

dξ1 = (P (

√
2kπ

L
)ξ1 −

3

2
ξ31 − 3ξ1ξ

2
2)dt+ σξ1 ◦ dWt, (3.41)

dξ2 = (P (

√
2kπ

L
)ξ2 −

3

2
ξ32 − 3ξ21ξ2)dt+ σξ2 ◦ dWt. (3.42)



EJDE-2023/20 STOCHASTIC ATTRACTOR BIFURCATION OF 2D S-H EQUATIONS 11

For L near
√

2kL1 and L >
√

2kL1 , this system has the following 8 nontrivial
solutions:

S1,2
L (θtω) = (±

√
1

3
aL(θtω), 0),

S3,4
L (θtω) = (0,±

√
1

3
aL(θtω)),

S5
L(θtω) = (

1

3
aL(θtω),

1

3
aL(θtω)),

S6
L(θtω) = (−1

3
aL(θtω),−1

3
aL(θtω)),

S7
L(θtω) = (

1

3
aL(θtω),−1

3
aL(θtω)),

S8
L(θtω) = (−1

3
aL(θtω),

1

3
aL(θtω)),

where

aL(ω) = (

∫ 0

−∞
e2P (

√
2kπ
L )τ+2σWτ (ω)dτ)−1/2.

For the system, we have the following theorems, whose proof is essentially the
same as in [6].

Theorem 3.2. The reduced system (3.41)-(3.42) undergoes a stochastic supercrit-

ical bifurcation at L =
√

2kL1 in the pullback sense. More precisely, when L near√
2kL1,

(a) For L <
√

2kL1, the origin is globally asymptotically stable.

(b) For L >
√

2kL1, the random compact set

AL(ω) = {SiL(ω), i = 1, 2, . . . , 8}

is a random pullback attractor.

As in [17], let ψ denote the flow associated with the system, and (ξ1(t), ξ2(t)) =
ψ(t, θ−tω)(ξ01 , ξ

0
2) be a solution of the SDE with initial condition (ξ01 , ξ

0
2). Using

standard arguments in ODE given by [17], we have

lim
t→+∞

ψ(t, θ−tω)(ξ01 , ξ
0
2) =

{
(±
√

1
3aL(ω), 0), if ξ01 > ξ02 ,

( 1
3aL(ω), 13aL(ω), if ξ01 = ξ02 .

The other cases can be obtained in a similar fashion. In a similar manner, we have
the following results.

Theorem 3.3. The reduced system (3.41)-(3.42) undergoes a stochastic subcritical

bifurcation at L =
√

2kL2 in the pullback sense. More precisely, when L near√
2kL2,

(a) For L <
√

2kL2, the random compact set AL(ω) is a random pullback
attractor.

(b) For L >
√

2kL2, the origin is globally asymptotically stable.

Remark 3.4. We can conclude that if α is small enough, the equation under-
goes stochastic supercritical bifurcation at L =

√
2kL1, and stochastic subcritical

bifurcation at L =
√

2kL2.



12 Q. XIAO, H. GAO EJDE-2023/20

Now we consider the stochastic Swift-Hohenberg equation with multiplicative
noise in Ito sense. As for the stochastic Swift-Hohenberg equation with multiplica-
tive noise in Stratonovich sense, we can obtain the reduced equation, up to the
leading order,

dξ1 = (P (

√
2kπ

L
)ξ1 −

3

2
ξ31 − 3ξ1ξ

2
2)dt+ σξ1dWt, (3.43)

dξ2 = (P (

√
2kπ

L
)ξ2 −

3

2
ξ32 − 3ξ21ξ2)dt+ σξ2dWt. (3.44)

For L near
√

2kL1 and L >
√

2kL1, this system has the following 8 nontrivial
solutions:

S̃1,2
L (θtω) = (±

√
1

3
ãL(θtω), 0),

S̃3,4
L (θtω) = (0,±

√
1

3
ãL(θtω)),

S̃5
L(θtω) = (

1

3
ãL(θtω),

1

3
ãL(θtω)),

S̃6
L(θtω) = (−1

3
ãL(θtω),−1

3
ãL(θtω)),

S̃7
L(θtω) = (

1

3
ãL(θtω),−1

3
ãL(θtω)),

S̃8
L(θtω) = (−1

3
ãL(θtω),

1

3
ãL(θtω)),

where

ãL(ω) =
(∫ 0

−∞
e2(P (

√
2kπ
L )−σ22 )τ+2σWτ (ω)dτ

)−1/2
. (3.45)

Theorem 3.5. When L is near
√

2kL1, for the reduced system (3.43)-(3.44), we
have:

(a) If L <
√

2kL1, the origin is globally asymptotically stable.

(b) If L >
√

2kL1, and P (
√
2kπ
L ) < σ2

2 , the origin is globally asymptotically
stable.

(c) If L >
√

2kL1, and P (
√
2kπ
L ) > σ2

2 , then the random compact set

ÃL(ω) = {S̃iL(ω), i = 1, 2, . . . , 8}
is a random pullback attractor.

In a similar manner, we have the following results.

Theorem 3.6. For the reduced system (3.43)-(3.44), we have:

(a) If L <
√

2kL2, P (
√
2kπ
L ) > σ2

2 , then the random compact set ÃL(ω) is a
random pullback attractor.

(b) If L <
√

2kL2, and P (
√
2kπ
L ) < σ2

2 , then the origin is globally asymptotically
stable.

(c) If L >
√

2kL2, 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 =

√
2kL1 and sub-

critical bifurcation at L =
√

2kL2. However, for the stochastic Swift-Hohenberg



EJDE-2023/20 STOCHASTIC ATTRACTOR BIFURCATION OF 2D S-H EQUATIONS 13

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
√
m2 + n2L1

and
√
m2 + n2L2 in the case the intervals Im,n do not overlap, and K = m2+n2 has

only two solutions (m,n) = (k, 0) and (0, k); that is when the attractor bifurcates
near the points kL1 and kL2.

Notice that the space H1 and H can be decomposed into

H1 = Hc
2 ⊕Hs

2 , H = Hc
2 ⊕ H̃s,

where Hc
2 = span{ek,0, e0,k} and H̃s is the closure of Hs

2 in H.
We will present a stochastic reduction procedure based on parameterizing mani-

folds (PM) associated with (1.3). A stochastic parameterizing manifolds ([6, 7]), as

the graph of a random continuous function h̃α(ξ, ω) from Hc
1 to H̃s, and for each

realization ω, the function is defined for ξ ∈ Hc
1 .

Projecting equation (2.4) onto the subspace Hc
1 , we obtain

dũc = (L̃cαũc + P̃cG(ũc + ũs))dt+ σũc ◦ dWt,

where ũs = P̃su is the unresolved variable, and P̃s, P̃c are respectively canonical

projections from H to Hc
1 and H̃s. To obtain a closed form of the above equation,

the unresolved variables ũs is parameterized in terms of the resolved variables ũc
through a random continuous function h̃α(ξ, ω) : Hc

1 × Ω→ H̃s.
The PM-based reduced equation for the resolved modes is

dξ = (L̃cαξ + P̃cG(ξ + h̃α(ξ, θtω)))dt+ σξ ◦ dWt, (4.1)

where ξ ∈ Hc
1 .

Instead of (4.1), we consider the reduced equation

dξt = (L̃cαξt + P̃cG(ξt + ũ(2)s [ξ(t, ω)](t+ T, θ−Tω; 0)))dt+ σξt ◦ dWt,

ξ(0, ω) = φ, t > 0,
(4.2)

where the notation ξt emphasized the t-dependence of the variable ξt, φ = P̃cu0,

and ũ
(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,

dũ(1)c = L̃cαũ(1)c ds+ σũ(1)c ◦ dWτ , τ ∈ [t− T, t], (4.3)

dũ(2)c = (L̃cαũ(2)c + P̃cG(ũ(1)c (τ − T, ω)))dτ + σũ(2)c ◦ dWτ−T , τ ∈ [t, t+ T ], (4.4)

dũ(2)s = (L̃sαũ(2)s + P̃sG(ũ(2)c (τ − T, ω)))dτ + σũ(2)s ◦ dWτ−T , τ ∈ [t, t+ T ]. (4.5)



14 Q. XIAO, H. GAO EJDE-2023/20

with

ũ(1)c (τ, ω)
∣∣
τ=t

= ξ(t, ω),

ũ(2)c (τ, ω)
∣∣
τ=t

= ξ(t, ω),

ũ(2)s (τ, θ−Tω)
∣∣
τ=t

= 0,

where L̃cα = P̃cLα and L̃sα = P̃sLα.
Since ũ1c , ũ

2
c ∈ Hc

2 , we write

ũ(1)c (τ, ω) = x̃
(1)
1 (τ, ω)ek,0 + x̃

(1)
2 (τ, ω)e0,k, (4.6)

ũ(2)c (τ, ω) = x̃
(2)
1 (τ, ω)ek,0 + x̃

(2)
2 (τ, ω)e0,k, (4.7)

ξ(τ, ω) = ξ̃1(τ, ω)ek,0 + ξ̃1(τ, ω)e0,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

dξ̃1 = (P (
kπ

L
)ξ̃1 −

3

2
ξ̃31 − 3ξ̃1ξ̃

2
2)dt+ σξ̃1 ◦ dWt, (4.9)

dξ̃2 = (P (
kπ

L
)ξ̃2 −

3

2
ξ̃32 − 3ξ̃21 ξ̃2)dt+ σξ̃2 ◦ dWt. (4.10)

When L is near kL1 and L > kL1, this system has the following 8 nontrivial
solutions:

S̃1,2
L (θtω) = (±

√
1

3
ãL(θtω), 0),

S̃3,4
L (θtω) = (0,±

√
1

3
ãL(θtω)),

S̃5
L(θtω) = (

1

3
ãL(θtω),

1

3
ãL(θtω)),

S̃6
L(θtω) = (−1

3
ãL(θtω),−1

3
ãL(θtω)),

S̃7
L(θtω) = (

1

3
ãL(θtω),−1

3
ãL(θtω)),

S̃8
L(θtω) = (−1

3
ãL(θtω),

1

3
ãL(θtω)),

where

ãL(ω) =
(∫ 0

−∞
e2P ( kπL )τ+2σWτ (ω)dτ

)−1/2
.

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 = kL1 in the pullback sense. More precisely, for L near kL1,

(a) If L < kL1, the origin is globally asymptotically stable.
(b) If L > kL1, the random compact set

ÃL(ω) = {S̃iL(ω), i = 1, 2, . . . , 8}

is a random pullback attractor.



EJDE-2023/20 STOCHASTIC ATTRACTOR BIFURCATION OF 2D S-H EQUATIONS 15

Theorem 4.2. The reduced system (4.9)-(4.10) undergoes a stochastic subcritical
bifurcation at L = kL2 in the pullback sense. More precisely, for L near kL2,

(a) If L < kL2, the random compact set ÃL(ω) is a random pullback attractor.
(b) If L > kL2, 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,

dξ̃1 = (P (
kπ

L
)ξ̃1 −

3

2
ξ̃31 − 3ξ̃1ξ̃

2
2)dt+ σξ̃1dWt, (4.11)

dξ̃2 = (P (
kπ

L
)ξ̃2 −

3

2
ξ̃32 − 3ξ̃21 ξ̃2)dt+ σξ̃2dWt. (4.12)

When L is near kL1 and L > kL1, this system has the following 8 nontrivial
solutions:

S1,2
L (θtω) = (±

√
1

3
aL(θtω), 0),

S3,4
L (θtω) = (0,±

√
1

3
aL(θtω)),

S5
L(θtω) = (

1

3
aL(θtω),

1

3
aL(θtω)),

S6
L(θtω) = (−1

3
aL(θtω),−1

3
aL(θtω)),

S7
L(θtω) = (

1

3
aL(θtω),−1

3
aL(θtω)),

S8
L(θtω) = (−1

3
aL(θtω),

1

3
aL(θtω)).

where

aL(ω) =
(∫ 0

−∞
e2(P ( kπL )−σ22 )τ+2σWτ (ω)dτ

)−1/2
.

We have the following results.

Theorem 4.3. For L near kL1, then for the reduced system (4.11)-(4.12), we have:

(a) If L < kL1, the origin is globally asymptotically stable.

(b) If L > kL1, and P (kπL ) < σ2

2 , the origin is globally asymptotically stable.

(c) If L > kL1, and P (kπL ) > σ2

2 , then the random compact set

AL(ω) = {Si
L(ω), i = 1, 2, . . . , 8.}.

is a random pullback attractor.

Theorem 4.4. For L near kL2, then for the reduced system (4.11)-(4.12), we have:

(a) If L < kL2, P (kπL ) > σ2

2 , then the random compact set AL(ω) is a random
pullback attractor.

(b) If L < kL2, and P (kπL ) < σ2

2 , then the origin is globally asymptotically
stable.

(c) If L > kL2, then the origin is globally asymptotically stable.



16 Q. XIAO, H. GAO EJDE-2023/20

Remark 4.5. For the deterministic Swift-Hohenberg equation, we can conclude
that the equation undergoes a supercritical bifurcation at L = kL1 and subcritical
bifurcation at L = kL2. 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
√
m2 + n2L1

and
√
m2 + n2L2 in the case the intervals Im,n do not overlap, and for fixed K.

K = m2 + n2 has only four solutions (m,±n) and (n,±m) in Z, e.g. 5 = m2 + n2

has four solutions (1,±2) and (2,±1) in Z. Here m 6= n, mn 6= 0, as these two
cases have been discussed in the previous two sections.

In this case, the space H1 and H can be decomposed into

H1 = Hc
3 ⊕Hs

3 , H = Hc
3 ⊕Hs,

where

Hc
3 = span{em,n, em,−n, en,m, en,−m}.

Projecting equation (2.4) onto the subspace Hc
3 , we obtain

dvc = (Lcαvc + PcG(vc + us))dt+ σuc ◦ dWt,

where vc = Pcu, and vs = Psu is the unresolved variable, and Pc, Ps are respectively
canonical projections from H to Hc

3 and Hs. To obtain a closed form of the above
equation, the unresolved variables us is parameterized in terms of the resolved
variables vc through a random continuous function hα(ξ, ω) : Hc

3 × Ω→ Hs.
The PM-based reduced equation for the resolved modes is

dξ = (Lcαξ + PcG(ξ + hα(ξ, θtω)))dt+ σξ ◦ dWt,

where ξ ∈ Hc
3 . Using an approximation of hα(ξ, ω) via the pullback characterization

as in section 3, we instead consider the reduced equation

dξt = (Lcαξt + PcG(ξt + v(4)s [ξ(t, ω)](t+ T, θ−Tω; 0)))dt+ σξt ◦ dWt,

ξ(0, ω) = ϕ, t > 0,
(5.1)

where T is sufficiently large, ϕ = Pcu0, v
(4)
s is used to approximate the stochas-

tic inertial manifold and is obtained from the following backward-forward systems
(5.2)-(5.6).

For a given t > 0 and T sufficiently large, let us consider the 4-layer auxiliary
backward-forward system

dv(1)c = Lcαv(1)c dτ + σv(1)c ◦ dWτ , τ ∈ [t− T, t], (5.2)

dv(2)c = (Lcαv(2)c + PcG(v(1)c )dτ + σv(2)c ◦ dWτ , τ ∈ [t− T, t], (5.3)

dv(3)c = (Lcαv(4)c + PcG(v(2)c )dτ + σv(3)c ◦ dWτ , τ ∈ [t− T, t], (5.4)

dv(4)c = (Lcαv(4)c + PcG(v(3)c )dτ + σv(4)c ◦ dWτ , τ ∈ [t− T, t], (5.5)

dv(4)s = (Lsαv(4)s + PcG(v(4)c (τ − T, ω)))dτ + σv(4)s ◦ dWτ−T , τ ∈ [t, t+ T ]. (5.6)

with

v(1)c (τ, ω)
∣∣
τ=t

= ξ(t, ω),



EJDE-2023/20 STOCHASTIC ATTRACTOR BIFURCATION OF 2D S-H EQUATIONS 17

v(2)c (τ, ω)
∣∣
τ=t

= ξ(t, ω),

v(3)c (τ, ω)
∣∣
τ=t

= ξ(t, ω),

v(4)c (τ, ω)
∣∣
τ=t

= ξ(t, ω),

v(4)s (τ, θ−Tω)
∣∣
τ=t

= 0.

Since ξ ∈ Hc
3 , we write

ξ(τ, ω) = ξ1(τ, ω)em,n + ξ2(τ, ω)em,−n + ξ3(τ, ω)en,m

+ ξ4(τ, ω)en,−m.
(5.7)

As in section 3, by computations, we obtain the reduced model on the PM, up
to the leading order,

dξ1 = (P (

√
m2 + n2π

L
)ξ1 −

3

2
ξ31 − 3ξ1ξ

2
2 − 3ξ1ξ

2
3 − 3ξ1ξ

2
4)dt+ σξ1 ◦ dWt, (5.8)

dξ2 = (P (

√
m2 + n2π

L
)ξ2 −

3

2
ξ32 − 3ξ21ξ2 − 3ξ2ξ

2
3 − 3ξ2ξ

2
4)dt+ σξ2 ◦ dWt, (5.9)

dξ3 = (P (

√
m2 + n2π

L
)ξ3 −

3

2
ξ33 − 3ξ21ξ3 − 3ξ22ξ3 − 3ξ3ξ

2
4)dt+ σξ3 ◦ dWt, (5.10)

dξ4 = (P (

√
m2 + n2π

L
)ξ4 −

3

2
ξ34 − 3ξ21ξ4 − 3ξ22ξ4 − 3ξ23ξ4)dt+ σξ4 ◦ dWt. (5.11)

For L near
√
m2 + n2L1 and L >

√
m2 + n2L1, this system has the following 80

nontrivial solutions:

S1,2L (θtω) = (±
√

1

3
aL(θtω), 0, 0, 0),

S3,4L (θtω) = (0,±
√

1

3
aL(θtω), 0, 0),

S5,6L (θtω) = (0, 0,±
√

1

3
aL(θtω), 0),

S7,8L (θtω) = (0, 0, 0,±
√

1

3
aL(θtω)),

S9,10L (θtω) = (
1

3
aL(θtω),±1

3
aL(θtω), 0, 0),

S11,12L (θtω) = (−1

3
aL(θtω),±1

3
aL(θtω), 0, 0),

S13,14L (θtω) = (
1

3
aL(θtω), 0,±1

3
aL(θtω), 0),

S15,16L (θtω) = (−1

3
aL(θtω), 0,±1

3
aL(θtω), 0),

S17,18L (θtω) = (
1

3
aL(θtω), 0, 0,±1

3
aL(θtω)),

S19,20L (θtω) = (−1

3
aL(θtω), 0, 0,±1

3
aL(θtω)),

S21,22L (θtω) = (0,
1

3
aL(θtω),±1

3
aL(θtω), 0),

S23,24L (θtω) = (0,−1

3
aL(θtω),±1

3
aL(θtω), 0),



18 Q. XIAO, H. GAO EJDE-2023/20

S25,26L (θtω) = (0,
1

3
aL(θtω), 0,±1

3
aL(θtω)),

S27,28L (θtω) = (0,−1

3
aL(θtω), 0,±1

3
aL(θtω)),

S29,30L (θtω) = (0, 0,
1

3
aL(θtω),±1

3
aL(θtω)),

S31,32L (θtω) = (0, 0,−1

3
aL(θtω),±1

3
aL(θtω)),

S33,34L (θtω) = (±
√

1

15
aL(θtω),

√
1

15
aL(θtω),

√
1

15
aL(θtω), 0),

S35,36L (θtω) = (±
√

1

15
aL(θtω),−

√
1

15
aL(θtω),

√
1

15
aL(θtω), 0),

S37,38L (θtω) = (±
√

1

15
aL(θtω),

√
1

15
aL(θtω),−

√
1

15
aL(θtω), 0),

S39,40L (θtω) = (±
√

1

15
aL(θtω),−

√
1

15
aL(θtω),−

√
1

15
aL(θtω), 0),

S41,42L (θtω) = (±
√

1

15
aL(θtω),

√
1

15
aL(θtω), 0,

√
1

15
aL(θtω)),

S43,44L (θtω) = (±
√

1

15
aL(θtω),−

√
1

15
aL(θtω), 0,

√
1

15
aL(θtω)),

S45,46L (θtω) = (±
√

1

15
aL(θtω),

√
1

15
aL(θtω), 0,−

√
1

15
aL(θtω)),

S47,48L (θtω) = (±
√

1

15
aL(θtω),−

√
1

15
aL(θtω), 0,−

√
1

15
aL(θtω)),

S49,50L (θtω) = (±
√

1

15
aL(θtω), 0,

√
1

15
aL(θtω),

√
1

15
aL(θtω)),

S51,52L (θtω) = (±
√

1

15
aL(θtω), 0,−

√
1

15
aL(θtω),

√
1

15
aL(θtω)),

S53,54L (θtω) = (±
√

1

15
aL(θtω), 0,

√
1

15
aL(θtω),−

√
1

15
aL(θtω)),

S55,56L (θtω) = (±
√

1

15
aL(θtω), 0,−

√
1

15
aL(θtω),−

√
1

15
aL(θtω)),

S57,58L (θtω) = (0,±
√

1

15
aL(θtω),

√
1

15
aL(θtω),

√
1

15
aL(θtω)),

S59,60L (θtω) = (0,±
√

1

15
aL(θtω),−

√
1

15
aL(θtω),

√
1

15
aL(θtω)),

S61,62L (θtω) = (0,±
√

1

15
aL(θtω),

√
1

15
aL(θtω),−

√
1

15
aL(θtω)),

S63,64L (θtω) = (0,±
√

1

15
aL(θtω),−

√
1

15
aL(θtω),−

√
1

15
aL(θtω)),

S65,66L (θtω) = (±
√

1

21
aL(θtω),

√
1

21
aL(θtω),

√
1

21
aL(θtω),

√
1

21
aL(θtω)),



EJDE-2023/20 STOCHASTIC ATTRACTOR BIFURCATION OF 2D S-H EQUATIONS 19

S67,68L (θtω) = (±
√

1

21
aL(θtω),−

√
1

21
aL(θtω),

√
1

21
aL(θtω),

√
1

21
aL(θtω)),

S69,70L (θtω) = (±
√

1

21
aL(θtω),

√
1

21
aL(θtω),−

√
1

21
aL(θtω),

√
1

21
aL(θtω)),

S71,72L (θtω) = (±
√

1

21
aL(θtω),

√
1

21
aL(θtω),

√
1

21
aL(θtω),−

√
1

21
aL(θtω)),

S73,74L (θtω) = (±
√

1

21
aL(θtω),−

√
1

21
aL(θtω),−

√
1

21
aL(θtω),

√
1

21
aL(θtω)),

S75,76L (θtω) = (±
√

1

21
aL(θtω),−

√
1

21
aL(θtω),

√
1

21
aL(θtω),−

√
1

21
aL(θtω)),

S77,78L (θtω) = (±
√

1

21
aL(θtω),

√
1

21
aL(θtω),

√
− 1

21
aL(θtω),−

√
1

21
aL(θtω)),

S79,80L (θtω) = (±
√

1

21
aL(θtω),−

√
1

21
aL(θtω),−

√
1

21
aL(θtω),−

√
1

21
aL(θtω)),

where

aL(ω) = (

∫ 0

−∞
e2P (

√
m2+n2π
L )τ+2σWτ (ω)dτ)−1/2. (5.12)

Let Φ denote the flow associated with the system, and

(ξ1(t), ξ2(t), ξ3(t), ξ4(t)) = Φ(t, θ−tω)(ξ01 , ξ
0
2 , ξ

0
3 , ξ

0
4)

be a solution of the SDE with initial condition satisfying ξ01 ≥ ξ02 > ξ03 > ξ04 > 0.
Using standard argument in ordinary differential equations, we have

lim
t→+∞

Φ(t, θ−tω)(ξ01 , ξ
0
2 , ξ

0
3 , ξ

0
4) =

{
(±
√

1
3aL(ω), 0, 0, 0), if ξ01 > ξ02 ,

( 1
3aL(ω), 13aL(ω), 0, 0), if ξ01 = ξ02 .

The other cases can be obtained in a similar fashion. In a similar manner, we have
the following results.

Theorem 5.1. The reduced system (5.8)-(5.11) undergoes a stochastic supercritical

bifurcation at L =
√
m2 + n2L1 in the pullback sense. More precisely, when L near√

m2 + n2L1,

(a) For L <
√
m2 + n2L1, the origin is globally asymptotically stable.

(b) For L >
√
m2 + n2L1, the random compact set

AL(ω) = {SiL(ω), i = 1, 2, . . . , 80.}

is a random pullback attractor.

Theorem 5.2. The reduced system (5.8)-(5.11) undergoes a stochastic subcritical

bifurcation at L =
√
m2 + n2L2 in the pullback sense. More precisely, when L near√

m2 + n2L2 ,

(a) For L <
√
m2 + n2L2, the random compact set AL(ω) is a random pullback

attractor.
(b) For L >

√
m2 + n2L2, 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.



20 Q. XIAO, H. GAO EJDE-2023/20

6. Attractor bifurcation analysis when
√

2L1 and L2 coincide

In this section, we analyze the stochastic attractor bifurcation when the bifur-
cation points L2 and

√
2L1 are close together, i.e., when α = 1

9 + ε where α = 1
9

satisfies L2(α) =
√

2L1(α) and ε is positive and small. In this case, the space H1

and H can be decomposed into

H1 = Hc
4 ⊕Hs

4 , H = Hc
4 ⊕Hs,

where

Hc
4 = span{e1,0, e0,1, e1,1, e1,−1}.

Projecting the above equation onto the subspace Hc
4 , we obtain

dṽc = (Lcαṽc + PcG(ṽc + ṽs))dt+ σṽc ◦ dWt,

where ṽc = Pcu, and ṽs = Psu is the unresolved variable, and Pc, Ps are respectively
canonical projections from H to Hc

4 and Hs. To obtain a closed form of the above
equation, the unresolved variables ṽs is parameterized in terms of the resolved
variables ṽc through a random continuous function hα(ξ, ω) : Hc

4 × Ω→ Hs.
The PM-based reduced equation for the resolved modes is

dξ = (Lcαξ + PcG(ξ + hα(ξ, θtω)))dt+ σξ ◦ dWt,

where ξ ∈ Hc
4 . Using an approximation of hα(ξ, ω) via the pullback characterization

as in section 3, we instead consider the reduced equation

dξt = (Lcαξt + PcG(ξt + v(4)s [ξ(t, ω)](t+ T, θ−Tω; 0)))dt+ σξt ◦ dWt,

ξ(0, ω) = ϕ, t > 0,
(6.1)

where T is sufficiently large, ϕ = Pcu0, ṽ
(4)
s is used to approximate the stochas-

tic inertial manifold and is obtained from the following backward-forward systems
(6.2)-(6.6).

For a given t > 0 and T sufficiently large, let us consider the following 4-layer
auxiliary backward-forward system

dṽ(1)c = Lcαṽ(1)c dτ + σṽ(1)c ◦ dWτ , τ ∈ [t− T, t], (6.2)

dṽ(2)c = (Lcαṽ(2)c + PcG(ṽ(1)c )dτ + σṽ(2)c ◦ dWτ , τ ∈ [t− T, t], (6.3)

dṽ(3)c = (Lcαṽ(4)c + PcG(ṽ(2)c )dτ + σṽ(3)c ◦ dWτ , τ ∈ [t− T, t], (6.4)

dṽ(4)c = (Lcαṽ(4)c + PcG(ṽ(3)c )dτ + σṽ(4)c ◦ dWτ , τ ∈ [t− T, t], (6.5)

dṽ(4)s = (Lsαṽ(4)s + PcG(ṽ(4)c (τ − T, ω)))dτ + σṽ(4)s ◦ dWτ−T , τ ∈ [t, t+ T ] (6.6)

with

ṽ(1)c (τ, ω)
∣∣
τ=t

= ξ(t, ω),

ṽ(2)c (τ, ω)
∣∣
τ=t

= ξ(t, ω),

ṽ(3)c (τ, ω)
∣∣
τ=t

= ξ(t, ω),

ṽ(4)c (τ, ω)
∣∣
τ=t

= ξ(t, ω),

ṽ(4)s (τ, θ−Tω)
∣∣
τ=t

= 0.

Since ξ ∈ Hc
4 , we write

ξ(τ, ω) = ξ1(τ, ω)e1,0 + ξ2(τ, ω)e0,1 + ξ3(τ, ω)e1,1 + ξ4(τ, ω)e1,−1. (6.7)



EJDE-2023/20 STOCHASTIC ATTRACTOR BIFURCATION OF 2D S-H EQUATIONS 21

By computations we obtain the reduced model on the PM, up to the leading order,

dξ1 = (P (
π

L
)ξ1 −

3

2
ξ31 − 3ξ1ξ

2
2 − 3ξ1ξ

2
3 − 3ξ1ξ

2
4 − 3ξ1ξ3ξ4)dt+ σξ1 ◦ dWt,

dξ2 = (P (
π

L
)ξ2 −

3

2
ξ32 − 3ξ21ξ2 − 3ξ2ξ

2
3 − 3ξ2ξ

2
4 + 3ξ2ξ3ξ4)dt+ σξ2 ◦ dWt.

dξ3 = (P (

√
2π

L
)ξ3 −

3

2
ξ33 − 3ξ21ξ3 − 3ξ22ξ3 − 3ξ3ξ

2
4 −

3

2
ξ21ξ4 +

3

2
ξ22ξ4)dt+ σξ3 ◦ dWt,

dξ4 = (P (

√
2π

L
)ξ4 −

3

2
ξ34 − 3ξ21ξ4 − 3ξ22ξ4 − 3ξ23ξ4 −

3

2
ξ21ξ3 +

3

2
ξ22ξ3)dt+ σξ4 ◦ dWt.

From the above, we obtain the approximation representation of manifold and the
corresponding reduced systems for stochastic Swift-Hohenberg equation when L2

and
√

2L1 are close together. The performances achieved by the above reduced
system can approximate dynamics on the Hc

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.

Acknowledgments. This research was supported by National Natural Science
Foundation of China (No. 11801270, 12171084).

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Qingkun Xiao
College of Sciences, Nanjing Agricultural University, Nanjing 210095, China

Email address: xiaoqk@njau.edu.cn

Hongjun Gao (corresponding author)

School of Mathematics, Southeast University, Nanjing 211189, China

Email address: hjgao@seu.edu.cn


	1. Introduction
	2. Mathematical setting
	3. Analysis of the case (m,n)=(k,k) and (k,-k)
	4. Analysis of the case (m,n)=(k,0) and (0,k)
	5. Analysis of the case (m,n) and (n,m)
	6. Attractor bifurcation analysis when 2L1 and L2 coincide
	Acknowledgments

	References