# Attractor Bifurcation for Extended Fisher-Kolmogorov Equation

Authors: Honglian You; Rong Yuan; Ziheng Zhang
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

We consider the asymptotic stability and attractor bifurcation of the extended Fisher-Kolmogorov equation on the one-dimensional domain(0,π) with Dirichlet or periodic boundary conditions. The novelty of this paper is that, based on a new method called attractor bifurcation, we investigate the existence of an attractor bifurcated from the trivial solution and give an explicit description of the bifurcated attractor. Moreover, the stability of the bifurcated branches is discussed.


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## Body

## 1. Introduction

Fourth-order parabolic differential equations of the form(1)∂u∂t=-γ∂4u∂x4+β∂2u∂x2+f(u),γ>0,β∈ℝ,
where t>0, f(u) is a nonlinear function, arise from many physics models such as the theory of phase transitions [1], nonlinear optics [2], and shallow water waves [3]. The prototypical example of the nonlinearity f is f(u)=u-u3; that is,
(2)∂u∂t=-γ∂4u∂x4+β∂2u∂x2+u-u3,γ>0,β∈ℝ.
When β>0, (2) is usually named extended Fisher-Kolmogorov (EFK) equation [4], while for β<0 the name Swift-Hohenberg (SH) equation [5] is more appropriate. In this paper, we restrict our attention to the case β>0. For this case, (2) is a natural extension of the classical Fisher-Kolmogorov (FK) or Allen-Cahn equation [6, 7]. It has been proposed as a model equation for phase transitions in the neighborhood of a Lifshitz point [8] and is frequently used as a model system for the study of pattern formation from an unstable spatially homogeneous state; see [1, 4].Due to the significance of fourth-order differential equations, substantial attention has been paid to (1). These results are mainly on basic properties of solutions (e.g., the existence, uniqueness, boundedness, and periodicity) [9–13], traveling waves [14], kinks [15, 16], asymptotic properties [17, 18], and global attractors [19]. In addition, (1) or similar equations play an important role in a variety of physics systems. Nevertheless we do not attempt to give a detailed account; instead refer the reader to [20] consulting a host of another related research.To understand the behavior of solutions of (1), it is essential to understand the stationary (time-independent) solutions (or steady states), that is, the solutions of (1) satisfying the fourth-order differential equations of the form
(3)-γu′′′′+βu′′+f(u)=0.
Typically, the stationary problem displays a multitude of periodic, homoclinic, and heteroclinic solutions. The steady-state equation of (1), that is, (3), has been analysed using shooting methods [21] and with the help of variational methods [22]. An extensive literature on this subject exists, and we refer to [3, 23–26] and the reference listed therein.In the recent paper [27], based on a variational gluing method, the authors considered the set of bifurcation stationary solutions of (2) for all L>0 with the Neumann boundary condition; that is,
(4)-γu′′′′+βu′′+u-u3=0,x∈(0,L),ux(0)=uxxx(0)=ux(L)=uxxx(L)=0.
Without loss of generality, they set β=1 and described the transition at the bifurcation point γ=1/8 and gave a precise description of how the attractor is changing for γ=1/8+ϵ, 0<ϵ≪1 (see Section  8 in [27]).Normally, to study (3), γ, β, the potential F (F(u)=∫0u‍f(s)ds), the interval-length L, and the boundary conditions are viewed as parameters. Motivated by the above works, in this paper we introduce new parameters and study the dynamical behavior of (2). Explicitly, we focus our attention on the asymptotic behavior and attractor bifurcation (the structure and the local stability of the bifurcated steady states) of the following extended Fisher-Kolmogorov (EFK) equation with β=1 (this can be done via a variable substitution t=τ/β):
(5)∂u∂t=-γ∂4u∂x4+∂2u∂x2+λu-λu3,x∈(0,π),
where γ>0 is a constant and λ>0 is the bifurcated parameter, using a new bifurcation theory (see Section 2) called attractor bifurcation developed recently by Ma and Wang [28, 29]. We are mainly dedicated to considering (5) associated to Dirichlet boundary condition
(6)u(0,t)=u(π,t)=0,u′′(0,t)=u′′(π,t)=0,
or odd-periodic boundary condition
(7)u(x+π,t)=u(x,t),u(-x,t)=-u(x,t),
or periodic boundary condition
(8)u(x+π,t)=u(x,t),
respectively. For the three cases, the existence and local stability of bifurcated branches are described completely (see the main results in Section 3). At the end of this paragraph, we should point out that this new bifurcation theory has been proved to be very practical. In fact, with the help of this new theory, many long-time standing bifurcation problems for a variety of PDEs arising from science and engineering, such as Boussinesq equations [30], Rayleigh-Bénard convections [31], Swift-Hohenberg (SH) equations [32], Ginzburg-Landau (GL) equations [33, 34], and Kuramoto-Sivashinsky equations [28, 35], have been extensively taken into account.The paper is organized as follows. In Section2, we recall some preliminaries in terms of the attractor bifurcation and the center manifold reduction process. In Section 3, the main body of this paper, we are dedicated to considering the attractor bifurcation of (5) associated to Dirichlet, odd-periodic, or periodic boundary conditions, respectively. Firstly, we calculate the corresponding eigenvalues of the linear part with different boundary conditions and determine all critical values. Then as the parameter λ crosses the first critical value, we show that (5) bifurcates from the trivial solution an attractor 𝒜λ. Moreover, we demonstrate an explicit description of 𝒜λ that, with Dirichlet or odd-periodic boundary conditions, it consists of two steady states, while with periodic boundary condition, it is homeomorphic to S1. Finally, we extend these results to other critical values.


## 2. Preliminary Results

### 2.1. Attractor Bifurcation Theory

We first recall the general theory on attractor bifurcation of abstract evolution equations; see [28] (also [29]).LetH and H1 be two Hilbert spaces and H1↪H a dense and compact inclusion. Consider the following abstract nonlinear evolution equation
(9)dudt=Lλu+G(u,λ),u(0)=ϕ,
where u:[0,+∞)→H is the unknown function, λ∈ℝ is the system parameter, and Lλ:H1→H are parameterized linear completely continuous fields continuously depending on λ∈ℝ, which satisfy
(10)Lλ=-A+Bλasectorialoperator,A:H1⟶Halinearhomeomorphism,Bλ:H1⟶Hparameterizedlinearcompactoperator.
Then Lλ generates an analytic semigroup {e-tLλ}t≥0, and we can define fractional power operators Lλα for any 0≤α≤1 with domain Hα=D(Lλα) such that Hα1⊂Hα2 if α1>α2 and H0=H. Furthermore, we assume that the nonlinear terms G(·,λ):Hα→H for some 0≤α<1 are family of parameterized Cr (r≥1) bounded operators depending continuously on the parameter λ∈ℝ such that
(11)G(u,λ)=o(∥u∥Hα),∀λ∈ℝ.Definition 1 (see [28, Definition  5.1]).
(1) We say that (9) bifurcates from (u,λ)=(0,λ0) an invariant set Ωλ, if there exists a sequence of invariant sets {Ωλn} such that 0∉Ωλn, and
(12)limn→+∞λn=λ0,limn→+∞d(Ωλn,0)=limn→+∞maxu∈Ωλn∥u∥H1=0.
(2) If the invariant setsΩλ are attractors of (9), then the bifurcation is called attractor bifurcation.
(3) If the invariant setsΩλ are attractors of (9), which are homotopic equivalent to an m-dimensional sphere, then we say that (9) has an Sm-attractor bifurcation at (0,λ0).Theorem 2 (see [28, Theorem  6.1]).
Assume that conditions (10) and (11) hold true. Let the eigenvalues (counting the multiplicity) of Lλ be given by β1(λ),β2(λ),…,βk(λ),…∈ℂ, where ℂ is the complex space. Suppose that(13)Reβi(λ){<0,ifλ<λ0,=0,ifλ=λ0,>0ifλ>λ0,1≤i≤m,(14)Reβi(λ0)<0,i≥m+1.
Let the eigenspace of Lλ at λ0 be
(15)E0=⋃i=1m‍⋃k=1+∞‍{u∈H1:(Lλ-βi(λ0))ku=0},
and let u=0 be a locally asymptotically stable equilibrium point of (9) at λ=λ0. Then the following assertions hold true. (1)
Equation (9) bifurcates from (u,λ)=(0,λ0) an attractor 𝒜λ for λ>λ0, with m-1≤dim𝒜λ≤m, which is connected as m≥2.(2)
The attractor𝒜λ is a limit of a sequence of m-dimensional annulus Mk with Mk+1⊂Mk; particularly if 𝒜λ is a finite simplicial complex, then 𝒜λ has the homotopy type of the (m-1)-dimensional sphere Sm-1.(3)
For anyuλ∈𝒜λ, uλ can be expressed as
(16)uλ=vλ+o(∥vλ∥H),vλ∈E0.(4)
If the number of the equilibrium points of (9) in 𝒜λ is finite, then one has the index formula
(17)∑ui∈𝒜λ‍ind[-(Lλ+G),ui]={2,ifmisodd,0,ifmiseven.(5)
Ifu=0 is globally asymptotically stable for (9) at λ=λ0, then for any bounded open set U⊂H with 0∈U, there is an ϵ>0 such that as λ0<λ<λ0+ϵ, the attractor 𝒜λ attracts U∖Γ in H, where Γ is the stable manifold of u=0 with codimension m. In particular, if (9) has a global attractor for all λ near λ0, then ϵ can be chosen independently of  U.From the previous theorem we see that the asymptotic stability of the equilibriumu=0 for λ=λ0 is crucial. In the following, we state a theorem on the asymptotic stability of u=0 for (9)λ=λ0, that is, the critical state.Theorem 3 (see [28, Theorems  3.16 and 3.17]).
LetLλ0:H1→H be symmetric with eigenvalues {βk}k=1∞ satisfying
(18)βi=0,if1≤i≤m,βi<0,ifi≥m+1.
Suppose that G(·,λ0):H1→H fulfills
(19)〈G(u,λ0),u〉<0,∀u∈H1,u≠0,
where 〈·,·〉 denotes the inner product of H. Then u=0 is globally asymptotically stable.When the first eigenvalue is simple, that is,m=1, the following result describes the structure of bifurcated attractor more precisely.Theorem 4 (see [28, Theorem  5.5 and Remark  6.1]).
Let (10)–(14) hold true with m=1. Assume that G(u,λ) is analytic at u=0 and u=0 is locally asymptotically stable for (9)λ=λ0. Then there exists an open set U⊂H with 0∈U such that if λ>λ0, (9) bifurcates from (0,λ0) exactly two equilibrium points u1 and u2∈U, and the open set U is decomposed into two open sets U1λ and U2λ, satisfying the following properties:(1)
U¯=U¯1λ+U¯2λ, U1λ⋂‍U2λ=∅,(2)
0∈∂U1λ⋂‍∂U2λ, ui∈Uiλ, i=1,2,(3)
for anyϕ∈Uiλ, i=1,2,
(20)limt→+∞∥u(t,ϕ)-ui∥H=0,where u(t,ϕ) is the solution of (9) initiated with ϕ.Ifm=1 in (13) and condition (14) is replaced by
(21)Reβi(λ0)>0,if1<i≤n+1,Reβi(λ0)<0,ifi>n+1,
the bifurcated singular points of (9) are saddle points, of which the Morse index is obtained explicitly. In the following, we first give the definition of Morse index for nondegenerate singular points.Definition 5 (see [28, Definition  3.5]).
Letu0∈H1 be a nondegenerate singular point of
(22)Lλu+G(u,λ)=0,
and all eigenvalues of
(23)Lλ+Du(u0,λ):H1⟶H
have nonzero real part. Then the Morse index of u0 is defined by “the number of eigenvalues of (23) having positive real part.”Theorem 6 (see [28, Theorem  6.6]).
Assume that (13), (21), and
(24)G(u,λ)=Gk(u,λ)+o(∥u∥k),k≥2aninteger,
hold and
(25)α:=〈Gk(e1,λ0),e1〉≠0,
where e1 is the eigenvector of Lλ corresponding to β1(λ). Then the bifurcated singular points of (9) from (0,λ0) have Morse index n+1 on λ<λ0 and have Morse index n on λ>λ0. Moreover, u=0 has Morse index n on λ<λ0 and has Morse index n+1 on λ>λ0.If the multiplicity of the first eigenvalue is2, then there exists a 2-dimensional center manifold, and we can reduce the infinite dimensional system to this center manifold to investigate the attractor bifurcation.Theorem 7 (see [28, Theorem  5.10]).
Letv be a 2-dimensional Cr (r≥1) vector field given by
(26)vλ(y)=λy-G(y,λ)
for y∈ℝ2. Here
(27)G(y,λ)=Gk(y,λ)+o(|y|k),
where Gk(y,λ):=Gk(y,…,y︸k,λ)is a k-multilinear field, which satisfies
(28)C1|y|k+1≤〈Gk(y,λ),y〉≤C2|y|k+1,
for some constants C2>C1>0, k=2p+1, and p≥1. Then vλ bifurcates from (x,λ)=(0,0) on λ>0 an attractor 𝒜λ, which is homeomorphic to S1. Moreover, one and only one of the following is true:(1)
𝒜λ is a periodic orbit,(2)
𝒜λ consists of only singular points, or(3)
𝒜λ contains at most 2(k+1)=4(p+1) singular points and has 4N+n  (N+n≥1) singular points, 2N of which are saddle points, 2N of which are stable node points (possibly degenerate), and n of which have index zero.


### 2.2. Reduction Procedure

During dealing with the bifurcation problem, the reduction of the equation to its local center manifold plays a crucial role; see [28, Section  3.2] for more information. Consider (9) and assume that eigenvalues of Lλ satisfy(29)Reβi(λ){<0,ifλ<λ0,=0,ifλ=λ0,1≤i≤m,>0,ifλ>λ0,Reβi(λ0)≠0,i≥m+1,(30)G(u,λ)=∑n=k∞‍Gn(u,λ),
for some k≥2, where Gn:H1×⋯×H1→H is an n-multilinear mapping and Gn(u,λ)=Gn(u,…,u,λ).Let{ei(λ)}i=1∞ be the eigenvectors corresponding to {βi(λ)}i=1∞. Under the above assumptions there is a center manifold near λ=λ0 with the center manifold function given by
(31)Φ(·,λ):Ω⟶Em⊥,
where Ω⊂Em is a neighborhood of u=0 and
(32)Em={∑i=1m‍xiei(λ):(x1,…,xm)∈ℝm},Em⊥={u∈H:〈u,ei(λ)〉=0,1≤i≤m}.
Then (9) can be reduced to the center manifold in the following form:
(33)dxdt=Jmλx+g(x,λ),
where x=(x1,…,xm)T∈ℝm, Jmλ is the Jordan matrix corresponding to eigenvalues β1(λ),…,βm(λ) of Lλ, and
(34)g(x,λ)=(1〈e1(λ),e1(λ)〉〈G(x+Φ(x,λ),λ),e1(λ)〉,…,1〈em(λ),em(λ)〉×〈G(x+Φ(x,λ),λ),em(λ)〉1〈e1(λ),e1(λ)〉)T.
By (30), we get the first approximation of (33), which is given by
(35)dxdt=Jmλx+Fk(x,λ)+o(|x|k),
where
(36)Fk(x,λ)=(1〈e1(λ),e1(λ)〉〈Gk(∑i=1m‍xiei(λ),λ),e1(λ)〉,…,1〈em(λ),em(λ)〉×〈Gk(∑i=1m‍xiei(λ),λ),em(λ)〉)T.


## 2.1. Attractor Bifurcation Theory

We first recall the general theory on attractor bifurcation of abstract evolution equations; see [28] (also [29]).LetH and H1 be two Hilbert spaces and H1↪H a dense and compact inclusion. Consider the following abstract nonlinear evolution equation
(9)dudt=Lλu+G(u,λ),u(0)=ϕ,
where u:[0,+∞)→H is the unknown function, λ∈ℝ is the system parameter, and Lλ:H1→H are parameterized linear completely continuous fields continuously depending on λ∈ℝ, which satisfy
(10)Lλ=-A+Bλasectorialoperator,A:H1⟶Halinearhomeomorphism,Bλ:H1⟶Hparameterizedlinearcompactoperator.
Then Lλ generates an analytic semigroup {e-tLλ}t≥0, and we can define fractional power operators Lλα for any 0≤α≤1 with domain Hα=D(Lλα) such that Hα1⊂Hα2 if α1>α2 and H0=H. Furthermore, we assume that the nonlinear terms G(·,λ):Hα→H for some 0≤α<1 are family of parameterized Cr (r≥1) bounded operators depending continuously on the parameter λ∈ℝ such that
(11)G(u,λ)=o(∥u∥Hα),∀λ∈ℝ.Definition 1 (see [28, Definition  5.1]).
(1) We say that (9) bifurcates from (u,λ)=(0,λ0) an invariant set Ωλ, if there exists a sequence of invariant sets {Ωλn} such that 0∉Ωλn, and
(12)limn→+∞λn=λ0,limn→+∞d(Ωλn,0)=limn→+∞maxu∈Ωλn∥u∥H1=0.
(2) If the invariant setsΩλ are attractors of (9), then the bifurcation is called attractor bifurcation.
(3) If the invariant setsΩλ are attractors of (9), which are homotopic equivalent to an m-dimensional sphere, then we say that (9) has an Sm-attractor bifurcation at (0,λ0).Theorem 2 (see [28, Theorem  6.1]).
Assume that conditions (10) and (11) hold true. Let the eigenvalues (counting the multiplicity) of Lλ be given by β1(λ),β2(λ),…,βk(λ),…∈ℂ, where ℂ is the complex space. Suppose that(13)Reβi(λ){<0,ifλ<λ0,=0,ifλ=λ0,>0ifλ>λ0,1≤i≤m,(14)Reβi(λ0)<0,i≥m+1.
Let the eigenspace of Lλ at λ0 be
(15)E0=⋃i=1m‍⋃k=1+∞‍{u∈H1:(Lλ-βi(λ0))ku=0},
and let u=0 be a locally asymptotically stable equilibrium point of (9) at λ=λ0. Then the following assertions hold true. (1)
Equation (9) bifurcates from (u,λ)=(0,λ0) an attractor 𝒜λ for λ>λ0, with m-1≤dim𝒜λ≤m, which is connected as m≥2.(2)
The attractor𝒜λ is a limit of a sequence of m-dimensional annulus Mk with Mk+1⊂Mk; particularly if 𝒜λ is a finite simplicial complex, then 𝒜λ has the homotopy type of the (m-1)-dimensional sphere Sm-1.(3)
For anyuλ∈𝒜λ, uλ can be expressed as
(16)uλ=vλ+o(∥vλ∥H),vλ∈E0.(4)
If the number of the equilibrium points of (9) in 𝒜λ is finite, then one has the index formula
(17)∑ui∈𝒜λ‍ind[-(Lλ+G),ui]={2,ifmisodd,0,ifmiseven.(5)
Ifu=0 is globally asymptotically stable for (9) at λ=λ0, then for any bounded open set U⊂H with 0∈U, there is an ϵ>0 such that as λ0<λ<λ0+ϵ, the attractor 𝒜λ attracts U∖Γ in H, where Γ is the stable manifold of u=0 with codimension m. In particular, if (9) has a global attractor for all λ near λ0, then ϵ can be chosen independently of  U.From the previous theorem we see that the asymptotic stability of the equilibriumu=0 for λ=λ0 is crucial. In the following, we state a theorem on the asymptotic stability of u=0 for (9)λ=λ0, that is, the critical state.Theorem 3 (see [28, Theorems  3.16 and 3.17]).
LetLλ0:H1→H be symmetric with eigenvalues {βk}k=1∞ satisfying
(18)βi=0,if1≤i≤m,βi<0,ifi≥m+1.
Suppose that G(·,λ0):H1→H fulfills
(19)〈G(u,λ0),u〉<0,∀u∈H1,u≠0,
where 〈·,·〉 denotes the inner product of H. Then u=0 is globally asymptotically stable.When the first eigenvalue is simple, that is,m=1, the following result describes the structure of bifurcated attractor more precisely.Theorem 4 (see [28, Theorem  5.5 and Remark  6.1]).
Let (10)–(14) hold true with m=1. Assume that G(u,λ) is analytic at u=0 and u=0 is locally asymptotically stable for (9)λ=λ0. Then there exists an open set U⊂H with 0∈U such that if λ>λ0, (9) bifurcates from (0,λ0) exactly two equilibrium points u1 and u2∈U, and the open set U is decomposed into two open sets U1λ and U2λ, satisfying the following properties:(1)
U¯=U¯1λ+U¯2λ, U1λ⋂‍U2λ=∅,(2)
0∈∂U1λ⋂‍∂U2λ, ui∈Uiλ, i=1,2,(3)
for anyϕ∈Uiλ, i=1,2,
(20)limt→+∞∥u(t,ϕ)-ui∥H=0,where u(t,ϕ) is the solution of (9) initiated with ϕ.Ifm=1 in (13) and condition (14) is replaced by
(21)Reβi(λ0)>0,if1<i≤n+1,Reβi(λ0)<0,ifi>n+1,
the bifurcated singular points of (9) are saddle points, of which the Morse index is obtained explicitly. In the following, we first give the definition of Morse index for nondegenerate singular points.Definition 5 (see [28, Definition  3.5]).
Letu0∈H1 be a nondegenerate singular point of
(22)Lλu+G(u,λ)=0,
and all eigenvalues of
(23)Lλ+Du(u0,λ):H1⟶H
have nonzero real part. Then the Morse index of u0 is defined by “the number of eigenvalues of (23) having positive real part.”Theorem 6 (see [28, Theorem  6.6]).
Assume that (13), (21), and
(24)G(u,λ)=Gk(u,λ)+o(∥u∥k),k≥2aninteger,
hold and
(25)α:=〈Gk(e1,λ0),e1〉≠0,
where e1 is the eigenvector of Lλ corresponding to β1(λ). Then the bifurcated singular points of (9) from (0,λ0) have Morse index n+1 on λ<λ0 and have Morse index n on λ>λ0. Moreover, u=0 has Morse index n on λ<λ0 and has Morse index n+1 on λ>λ0.If the multiplicity of the first eigenvalue is2, then there exists a 2-dimensional center manifold, and we can reduce the infinite dimensional system to this center manifold to investigate the attractor bifurcation.Theorem 7 (see [28, Theorem  5.10]).
Letv be a 2-dimensional Cr (r≥1) vector field given by
(26)vλ(y)=λy-G(y,λ)
for y∈ℝ2. Here
(27)G(y,λ)=Gk(y,λ)+o(|y|k),
where Gk(y,λ):=Gk(y,…,y︸k,λ)is a k-multilinear field, which satisfies
(28)C1|y|k+1≤〈Gk(y,λ),y〉≤C2|y|k+1,
for some constants C2>C1>0, k=2p+1, and p≥1. Then vλ bifurcates from (x,λ)=(0,0) on λ>0 an attractor 𝒜λ, which is homeomorphic to S1. Moreover, one and only one of the following is true:(1)
𝒜λ is a periodic orbit,(2)
𝒜λ consists of only singular points, or(3)
𝒜λ contains at most 2(k+1)=4(p+1) singular points and has 4N+n  (N+n≥1) singular points, 2N of which are saddle points, 2N of which are stable node points (possibly degenerate), and n of which have index zero.


## 2.2. Reduction Procedure

During dealing with the bifurcation problem, the reduction of the equation to its local center manifold plays a crucial role; see [28, Section  3.2] for more information. Consider (9) and assume that eigenvalues of Lλ satisfy(29)Reβi(λ){<0,ifλ<λ0,=0,ifλ=λ0,1≤i≤m,>0,ifλ>λ0,Reβi(λ0)≠0,i≥m+1,(30)G(u,λ)=∑n=k∞‍Gn(u,λ),
for some k≥2, where Gn:H1×⋯×H1→H is an n-multilinear mapping and Gn(u,λ)=Gn(u,…,u,λ).Let{ei(λ)}i=1∞ be the eigenvectors corresponding to {βi(λ)}i=1∞. Under the above assumptions there is a center manifold near λ=λ0 with the center manifold function given by
(31)Φ(·,λ):Ω⟶Em⊥,
where Ω⊂Em is a neighborhood of u=0 and
(32)Em={∑i=1m‍xiei(λ):(x1,…,xm)∈ℝm},Em⊥={u∈H:〈u,ei(λ)〉=0,1≤i≤m}.
Then (9) can be reduced to the center manifold in the following form:
(33)dxdt=Jmλx+g(x,λ),
where x=(x1,…,xm)T∈ℝm, Jmλ is the Jordan matrix corresponding to eigenvalues β1(λ),…,βm(λ) of Lλ, and
(34)g(x,λ)=(1〈e1(λ),e1(λ)〉〈G(x+Φ(x,λ),λ),e1(λ)〉,…,1〈em(λ),em(λ)〉×〈G(x+Φ(x,λ),λ),em(λ)〉1〈e1(λ),e1(λ)〉)T.
By (30), we get the first approximation of (33), which is given by
(35)dxdt=Jmλx+Fk(x,λ)+o(|x|k),
where
(36)Fk(x,λ)=(1〈e1(λ),e1(λ)〉〈Gk(∑i=1m‍xiei(λ),λ),e1(λ)〉,…,1〈em(λ),em(λ)〉×〈Gk(∑i=1m‍xiei(λ),λ),em(λ)〉)T.


## 3. Attractor Bifurcation of the EFK Equations

### 3.1. Dirichlet Boundary Condition

Consider the extended Fisher-Kolmogorov equation (5) with the Dirichlet boundary condition
(37)∂u∂t=-γ∂4u∂x4+∂2u∂x2+λu-λu3,x∈(0,π),λ>0,u(0,t)=u(π,t)=0,u′′(0,t)=u′′(π,t)=0,u(x,0)=u0(x).
In order to apply the bifurcation theory introduced in Section 2 to discuss the attractor bifurcation of (37), firstly we choose the Hilbert spaces H1 and H as follows:
(38)H1∶={u∈H4(0,π):u(0)=u(π)=0,22222u′′(0)=u′′(π)=0},H∶=L2(0,π),
and denote by ∥·∥H the norm on H which is induced by the inner product
(39)〈u,v〉=∫0π‍u(x)v(x)dx,∀u,v∈H.
It is well known that the embedding of H1↪H is compact and dense. Then (37) can be transformed into the following abstract evolution equation in H:
(40)dudt=Lλu+G(u,λ),u(0)=u0,
where Lλ:=-A+Bλ, for all λ>0, with A and Bλ:H1→H defined as follows:
(41)Au=γ∂4u∂x4-∂2u∂x2,Bλu=λu,
and, for some α∈[0,1), G(·,λ):Hα→H is of the form
(42)G(u,λ)=-λu3.
An easy calculation shows that the eigenvalues of -A:H1→H are
(43)δn=-n2(1+n2γ),n=1,2,…
with eigenvectors en(x)=sinnx. Hence, -A is a linear homeomorphism. On the other hand, since H1 is compactly imbedded in H, Bλ is a compact operator. Therefore, Lλ=-A+Bλ is a linear completely continuous filed.Proposition 8.
For anyλ>0, Lλ is a sectorial operator.Proof.
It is obvious that the spectrum of-A consists of eigenvalues; that is,
(44)σ(-A)={-n2(1+n2γ),n=1,2,…}.
Let u=∑n=1∞‍unen. Then
(45)(δI+A)-1u=∑n=1∞‍(δ-δn)-1unen,ifδ>δ1,
which yields that
(46)∥(δI+A)-1∥≤1δ-δ1,ifδ>δ1.
Therefore, by [36, page 12, Corollary  3.8],
(47)∥T(t)∥≤eδ1t≤1,∀t≥0,
where T(t), t≥0, is the C0-semigroup generated by -A. On the other hand, for δ∈ℂ with Reδ>0 and Imδ≠0, we have
(48)∥(δI+A)-1∥≤1|δ-δ1|≤1|Imδ|.
By [36, page 61, Theorem  5.2], -A is a sectorial operator. Furthermore, Bλ:H1→H is a linear bounded operator. Therefore, -A+Bλ is a sectorial operator.Lemma 9.
For any1/8<α<1, G(u,λ)=o(∥u∥Hα), for all λ>0.Proof.
For anyu∈Hα, by [37, Theorem  1.6.1], we have
(49)∥G(u,λ)∥H2=λ2∫0π‍u6dx≤πλ2∥u∥C06≤C12πλ2∥u∥Hα6,
where ∥·∥C0 denotes the supremum norm on [0,π], C1>0 is some constant, and α∈[0,1) satisfies 4α-1/2>0, that is, 1/8<α<1.From the hypotheses in Theorem2, we know that the eigenvalues of Lλ are vital to give rise to the bifurcation. After a simple computation, the eigenvalues and the eigenvectors of Lλ are given as follows.Proposition 10.
The eigenvalues ofLλ are
(50)βn(λ)=λ-n2(1+n2γ),n=1,2,…,
and the corresponding eigenvectors are
(51)en(x)=sinnx,n=1,2,….
Moreover, {en(x)}n=1∞ forms an orthogonal basis of H1, as well as H.From (50) we know that every eigenvalue is simple and the first eigenvalue is β1(λ)=λ-(1+γ). Take the first critical value λ1:=1+γ. Now we give the result on bifurcation for (37) when λ crosses λ1.Theorem 11.
For (37), one has the following assertions. (1)
As0<λ≤1+γ, u=0 is globally asymptotically stable.(2)
Equation (37) bifurcates from (0,1+γ) an attractor 𝒜λ on the right side of λ=1+γ, which consists of exactly two steady states u1λ and u2λ expressed as follows:
(52)u1λ=α(λ)sinx+o(α(λ)),u2λ=-α(λ)sinx+o(α(λ)),α(λ)=4(λ-(1+γ))3λ.(3)
For someϵ>0, if 1+γ<λ<1+γ+ϵ, there exists an open set U⊂H containing 0, which can be decomposed by the stable manifold of u=0 into two open sets U1λ and U2λ with the property that uiλ∈Uiλ, i=1,2, and
(53)limt→+∞∥u(t,ϕ)-uiλ∥H=0,ϕ∈Uiλ,
where u(t,ϕ) is the solution of (37) initiated with ϕ.Proof.
For any0≠u∈H1 and λ>0,
(54)〈G(u,λ),u〉=-λ∫0π‍u4dx<0.
By Theorem 3, u=0 is globally asymptotically stable as 0<λ≤1+γ.
The existence of𝒜λ follows from Propositions 8 and 10, Lemma 9, and Theorem 2 directly. Moreover, Theorem 4 indicates that the bifurcated attractor consists of exactly two steady states and (3) is also true. Now we give the formula of two steady states by reducing (40) to its local center manifold.
Letu(x)=∑n=1∞‍unsinnx∈H1. Then
(55)dundt=βn(λ)un+2π〈G(u,λ),sinnx〉,
which deduces that the steady-state equation is of the form
(56)βn(λ)un+2π〈G(u,λ),sinnx〉=0.
Since
(57)G(u,λ)=-λu3=:G3(u,λ),
where G3:H1×H1×H1→H is a 3-multilinear mapping, we have
(58)〈G(u,λ),sinnx〉=〈G3(∑i=1∞‍uisinix,∑j=1∞‍ujsinjx,∑k=1∞‍uksinkx),sinnx〉=∑i,j,k=1∞uiujuk〈G3(sinix,sinjx,sinkx,λ),sinnx〉=-λ∑i,j,k=1∞uiujuk∫0π(sinixsinjxsinkx)sinnxdx=λ4∑i,j,k=1∞uiujuk∫0π(sin(i+j+k)x-sin(i+j-k)x222222222222222226-sin(i-j+k)x222222222222222226-sin(j+k-i)x)sinnxdx={λ4∑i+j=k+nuiujuk(-3∫0π‍sin2nxdx),n=1,2,λ4(∑i+j=k=nuiujuk(∫0π‍sin2nxdx)222222+∑i+j=k+nuiujuk(-3∫0π‍sin2nxdx)),n≥3,={-3πλ8∑i+j=k+nuiujuk,n=1,2,-πλ8(3∑i+j=k+nuiujuk-∑i+j=k=nuiujuk),n≥3.
Forn=1 in (56), we have
(59)β1(λ)u1-2π·3πλ8∑i+j=k+1‍uiujuk=0,
which yields that
(60)u1=3λ4β1(λ)∑i+j=k+1‍uiujuk.
Similarly, one deduces that
(61)u2=3λ4β2(λ)∑i+j=k+2‍uiujuk,u3=λ4β3(λ)(3∑i+j=k+3‍uiujuk-u13),un=λ4βn(λ)(3∑i+j=k+n‍uiujuk-∑i+j+k=n‍uiujuk),n≥4.
Hence, by induction, we have
(62)u2=o(|u1|3),u3=-λ4β3(λ)u13+o(|u1|3),un=cnu1n+o(|u1|n),n≥4,
where cn is a constant. Then we obtain the steady-state bifurcation equation of (37) as follows:
(63)4β1(λ)u1-3λu13+o(|u1|3)=0,
and after an easy computation, we have
(64)u1=±4(λ-(1+γ))3λ+o(|u1|).
By (3) in Theorem 2, the elements in the bifurcated attractor 𝒜λ are
(65)u1sinx+o(∥u1sinx∥H)=±4(λ-(1+γ))3λsinx+o(4(λ-(1+γ))3λ).Asλ crosses the nth critical value λn=n2(1+n2γ), n=2,3,…, the local stability of the bifurcated steady states does not hold true. Nevertheless, as a direct consequence of Theorem 6, we have the following result.Theorem 12.
Equation (37) bifurcates from (0,n2(1+n2γ)), n=2,3,…, two saddle points with Morse index n on λ<n2(1+n2γ) and Morse index n-1 on λ>n2(1+n2γ).


### 3.2. Periodic Boundary Condition

In this section we are concerned with two cases: the odd-periodic boundary condition or the periodic boundary condition. Now we deal with the first case(66)∂u∂t=-γ∂4u∂x4+∂2u∂x2+λu-λu3,x∈(0,π),λ>0,u(x,t)=u(x+π,t),u(-x,t)=-u(x,t),u(x,0)=u0(x).
Here we choose the Hilbert spaces H~1 and H~ as follows:
(67)H~1={∫0π‍u∈H4(0,π):u(x)=u(x+π),2222u(-x)=-u(x),∫0π‍u(x)dx=0},H~={∫0π‍u∈L2(0,π):u(x)=u(x+π),2222u(-x)=-u(x),∫0π‍u(x)dx=0}.
It is obvious that H~1↪H~ is compact and dense. Similar to that in Section 3.1, A and Bλ are defined by
(68)Au=γ∂4u∂x4-∂2u∂x2,Bλu=λu,
and, for some α∈[0,1), G(·,λ):H~α→H~ is of the form
(69)G(u,λ)=-λu3.
Denote Lλ:=-A+Bλ, for all λ>0. Then (66) can be deformed into the following abstract evolution equation in H~1:
(70)dudt=Lλu+G(u,λ),u(0)=u0,
in which Lλ is a linear completely continuous field and a sectorial operator and symmetric and G(u,λ)=o(∥u∥H~α)(1/8<α<1), for all λ>0.Proposition 13.
The eigenvalues ofLλ in H~ are
(71)βn(λ)=λ-4n2(1+4γn2),n=1,2,…,
and the corresponding eigenvectors are
(72)en(x)=sin2nx,n=1,2,….
In addition, {en(x)}n=1∞ forms an orthogonal basis of H~1, as well as H~.Now we are in the position to state the bifurcation result for (66) at the first eigenvalue.Theorem 14.
For (66), the following assertions hold true. (1)
For0<λ≤4(1+4γ), u=0 is globally asymptotically stable.(2)
Equation (66) bifurcates from (0,4(1+4γ)), on the right side of λ=4(1+4γ), an attractor 𝒜λ, which consists of exactly two steady states u1λ and u2λ, expressed by
(73)u1λ=α(λ)sin2x+o(α(λ)),u2λ=-α(λ)sin2x+o(α(λ)),α(λ)=4(λ-4(1+4γ))3λ.(3)
For someϵ>0, if 4(1+4γ)<λ<4(1+4γ)+ϵ, there exists an open set U⊂H containing 0, which can be decomposed by the stable manifold of u=0 into two open sets U1λ and U2λ with the property that uiλ∈Uiλ, i=1,2, and
(74)limt→+∞∥u(t,ϕ)-uiλ∥H=0,ϕ∈Uiλ,
where u(t,ϕ) is the solution of (66) initiated with ϕ.Proof.
For the proof of (1) and the existence of𝒜λ, it is just the same as in Theorem 11. In what follows, we are devoted to describing the structure of 𝒜λ.
Letu∈H~1 be expressed as u(x)=∑n=1∞‍unsin2nx. Then we obtain that
(75)dundt=βn(λ)un+2π〈G(u,λ),sin2nx〉.
Proposition 13 shows that the eigenvalues here are simple. Therefore, with the help of Theorem 4, we know that the bifurcated attractor consists of exactly two steady states. Hence, we consider the following steady-state equation:
(76)βn(λ)un+2π〈G(u,λ),sin2nx〉=0.
By (57), we get(77)〈G(u,λ),sin2nx〉=〈G3(∑i=1∞‍uisin2ix,∑j=1∞‍ujsin2jx,∑k=1∞‍uksin2kx,λ),sin2nx〉=∑i,j,k=1∞‍uiujuk〈G3(sin2ix,sin2jx,sin2kx,λ),sin2nx〉=-λ∑i,j,k=1∞‍uiujuk∫0π‍(sin2ixsin2jxsin2kx)sin2nxdx=λ4∑i,j,k=1∞‍uiujuk∫0π‍(sin2(i+j+k)x-sin2(i+j-k)x222222222222222222-sin2(i-j+k)x222222222222222222-sin2(j+k-i)x)sin2nxdx={λ4∑i+j=k+n‍uiujuk(-3∫0π‍sin22nxdx),n=1,2,λ4(∑i+j+k=n‍uiujuk(∫0π‍sin22nxdx)333333+∑i+j=k+n‍uiujuk(-3∫0π‍sin22nxdx)),n≥3,={-3πλ8∑i+j=k+n‍uiujuk,n=1,2,-πλ8(3∑i+j=k+n‍uiujuk-∑i+j+k=n‍uiujuk),n≥3.
Similar to Theorem11, for n=1, we get the bifurcation equation
(78)4β1(λ)u1-3λu13+o(|u1|3)=0.
The remaining part is the same as in Theorem 11. So we omit the details.As far as thenth critical value is concerned, we have the following result.Theorem 15.
Equation  (66) bifurcates from (0,4n2(1+4n2γ)), n=2,3,…, two saddle points with Morse index n on λ<4n2(1+4n2γ) and Morse index n-1 on λ>4n2(1+4n2γ).In what follows, we turn our attention to the periodic boundary condition(79)∂u∂t=-γ∂4u∂x4+∂2u∂x2+λu-λu3,x∈(0,π),λ>0,u(x,t)=u(x+π,t),u(x,0)=u0(x).
Let
(80)H^1={u∈H4(0,π):u(x)=x(x+π),∫0π‍u(x)dx=0},H^={u∈L2(0,π):u(x)=x(x+π),∫0π‍u(x)dx=0}.
Define A and Bλ as follows:
(81)Au=γ∂4u∂x4-∂2u∂x2,Bλu=λu,
and, for α∈[0,1), G(·,λ):H^1→H^ is defined by
(82)G(u,λ)=-λu3.
Denote Lλ:=-A+Bλ, for all λ∈ℝ. Then (79) can be transformed into the following abstract evolution equation in H^1:
(83)dudt=Lλu+G(u,λ),u(0)=u0,
in which Lλ is a linear completely continuous field and a sectorial operator and symmetric, and G(u,λ)=o(∥u∥H^α) (1/8<α<1), for all λ>0.Proposition 16.
The eigenvalues ofLλ in H^ are
(84)β2n-1(λ)=β2n(λ)=λ-4n2(1+4γn2),n=1,2,…,
and the corresponding eigenvectors are
(85)e2n-1(x)=sin2nx,e2n(x)=cos2nx,n=1,2,….
Moreover, {e2n-1(x),e2n(x)}n=1∞ forms an orthogonal basis of  H^1 and H^.From (84) we know that every eigenvalue has multiplicity 2. Consider the first critical value λ1=4(1+4γ)>0. Then (79) has a 2-dimensional center manifold when λ=λ1. Reducing (79) to its center manifold and using Theorem 7, we get the following bifurcation result of (79) at (0,λ1).Theorem 17.
For (79), one has the following conclusions.(1)
For0<λ≤4(1+4γ), u=0 is globally asymptotically stable.(2)
Equation (79) bifurcates from (0,4(1+4γ)), on the right side of λ=4(1+4γ), an attractor 𝒜λ.(3)
𝒜λ≅S1 consists of steady states, which can be expressed as follows:
(86)𝒜λ={4(λ-4(1+4γ))3λα1(λ)sin2x+α2(λ)cos2x+o(|α1(λ)|,|α2(λ)|):2222α12(λ)+α22(λ)=4(λ-4(1+4γ))3λ>0}.Proof .
For the proof of (1) and (2), we can see Theorems11 or 14. Next we focus on proving (3). Since u(x)=∑n=1∞‍(unsin2nx+vncos2nx)∈H^1, reducing (79) to the 2-dimensional center manifold, we obtain that
(87)du1dt=β1(λ)u1+2π〈G(u,λ),sin2x〉,dv1dt=β1(λ)v1+2π〈G(u,λ),cos2x〉.
By (35), we get the first approximation of (87)
(88)du1dt=β1(λ)u1+2π〈G3(u1sin2x+v1cos2x,λ),sin2x〉+o(|y|3),dv1dt=β1(λ)v1+2π〈G3(u1sin2x+v1cos2x,λ),cos2x〉+o(|y|3),
where y=(u1,v1)T∈ℝ2. On the other hand,
(89)〈G3(u1sin2x+v1cos2x,λ),sin2x〉=-λ∫0π‍(u13sin32x+3u12v1sin22xcos2xvvvvvv+3u1v12sin2xcos22x+v13cos32x)sin2xdx=-3πλ8(u13+u1v12),〈G3(u1sin2x+v1cos2x,λ),cos2x〉=-λ∫0π‍(u13sin32x+3u12v1sin22xcos2x222222+3u1v12sin2xcos22x+v13cos32x)cos2xdx=-3πλ8(v13+v1u12).
Thus, we deduce that
(90)du1dt=β1(λ)u1-3λ4(u13+u1v12)+o(|y|3),dv1dt=β1(λ)v1-3λ4(v13+v1u12)+o(|y|3),
where y=(u1,v1)T∈ℝ2. Let
(91)G~(y,λ)=3λ4(u13+u1v12v13+v1u12)+o(|y|3).
Then G~(y,λ) satisfies Theorem 7 with k=3 and 0<C1<3λ/4<C2. Thus, if λ>4(1+4γ), (79) bifurcates an attractor 𝒜λ≅S1.
Finally, we prove that the attractor𝒜λ consists of steady states. It is clear that H~1⊂H^1, H~⊂H^, and (Lλ+G(·,λ))H~1⊂H~ that is, the subset H~ is invariant under Lλ+G(·,λ). Moreover, the eigenvalues of Lλ in H~1 are λ-4n2(1+4γn2), which are simple. By the arguments in odd-periodic boundary condition we know that as λ crosses the first critical value 4(1+4γ), Lλ+G(·,λ) bifurcates from (0,4(1+4γ)) two steady states
(92)uλ(x)=±α(λ)sin2x+o(α(λ)),α(λ)=4(λ-4(1+4γ))3λ.
Because of the invariance of (79) for the translation
(93)u(x,t)⟶u(x+θ,t),θ∈ℝ,
the functions
(94)uλ(x+θ)=±α(λ)sin2(x+θ)+o(α(λ)),θ∈ℝ
are steady states of (79). Moreover, the set
(95)Γ={α(λ)sin2(x+θ)+o(α(λ)):θ∈ℝ}
is a circle S1 in H^. Therefore,
(96)𝒜λ=Γ={4(λ-4(1+4λ))3λa1(λ)sin2x+a2(λ)cos2x+o(|a1(λ)|,|a2(λ)|):a12(λ)+a22(λ)=4(λ-4(1+4λ))3λ>0},
where
(97)α1(λ)=α(λ)cos2θ,α2(λ)=α(λ)sin2θ,θ∈ℝ.Asλ crosses the nth critical value 4n2(1+4γn2), similar to the proof of Theorem 17, we can obtain the following.Theorem 18.
Equation (79) bifurcates from (0,4n2(1+4γn2)), on the right side of λ=4n2(1+4γn2), an invariant set Σλ⊂H^. Furthermore, Σλ≅S1 consists of steady states of (79).


## 3.1. Dirichlet Boundary Condition

Consider the extended Fisher-Kolmogorov equation (5) with the Dirichlet boundary condition
(37)∂u∂t=-γ∂4u∂x4+∂2u∂x2+λu-λu3,x∈(0,π),λ>0,u(0,t)=u(π,t)=0,u′′(0,t)=u′′(π,t)=0,u(x,0)=u0(x).
In order to apply the bifurcation theory introduced in Section 2 to discuss the attractor bifurcation of (37), firstly we choose the Hilbert spaces H1 and H as follows:
(38)H1∶={u∈H4(0,π):u(0)=u(π)=0,22222u′′(0)=u′′(π)=0},H∶=L2(0,π),
and denote by ∥·∥H the norm on H which is induced by the inner product
(39)〈u,v〉=∫0π‍u(x)v(x)dx,∀u,v∈H.
It is well known that the embedding of H1↪H is compact and dense. Then (37) can be transformed into the following abstract evolution equation in H:
(40)dudt=Lλu+G(u,λ),u(0)=u0,
where Lλ:=-A+Bλ, for all λ>0, with A and Bλ:H1→H defined as follows:
(41)Au=γ∂4u∂x4-∂2u∂x2,Bλu=λu,
and, for some α∈[0,1), G(·,λ):Hα→H is of the form
(42)G(u,λ)=-λu3.
An easy calculation shows that the eigenvalues of -A:H1→H are
(43)δn=-n2(1+n2γ),n=1,2,…
with eigenvectors en(x)=sinnx. Hence, -A is a linear homeomorphism. On the other hand, since H1 is compactly imbedded in H, Bλ is a compact operator. Therefore, Lλ=-A+Bλ is a linear completely continuous filed.Proposition 8.
For anyλ>0, Lλ is a sectorial operator.Proof.
It is obvious that the spectrum of-A consists of eigenvalues; that is,
(44)σ(-A)={-n2(1+n2γ),n=1,2,…}.
Let u=∑n=1∞‍unen. Then
(45)(δI+A)-1u=∑n=1∞‍(δ-δn)-1unen,ifδ>δ1,
which yields that
(46)∥(δI+A)-1∥≤1δ-δ1,ifδ>δ1.
Therefore, by [36, page 12, Corollary  3.8],
(47)∥T(t)∥≤eδ1t≤1,∀t≥0,
where T(t), t≥0, is the C0-semigroup generated by -A. On the other hand, for δ∈ℂ with Reδ>0 and Imδ≠0, we have
(48)∥(δI+A)-1∥≤1|δ-δ1|≤1|Imδ|.
By [36, page 61, Theorem  5.2], -A is a sectorial operator. Furthermore, Bλ:H1→H is a linear bounded operator. Therefore, -A+Bλ is a sectorial operator.Lemma 9.
For any1/8<α<1, G(u,λ)=o(∥u∥Hα), for all λ>0.Proof.
For anyu∈Hα, by [37, Theorem  1.6.1], we have
(49)∥G(u,λ)∥H2=λ2∫0π‍u6dx≤πλ2∥u∥C06≤C12πλ2∥u∥Hα6,
where ∥·∥C0 denotes the supremum norm on [0,π], C1>0 is some constant, and α∈[0,1) satisfies 4α-1/2>0, that is, 1/8<α<1.From the hypotheses in Theorem2, we know that the eigenvalues of Lλ are vital to give rise to the bifurcation. After a simple computation, the eigenvalues and the eigenvectors of Lλ are given as follows.Proposition 10.
The eigenvalues ofLλ are
(50)βn(λ)=λ-n2(1+n2γ),n=1,2,…,
and the corresponding eigenvectors are
(51)en(x)=sinnx,n=1,2,….
Moreover, {en(x)}n=1∞ forms an orthogonal basis of H1, as well as H.From (50) we know that every eigenvalue is simple and the first eigenvalue is β1(λ)=λ-(1+γ). Take the first critical value λ1:=1+γ. Now we give the result on bifurcation for (37) when λ crosses λ1.Theorem 11.
For (37), one has the following assertions. (1)
As0<λ≤1+γ, u=0 is globally asymptotically stable.(2)
Equation (37) bifurcates from (0,1+γ) an attractor 𝒜λ on the right side of λ=1+γ, which consists of exactly two steady states u1λ and u2λ expressed as follows:
(52)u1λ=α(λ)sinx+o(α(λ)),u2λ=-α(λ)sinx+o(α(λ)),α(λ)=4(λ-(1+γ))3λ.(3)
For someϵ>0, if 1+γ<λ<1+γ+ϵ, there exists an open set U⊂H containing 0, which can be decomposed by the stable manifold of u=0 into two open sets U1λ and U2λ with the property that uiλ∈Uiλ, i=1,2, and
(53)limt→+∞∥u(t,ϕ)-uiλ∥H=0,ϕ∈Uiλ,
where u(t,ϕ) is the solution of (37) initiated with ϕ.Proof.
For any0≠u∈H1 and λ>0,
(54)〈G(u,λ),u〉=-λ∫0π‍u4dx<0.
By Theorem 3, u=0 is globally asymptotically stable as 0<λ≤1+γ.
The existence of𝒜λ follows from Propositions 8 and 10, Lemma 9, and Theorem 2 directly. Moreover, Theorem 4 indicates that the bifurcated attractor consists of exactly two steady states and (3) is also true. Now we give the formula of two steady states by reducing (40) to its local center manifold.
Letu(x)=∑n=1∞‍unsinnx∈H1. Then
(55)dundt=βn(λ)un+2π〈G(u,λ),sinnx〉,
which deduces that the steady-state equation is of the form
(56)βn(λ)un+2π〈G(u,λ),sinnx〉=0.
Since
(57)G(u,λ)=-λu3=:G3(u,λ),
where G3:H1×H1×H1→H is a 3-multilinear mapping, we have
(58)〈G(u,λ),sinnx〉=〈G3(∑i=1∞‍uisinix,∑j=1∞‍ujsinjx,∑k=1∞‍uksinkx),sinnx〉=∑i,j,k=1∞uiujuk〈G3(sinix,sinjx,sinkx,λ),sinnx〉=-λ∑i,j,k=1∞uiujuk∫0π(sinixsinjxsinkx)sinnxdx=λ4∑i,j,k=1∞uiujuk∫0π(sin(i+j+k)x-sin(i+j-k)x222222222222222226-sin(i-j+k)x222222222222222226-sin(j+k-i)x)sinnxdx={λ4∑i+j=k+nuiujuk(-3∫0π‍sin2nxdx),n=1,2,λ4(∑i+j=k=nuiujuk(∫0π‍sin2nxdx)222222+∑i+j=k+nuiujuk(-3∫0π‍sin2nxdx)),n≥3,={-3πλ8∑i+j=k+nuiujuk,n=1,2,-πλ8(3∑i+j=k+nuiujuk-∑i+j=k=nuiujuk),n≥3.
Forn=1 in (56), we have
(59)β1(λ)u1-2π·3πλ8∑i+j=k+1‍uiujuk=0,
which yields that
(60)u1=3λ4β1(λ)∑i+j=k+1‍uiujuk.
Similarly, one deduces that
(61)u2=3λ4β2(λ)∑i+j=k+2‍uiujuk,u3=λ4β3(λ)(3∑i+j=k+3‍uiujuk-u13),un=λ4βn(λ)(3∑i+j=k+n‍uiujuk-∑i+j+k=n‍uiujuk),n≥4.
Hence, by induction, we have
(62)u2=o(|u1|3),u3=-λ4β3(λ)u13+o(|u1|3),un=cnu1n+o(|u1|n),n≥4,
where cn is a constant. Then we obtain the steady-state bifurcation equation of (37) as follows:
(63)4β1(λ)u1-3λu13+o(|u1|3)=0,
and after an easy computation, we have
(64)u1=±4(λ-(1+γ))3λ+o(|u1|).
By (3) in Theorem 2, the elements in the bifurcated attractor 𝒜λ are
(65)u1sinx+o(∥u1sinx∥H)=±4(λ-(1+γ))3λsinx+o(4(λ-(1+γ))3λ).Asλ crosses the nth critical value λn=n2(1+n2γ), n=2,3,…, the local stability of the bifurcated steady states does not hold true. Nevertheless, as a direct consequence of Theorem 6, we have the following result.Theorem 12.
Equation (37) bifurcates from (0,n2(1+n2γ)), n=2,3,…, two saddle points with Morse index n on λ<n2(1+n2γ) and Morse index n-1 on λ>n2(1+n2γ).


## 3.2. Periodic Boundary Condition

In this section we are concerned with two cases: the odd-periodic boundary condition or the periodic boundary condition. Now we deal with the first case(66)∂u∂t=-γ∂4u∂x4+∂2u∂x2+λu-λu3,x∈(0,π),λ>0,u(x,t)=u(x+π,t),u(-x,t)=-u(x,t),u(x,0)=u0(x).
Here we choose the Hilbert spaces H~1 and H~ as follows:
(67)H~1={∫0π‍u∈H4(0,π):u(x)=u(x+π),2222u(-x)=-u(x),∫0π‍u(x)dx=0},H~={∫0π‍u∈L2(0,π):u(x)=u(x+π),2222u(-x)=-u(x),∫0π‍u(x)dx=0}.
It is obvious that H~1↪H~ is compact and dense. Similar to that in Section 3.1, A and Bλ are defined by
(68)Au=γ∂4u∂x4-∂2u∂x2,Bλu=λu,
and, for some α∈[0,1), G(·,λ):H~α→H~ is of the form
(69)G(u,λ)=-λu3.
Denote Lλ:=-A+Bλ, for all λ>0. Then (66) can be deformed into the following abstract evolution equation in H~1:
(70)dudt=Lλu+G(u,λ),u(0)=u0,
in which Lλ is a linear completely continuous field and a sectorial operator and symmetric and G(u,λ)=o(∥u∥H~α)(1/8<α<1), for all λ>0.Proposition 13.
The eigenvalues ofLλ in H~ are
(71)βn(λ)=λ-4n2(1+4γn2),n=1,2,…,
and the corresponding eigenvectors are
(72)en(x)=sin2nx,n=1,2,….
In addition, {en(x)}n=1∞ forms an orthogonal basis of H~1, as well as H~.Now we are in the position to state the bifurcation result for (66) at the first eigenvalue.Theorem 14.
For (66), the following assertions hold true. (1)
For0<λ≤4(1+4γ), u=0 is globally asymptotically stable.(2)
Equation (66) bifurcates from (0,4(1+4γ)), on the right side of λ=4(1+4γ), an attractor 𝒜λ, which consists of exactly two steady states u1λ and u2λ, expressed by
(73)u1λ=α(λ)sin2x+o(α(λ)),u2λ=-α(λ)sin2x+o(α(λ)),α(λ)=4(λ-4(1+4γ))3λ.(3)
For someϵ>0, if 4(1+4γ)<λ<4(1+4γ)+ϵ, there exists an open set U⊂H containing 0, which can be decomposed by the stable manifold of u=0 into two open sets U1λ and U2λ with the property that uiλ∈Uiλ, i=1,2, and
(74)limt→+∞∥u(t,ϕ)-uiλ∥H=0,ϕ∈Uiλ,
where u(t,ϕ) is the solution of (66) initiated with ϕ.Proof.
For the proof of (1) and the existence of𝒜λ, it is just the same as in Theorem 11. In what follows, we are devoted to describing the structure of 𝒜λ.
Letu∈H~1 be expressed as u(x)=∑n=1∞‍unsin2nx. Then we obtain that
(75)dundt=βn(λ)un+2π〈G(u,λ),sin2nx〉.
Proposition 13 shows that the eigenvalues here are simple. Therefore, with the help of Theorem 4, we know that the bifurcated attractor consists of exactly two steady states. Hence, we consider the following steady-state equation:
(76)βn(λ)un+2π〈G(u,λ),sin2nx〉=0.
By (57), we get(77)〈G(u,λ),sin2nx〉=〈G3(∑i=1∞‍uisin2ix,∑j=1∞‍ujsin2jx,∑k=1∞‍uksin2kx,λ),sin2nx〉=∑i,j,k=1∞‍uiujuk〈G3(sin2ix,sin2jx,sin2kx,λ),sin2nx〉=-λ∑i,j,k=1∞‍uiujuk∫0π‍(sin2ixsin2jxsin2kx)sin2nxdx=λ4∑i,j,k=1∞‍uiujuk∫0π‍(sin2(i+j+k)x-sin2(i+j-k)x222222222222222222-sin2(i-j+k)x222222222222222222-sin2(j+k-i)x)sin2nxdx={λ4∑i+j=k+n‍uiujuk(-3∫0π‍sin22nxdx),n=1,2,λ4(∑i+j+k=n‍uiujuk(∫0π‍sin22nxdx)333333+∑i+j=k+n‍uiujuk(-3∫0π‍sin22nxdx)),n≥3,={-3πλ8∑i+j=k+n‍uiujuk,n=1,2,-πλ8(3∑i+j=k+n‍uiujuk-∑i+j+k=n‍uiujuk),n≥3.
Similar to Theorem11, for n=1, we get the bifurcation equation
(78)4β1(λ)u1-3λu13+o(|u1|3)=0.
The remaining part is the same as in Theorem 11. So we omit the details.As far as thenth critical value is concerned, we have the following result.Theorem 15.
Equation  (66) bifurcates from (0,4n2(1+4n2γ)), n=2,3,…, two saddle points with Morse index n on λ<4n2(1+4n2γ) and Morse index n-1 on λ>4n2(1+4n2γ).In what follows, we turn our attention to the periodic boundary condition(79)∂u∂t=-γ∂4u∂x4+∂2u∂x2+λu-λu3,x∈(0,π),λ>0,u(x,t)=u(x+π,t),u(x,0)=u0(x).
Let
(80)H^1={u∈H4(0,π):u(x)=x(x+π),∫0π‍u(x)dx=0},H^={u∈L2(0,π):u(x)=x(x+π),∫0π‍u(x)dx=0}.
Define A and Bλ as follows:
(81)Au=γ∂4u∂x4-∂2u∂x2,Bλu=λu,
and, for α∈[0,1), G(·,λ):H^1→H^ is defined by
(82)G(u,λ)=-λu3.
Denote Lλ:=-A+Bλ, for all λ∈ℝ. Then (79) can be transformed into the following abstract evolution equation in H^1:
(83)dudt=Lλu+G(u,λ),u(0)=u0,
in which Lλ is a linear completely continuous field and a sectorial operator and symmetric, and G(u,λ)=o(∥u∥H^α) (1/8<α<1), for all λ>0.Proposition 16.
The eigenvalues ofLλ in H^ are
(84)β2n-1(λ)=β2n(λ)=λ-4n2(1+4γn2),n=1,2,…,
and the corresponding eigenvectors are
(85)e2n-1(x)=sin2nx,e2n(x)=cos2nx,n=1,2,….
Moreover, {e2n-1(x),e2n(x)}n=1∞ forms an orthogonal basis of  H^1 and H^.From (84) we know that every eigenvalue has multiplicity 2. Consider the first critical value λ1=4(1+4γ)>0. Then (79) has a 2-dimensional center manifold when λ=λ1. Reducing (79) to its center manifold and using Theorem 7, we get the following bifurcation result of (79) at (0,λ1).Theorem 17.
For (79), one has the following conclusions.(1)
For0<λ≤4(1+4γ), u=0 is globally asymptotically stable.(2)
Equation (79) bifurcates from (0,4(1+4γ)), on the right side of λ=4(1+4γ), an attractor 𝒜λ.(3)
𝒜λ≅S1 consists of steady states, which can be expressed as follows:
(86)𝒜λ={4(λ-4(1+4γ))3λα1(λ)sin2x+α2(λ)cos2x+o(|α1(λ)|,|α2(λ)|):2222α12(λ)+α22(λ)=4(λ-4(1+4γ))3λ>0}.Proof .
For the proof of (1) and (2), we can see Theorems11 or 14. Next we focus on proving (3). Since u(x)=∑n=1∞‍(unsin2nx+vncos2nx)∈H^1, reducing (79) to the 2-dimensional center manifold, we obtain that
(87)du1dt=β1(λ)u1+2π〈G(u,λ),sin2x〉,dv1dt=β1(λ)v1+2π〈G(u,λ),cos2x〉.
By (35), we get the first approximation of (87)
(88)du1dt=β1(λ)u1+2π〈G3(u1sin2x+v1cos2x,λ),sin2x〉+o(|y|3),dv1dt=β1(λ)v1+2π〈G3(u1sin2x+v1cos2x,λ),cos2x〉+o(|y|3),
where y=(u1,v1)T∈ℝ2. On the other hand,
(89)〈G3(u1sin2x+v1cos2x,λ),sin2x〉=-λ∫0π‍(u13sin32x+3u12v1sin22xcos2xvvvvvv+3u1v12sin2xcos22x+v13cos32x)sin2xdx=-3πλ8(u13+u1v12),〈G3(u1sin2x+v1cos2x,λ),cos2x〉=-λ∫0π‍(u13sin32x+3u12v1sin22xcos2x222222+3u1v12sin2xcos22x+v13cos32x)cos2xdx=-3πλ8(v13+v1u12).
Thus, we deduce that
(90)du1dt=β1(λ)u1-3λ4(u13+u1v12)+o(|y|3),dv1dt=β1(λ)v1-3λ4(v13+v1u12)+o(|y|3),
where y=(u1,v1)T∈ℝ2. Let
(91)G~(y,λ)=3λ4(u13+u1v12v13+v1u12)+o(|y|3).
Then G~(y,λ) satisfies Theorem 7 with k=3 and 0<C1<3λ/4<C2. Thus, if λ>4(1+4γ), (79) bifurcates an attractor 𝒜λ≅S1.
Finally, we prove that the attractor𝒜λ consists of steady states. It is clear that H~1⊂H^1, H~⊂H^, and (Lλ+G(·,λ))H~1⊂H~ that is, the subset H~ is invariant under Lλ+G(·,λ). Moreover, the eigenvalues of Lλ in H~1 are λ-4n2(1+4γn2), which are simple. By the arguments in odd-periodic boundary condition we know that as λ crosses the first critical value 4(1+4γ), Lλ+G(·,λ) bifurcates from (0,4(1+4γ)) two steady states
(92)uλ(x)=±α(λ)sin2x+o(α(λ)),α(λ)=4(λ-4(1+4γ))3λ.
Because of the invariance of (79) for the translation
(93)u(x,t)⟶u(x+θ,t),θ∈ℝ,
the functions
(94)uλ(x+θ)=±α(λ)sin2(x+θ)+o(α(λ)),θ∈ℝ
are steady states of (79). Moreover, the set
(95)Γ={α(λ)sin2(x+θ)+o(α(λ)):θ∈ℝ}
is a circle S1 in H^. Therefore,
(96)𝒜λ=Γ={4(λ-4(1+4λ))3λa1(λ)sin2x+a2(λ)cos2x+o(|a1(λ)|,|a2(λ)|):a12(λ)+a22(λ)=4(λ-4(1+4λ))3λ>0},
where
(97)α1(λ)=α(λ)cos2θ,α2(λ)=α(λ)sin2θ,θ∈ℝ.Asλ crosses the nth critical value 4n2(1+4γn2), similar to the proof of Theorem 17, we can obtain the following.Theorem 18.
Equation (79) bifurcates from (0,4n2(1+4γn2)), on the right side of λ=4n2(1+4γn2), an invariant set Σλ⊂H^. Furthermore, Σλ≅S1 consists of steady states of (79).



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