Attractor bifurcation for the extended Fisher-Kolmogorov equation with periodic boundary condition

  • Qiang Zhang1Email author and

    Affiliated with

    • Hong Luo2

      Affiliated with

      Boundary Value Problems20132013:169

      DOI: 10.1186/1687-2770-2013-169

      Received: 24 October 2012

      Accepted: 5 July 2013

      Published: 19 July 2013

      Abstract

      In this paper, we study the bifurcation and stability of solutions of the extended Fisher-Kolmogorov equation with periodic boundary condition. We prove that the system bifurcates from the trivial solution to an attractor as parameter crosses certain critical value. The topological structure of the attractor is also investigated.

      MSC:35B32, 35K35, 37G35.

      Keywords

      extended Fisher-Kolmogorov equation periodic boundary condition attractor bifurcation center manifold

      1 Introduction

      In this paper we work with the extended Fisher-Kolmogorov type equation with periodic boundary condition, which reads
      { u t = μ 4 x 4 u + α 2 x 2 u + λ u + g ( u ) , ( x , t ) R × ( 0 , ) , 0 2 π u ( x , t ) d x = 0 , t 0 , u ( x , t ) = u ( x + 2 k π , t ) , k Z , u ( x , 0 ) = u 0 , x R , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_Equ1_HTML.gif
      (1.1)
      where u = u ( x , t ) : R × [ 0 , ) R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_IEq1_HTML.gif is an unknown function, μ > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_IEq2_HTML.gif, α > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_IEq3_HTML.gif are constants, λ R + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_IEq4_HTML.gif is the system parameter. g ( s ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_IEq5_HTML.gif is a polynomial on s R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_IEq6_HTML.gif, which is given by
      g ( s ) = k = 2 p a k s k , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_Equa_HTML.gif

      where 2 p N http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_IEq7_HTML.gif and a k http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_IEq8_HTML.gif are given constants.

      The extended Fisher-Kolmogorov (EFK) equation has been proposed as a model for phase transitions and other bistable phenomena [13]. It has been extensively studied during past decades. Kalies and van der Vorst [4] considered the steady-state problem; by analyzing the variational structure, they proved the existence of heteroclinic connections, which are the critical points of a certain functional. Also, by the variational method, Tersian and Chaparova [5] derived the existence of periodic and homoclinic solutions. Peletier and Troy [6] were interested in the stationary spatially periodic patterns and showed that the structure of the solutions is enriched by increasing the coefficient of the fourth-order derivative term. The structure of the solution set was also investigated by van den Berg [7], who enumerated all the possible bounded stationary solutions provided this coefficient is small. Rottschäer and Wayne [8] showed that for every positive wavespeed there exists a traveling wave. And they also found the critical wavespeed to discriminate the monotonic solution from the oscillatory one. By an iteration procedure, Luo and Zhang [9] proved that equation (1.1) possesses a global attractor in the Sobolev space H k http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_IEq9_HTML.gif for all k > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_IEq10_HTML.gif provided that a p < 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_IEq11_HTML.gif and p is an odd number. We refer the interested readers to the references in [49] for other results on the EFK equation; see also, among others, [1013].

      Returning to problem (1.1), our main interest in the present paper is the bifurcation and stability of solutions. By using a notion of bifurcation called attractor bifurcation developed by Ma and Wang in [14, 15], a nonlinear attractor bifurcation theory for this problem is established. Work on the topic of attractor bifurcation also can be seen in [16, 17].

      The main objectives of this theory include:
      1. (1)

        existence of attractor bifurcation when the system parameter crosses some critical number,

         
      2. (2)

        dynamic stability of bifurcated solutions, and

         
      3. (3)

        the topological structure of the bifurcated attractor.

         
      Our main results can be summarized as follows.
      1. 1.

        If λ μ + α http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_IEq12_HTML.gif, the steady state u = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_IEq13_HTML.gif is locally asymptotically stable.

         
      2. 2.

        As λ crosses μ + α http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_IEq14_HTML.gif, i.e., there exists an ϵ > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_IEq15_HTML.gif such that for any μ + α < λ < λ + ϵ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_IEq16_HTML.gif, system (1.1) bifurcates from the trivial solution to an attractor Σ λ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_IEq17_HTML.gif.

         
      3. 3.

        Σ λ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_IEq17_HTML.gif is homeomorphic to S 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_IEq18_HTML.gif and consists of exactly one cycle of steady solutions of (1.1).

         

      Moreover, we apply this theory to a model of the population density for single-species and derive biological results.

      This article is organized as follows. The preliminaries are given in Section 2. The mathematical setting is presented in Section 3. The mathematical results are given in Section 4. In Section 5 we apply mathematical results to a model of the population density for single-species and derive biological results. In Section 6 we discuss some existing results and compare them with ours. Finally, Section 7 is devoted to the conclusions.

      2 Preliminaries

      We begin with the definition of attractor bifurcation which was first proposed by Ma and Wang in [14, 15].

      Let H and H 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_IEq19_HTML.gif be two Hilbert spaces, and let H 1 H http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_IEq20_HTML.gif be a dense and compact inclusion. We consider the following nonlinear evolution equations
      { d u d t = L λ u + G ( u ) , u ( 0 ) = u 0 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_Equ2_HTML.gif
      (2.1)
      where u : [ 0 , ) H http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_IEq21_HTML.gif is the unknown function, λ R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_IEq22_HTML.gif is the system parameter, and L λ : H 1 H http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_IEq23_HTML.gif are parameterized linear completely continuous fields depending continuously on λ, which satisfy
      { L λ = A + B λ a sectorial operator , A : H 1 H a linear homeomorphism , B λ : H 1 H parameterized linear compact operators . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_Equ3_HTML.gif
      (2.2)

      Since L λ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_IEq24_HTML.gif is a sectorial operator which generates an analytic semigroup S λ ( t ) = { e t L λ } t 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_IEq25_HTML.gif for any λ R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_IEq26_HTML.gif, we can define fractional power operators ( L λ ) μ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_IEq27_HTML.gif for 0 μ 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_IEq28_HTML.gif with domain H μ = D ( ( L λ ) μ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_IEq29_HTML.gif such that H μ 1 H μ 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_IEq30_HTML.gif if μ 1 > μ 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_IEq31_HTML.gif, and H 0 = H http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_IEq32_HTML.gif (see [18, 19]).

      In addition, we assume that the nonlinear terms G : H α H http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_IEq33_HTML.gif for some 0 α < 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_IEq34_HTML.gif are a family of parameterized C r http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_IEq35_HTML.gif bounded operators ( r 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_IEq36_HTML.gif) such that
      G ( u ) = o ( u H α ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_Equ4_HTML.gif
      (2.3)

      Definition 2.1 [15]

      A set Σ H http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_IEq37_HTML.gif is called an invariant set of (2.1) if S ( t ) Σ = Σ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_IEq38_HTML.gif for any t 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_IEq39_HTML.gif. An invariant set Σ H http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_IEq37_HTML.gif of (2.1) is said to be an attractor if Σ is compact, and there exists a neighborhood of W H http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_IEq40_HTML.gif of Σ such that for any φ W http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_IEq41_HTML.gif we have
      lim t dist H ( u ( t , φ ) , Σ ) = 0 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_Equb_HTML.gif

      where dist H ( u ( t , φ ) , Σ ) = inf v Σ u ( t , φ ) v H http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_IEq42_HTML.gif, t 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_IEq43_HTML.gif.

      Definition 2.2 [15]
      1. (1)
        We say that the solution to equation (2.1) bifurcates from ( u , λ ) = ( 0 , λ 0 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_IEq44_HTML.gif to an invariant set Σ λ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_IEq17_HTML.gif if there exists a sequence of invariant sets { Σ λ n } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_IEq45_HTML.gif of (2.1) such that 0 Σ λ n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_IEq46_HTML.gif, and
        lim n λ n = λ 0 , lim n max v Σ λ n v H = 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_Equc_HTML.gif
         
      2. (2)

        If the invariant sets Σ λ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_IEq17_HTML.gif are attractors of (2.1), then the bifurcation is called attractor bifurcation.

         

      To prove the main result, we introduce an important theorem.

      Let the eigenvalues (counting multiplicity) of L λ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_IEq24_HTML.gif be given by
      β k ( λ ) C ( k 1 ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_Equd_HTML.gif
      and the principle of exchange of stabilities holds true:
      Re β i ( λ ) { < 0 , if  λ < λ 0 , = 0 , if  λ = λ 0 ( 1 i m ) , > 0 , if  λ > λ 0 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_Equ5_HTML.gif
      (2.4)
      Re β j ( λ 0 ) < 0 , j m + 1 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_Equ6_HTML.gif
      (2.5)
      Let the eigenspace of L λ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_IEq24_HTML.gif at λ = λ 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_IEq47_HTML.gif be
      E 0 = 1 j m k = 1 { u , v H 1 | ( L λ 0 β j ( λ 0 ) ) k w = 0 , w = u + i v } . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_Eque_HTML.gif

      It is known that dim E 0 = m http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_IEq48_HTML.gif.

      The following attractor bifurcation theorem can be found in [15].

      Theorem 2.1 Let H 1 = H = R n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_IEq49_HTML.gif, conditions (2.4), (2.5) hold true, and u = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_IEq13_HTML.gif is a locally asymptotically stable equilibrium point of (2.1) at λ = λ 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_IEq47_HTML.gif. Then the following assertions hold true:
      1. (1)

        Equation (2.1) bifurcates from ( u , λ ) = ( 0 , λ 0 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_IEq44_HTML.gif to attractors Σ λ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_IEq17_HTML.gif for λ > λ 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_IEq50_HTML.gif, with dimension m 1 dim Σ λ m http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_IEq51_HTML.gif, which is connected as m > 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_IEq52_HTML.gif.

         
      2. (2)

        The attractor Σ λ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_IEq17_HTML.gif is a limit of a sequence of m-dimensional annuli A k http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_IEq53_HTML.gif with A k + 1 A k http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_IEq54_HTML.gif; especially, if Σ λ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_IEq17_HTML.gif is a finite simplicial complex, then Σ λ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_IEq17_HTML.gif has the homology type of the ( m 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_IEq55_HTML.gif-dimensional sphere S m 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_IEq56_HTML.gif.

         
      3. (3)
        For any u λ Σ λ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_IEq57_HTML.gif, u λ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_IEq58_HTML.gif can be expressed as
        u λ = v λ + o ( v λ H 1 ) , v λ E 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_Equf_HTML.gif
         
      4. (4)

        If u = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_IEq59_HTML.gif is globally asymptotically stable for (2.1) at λ = λ 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_IEq47_HTML.gif, then for any bounded open set U H http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_IEq60_HTML.gif with 0 U http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_IEq61_HTML.gif, there is an ϵ > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_IEq62_HTML.gif such that λ 0 < λ < λ 0 + ϵ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_IEq63_HTML.gif, the attractor Σ λ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_IEq17_HTML.gif attracts U Γ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_IEq64_HTML.gif in H, where Γ is the stable manifold of u = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_IEq13_HTML.gif with codimension m. In particular, if (2.1) has a global attractor for all λ near λ 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_IEq65_HTML.gif, then U = H http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_IEq66_HTML.gif.

         

      Remark 2.1 As H 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_IEq19_HTML.gif and H are infinite dimensional Hilbert spaces, if (2.1) satisfies conditions (2.2)-(2.5) and u = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_IEq59_HTML.gif is a locally (global) asymptotically stable equilibrium point of (2.1) at λ = λ 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_IEq47_HTML.gif, then the assertions (1)-(4) of Theorem 2.1 hold; see [14, 15].

      To get the structure of the bifurcated solutions, we introduce another theorem.

      Let v be a two-dimensional C r http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_IEq35_HTML.gif ( r 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_IEq36_HTML.gif) vector field given by
      v λ ( x ) = λ x F ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_Equ7_HTML.gif
      (2.6)
      for x R 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_IEq67_HTML.gif. Here
      F ( x ) = F k ( x ) + o ( | x | k ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_Equ8_HTML.gif
      (2.7)
      where F k http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_IEq68_HTML.gif is a k-multilinear field, which satisfies the inequality
      C 1 | x | k + 1 F k ( x ) , x C 2 | x | k + 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_Equ9_HTML.gif
      (2.8)

      for some constants 0 < C 1 < C 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_IEq69_HTML.gif and k = 2 m + 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_IEq70_HTML.gif, m 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_IEq71_HTML.gif.

      Theorem 2.2 (Theorem 5.10 in [15])

      Under conditions (2.7), (2.8), the vector field (2.6) bifurcates from ( x , λ ) = ( 0 , 0 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_IEq72_HTML.gif to an attractor Σ λ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_IEq17_HTML.gif for λ > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_IEq73_HTML.gif, which is homeomorphic to S 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_IEq18_HTML.gif. Moreover, one and only one of the following conclusions is true:
      1. (1)

        Σ λ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_IEq17_HTML.gif is a period orbit.

         
      2. (2)

        Σ λ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_IEq17_HTML.gif consists of infinitely many singular points.

         
      3. (3)

        Σ λ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_IEq17_HTML.gif contains at most 2 ( k + 1 ) = 4 ( m + 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_IEq74_HTML.gif singular points and has 4 N + n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_IEq75_HTML.gif ( N + n 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_IEq76_HTML.gif) singular points, 2N of which are saddle points, 2N of which are stable node points (possibly degenerate), and n of which have index zero.

         

      3 Mathematical setting

      Let
      H = L 2 ( 0 , 2 π ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_Equg_HTML.gif
      and
      H 1 = { u H 4 ( 0 , 2 π ) | u ( x + 2 π ) = u ( x ) , 0 2 π u d x = 0 } . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_Equh_HTML.gif
      We define L λ = A + B λ : H 1 H http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_IEq77_HTML.gif and G : H 1 H http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_IEq78_HTML.gif by
      { A u = μ d 4 d x 4 u + α d 2 d x 2 u , B λ u = λ u , G ( u ) = g ( u ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_Equ10_HTML.gif
      (3.1)
      Consequently, we have an operator equation which is equivalent to problem (1.1) as follows:
      { d u d t = L λ u + G ( u ) , u ( 0 ) = u 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_Equ11_HTML.gif
      (3.2)

      4 Mathematical results

      As mentioned in the introduction, we study in this manuscript attractor bifurcation of the EFK equation under the periodic boundary condition. Then we have the following bifurcation theorem.

      Theorem 4.1 For problem (1.1), if 2 a 2 2 + 45 μ a 3 + 9 α a 3 < 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_IEq79_HTML.gif is satisfied, then the following assertions hold true:
      1. (1)

        If λ μ + α http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_IEq12_HTML.gif, the steady state u = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_IEq13_HTML.gif is locally asymptotically stable.

         
      2. (2)

        If λ > μ + α http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_IEq80_HTML.gif, system (1.1) bifurcates from the trivial solution u = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_IEq13_HTML.gif to an attractor Σ λ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_IEq17_HTML.gif.

         
      3. (3)

        Σ λ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_IEq17_HTML.gif is homeomorphic to S 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_IEq18_HTML.gif and consists of exactly one cycle of steady solutions of (1.1).

         
      4. (4)
        Σ λ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_IEq17_HTML.gif can be expressed as
        Σ λ = { x ˜ cos ( x + θ ) + o ( | x ˜ | ) | θ R } , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_Equi_HTML.gif
         

      where x ˜ = 4 ( 16 μ + 4 α λ ) ( μ + α λ ) 3 a 3 ( 16 μ + 4 α λ ) + 2 a 2 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_IEq81_HTML.gif ( a 2 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_IEq82_HTML.gif), or x ˜ = 4 ( μ + α λ ) 3 a 3 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_IEq83_HTML.gif ( a 2 = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_IEq84_HTML.gif), and μ + α < λ < μ + α + ϵ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_IEq85_HTML.gif, ϵ is sufficiently small.

      Proof of Theorem 4.1 We shall prove Theorem 4.1 in four steps.

      Step 1. In this step, we study the eigenvalue problem of the linearized equation of (3.2) and find the eigenvectors and the critical value of λ.

      Consider the eigenvalue problem of the linear equation,
      L λ u = β u . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_Equ12_HTML.gif
      (4.1)
      It is not difficult to find that the eigenvalues and the normalized eigenvectors of (4.1) are
      { β 2 k 1 = β 2 k = λ μ k 4 α k 2 , k = 1 , 2 , , e 2 k 1 = sin k x π , e 2 k = cos k x π , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_Equ13_HTML.gif
      (4.2)
      under condition that we get the principle of exchange of stabilities
      β 1 ( λ ) = β 2 ( λ ) { < 0 , λ < μ + α , = 0 , λ = μ + α , > 0 , λ > μ + α , β j ( μ + α ) < 0 , j 3 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_Equj_HTML.gif

      Step 2. We verify that for any λ R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_IEq22_HTML.gif, operator L λ + G http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_IEq86_HTML.gif satisfies conditions (2.2) and (2.3).

      Thanks to the results in [9, 18, 19], we know that the operator L λ : H 1 H http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_IEq87_HTML.gif is a sectorial operator which implies that condition (2.2) holds true.

      It is easy to get the following inequality:
      G ( u ) H 2 = 0 2 π | g ( u ) | 2 d x C 0 2 π ( k = 2 p | u | 2 k ) d x C k = 2 p u L 2 k ( 0 , 2 π ) 2 k C k = 2 p u H 1 2 2 k , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_Equk_HTML.gif
      which implies that G ( u ) = o ( u H 1 2 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_IEq88_HTML.gif, where
      H 1 2 = { u H 2 ( 0 , 2 π ) | 0 2 π u d x = 0 , u ( x + 2 π ) = u ( x ) } , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_Equl_HTML.gif

      then condition (2.3) holds true.

      Step 3. In this part, we prove the existence of attractor bifurcation and analyze the topological structure of the attractor Σ λ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_IEq17_HTML.gif.

      Let E 1 λ = E 0 = span { e 1 , e 2 } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_IEq89_HTML.gif, E 2 λ = E 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_IEq90_HTML.gif. Let Φ be the center manifold function, in the neighborhood of ( u , λ ) = ( 0 , μ + α ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_IEq91_HTML.gif, we have
      u = y + Φ ( y ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_Equm_HTML.gif

      where y = x 1 e 1 + x 2 e 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_IEq92_HTML.gif.

      Then the reduction equations of (3.2) are as follows:
      { d x 1 d t = ( λ μ α ) x 1 + G ( u ) , e 1 , d x 2 d t = ( λ μ α ) x 2 + G ( u ) , e 2 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_Equ14_HTML.gif
      (4.3)

      To get the exact form of the reduction equations, we need to obtain the expression of G ( u ) , e 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_IEq93_HTML.gif and G ( u ) , e 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_IEq94_HTML.gif.

      Let G 2 : H 1 × H 1 H http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_IEq95_HTML.gif and G 3 : H 1 × H 1 × H 1 H http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_IEq96_HTML.gif be the bilinear and trilinear operators of G respectively, i.e.,
      G 2 ( u 1 , u 2 ) = a 2 u 1 u 2 , G 3 ( u 1 , u 2 , u 3 ) = a 3 u 1 u 2 u 3 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_Equn_HTML.gif
      Since
      G 2 ( y , y ) , e 1 = G 2 ( y , y ) , e 2 = 0 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_Equo_HTML.gif

      the first order approximation of (4.3) does not work. Now, we shall find out the second order approximation of (4.3). And the most important of all is to obtain the approximation expression of the center manifold function.

      By direct calculation, we have
      G 2 ( y , y ) , e k = { a 2 π x 1 x 2 , k = 3 , a 2 2 π x 2 2 a 2 2 π x 1 2 , k = 4 , 0 , k 3 , 4 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_Equp_HTML.gif
      According to the formula of Theorem 3.8 in [15] (or Remark 4.1), the center manifold function Φ, in the neighborhood of ( u , λ ) = ( 0 , μ + α ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_IEq97_HTML.gif, can be expressed as
      Φ ( y ) = k = 3 β k 1 G 2 ( y , y ) , e k e k + O ( ( | β 1 | 2 + | β 2 | 2 ) | y | 2 ) + o ( | y | 2 ) = ( λ 16 μ 4 α ) 1 a 2 2 π ( 2 x 1 x 2 e 3 + x 2 2 e 4 x 1 2 e 4 ) + O ( | λ μ α | 2 ( | x 1 | 2 + | x 2 | 2 ) ) + o ( | x 1 | 2 + | x 2 | 2 ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_Equq_HTML.gif
      In the following, we calculate G ( u ) , e j http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_IEq98_HTML.gif, j = 1 , 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_IEq99_HTML.gif.
      G ( u ) , e j = G 2 ( y , Φ ( y ) ) , e j + G 2 ( Φ ( y ) , y ) , e j + G 3 ( y , y , y ) , e j + O ( | λ μ α | 2 ( | x 1 | 3 + | x 2 | 3 ) ) + o ( | x 1 | 3 + | x 2 | 3 ) , j = 1 , 2 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_Equr_HTML.gif
      By direct calculation, we have
      G 2 ( y , Φ ( y ) ) , e 1 = G 2 ( Φ ( y ) , y ) , e 1 = ( λ 16 μ 4 α ) 1 a 2 2 4 π x 1 3 ( λ 16 μ 4 α ) 1 a 2 2 4 π x 1 x 2 2 + O ( | λ μ α | 2 ( | x 1 | 3 + | x 2 | 3 ) ) + o ( | x 1 | 3 + | x 2 | 3 ) , G 2 ( y , Φ ( y ) ) , e 2 = G 2 ( Φ ( y ) , y ) , e 2 = ( λ 16 μ 4 α ) 1 a 2 2 4 π x 2 3 ( λ 16 μ 4 α ) 1 a 2 2 4 π x 1 2 x 2 + O ( | λ μ α | 2 ( | x 1 | 3 + | x 2 | 3 ) ) + o ( | x 1 | 3 + | x 2 | 3 ) , G 3 ( y , y , y ) , e 1 = 3 a 3 4 π x 1 3 + 3 a 3 4 π x 1 x 2 2 , G 3 ( y , y , y ) , e 2 = 3 a 3 4 π x 2 3 + 3 a 3 4 π x 1 2 x 2 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_Equs_HTML.gif
      then we obtain the expression of G ( u ) , e j http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_IEq98_HTML.gif, j = 1 , 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_IEq99_HTML.gif.
      G ( u ) , e 1 = A x 1 3 + A x 1 x 2 2 + O ( | λ μ α | 2 ( | x 1 | 3 + | x 2 | 3 ) ) + o ( | x 1 | 3 + | x 2 | 3 ) , G ( u ) , e 2 = A x 1 2 x 2 + A x 2 3 + O ( | λ μ α | 2 ( | x 1 | 3 + | x 2 | 3 ) ) + o ( | x 1 | 3 + | x 2 | 3 ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_Equ15_HTML.gif
      (4.4)

      where A = ( λ 16 μ 4 α ) 1 a 2 2 2 π + 3 a 3 4 π http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_IEq100_HTML.gif.

      Putting (4.4) into (4.3), we have the reduction equations
      { d x 1 d t = ( λ μ α ) x 1 + A x 1 3 + A x 1 x 2 2 d x 1 d t = + O ( | λ μ α | 2 ( | x 1 | 3 + | x 2 | 3 ) ) + o ( | x 1 | 3 + | x 2 | 3 ) , d x 2 d t = ( λ μ α ) x 2 + A x 1 2 x 2 + A x 2 3 d x 2 d t = + O ( | λ μ α | 2 ( | x 1 | 3 + | x 2 | 3 ) ) + o ( | x 1 | 3 + | x 2 | 3 ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_Equ16_HTML.gif
      (4.5)

      For the case of λ < μ + α http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_IEq101_HTML.gif, it is obvious that u = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_IEq13_HTML.gif is locally asymptotically stable. For the case of λ = μ + α http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_IEq102_HTML.gif, if 2 a 2 2 + ( 45 μ + 9 α ) a 3 < 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_IEq103_HTML.gif, which implies that A < 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_IEq104_HTML.gif, then u = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_IEq13_HTML.gif is also locally asymptotically stable. Assertion (1) of Theorem 4.1 is proved.

      Since the following equality holds true:
      x 1 ( A x 1 3 + A x 1 x 2 2 ) + x 2 ( A x 1 2 x 2 + A x 2 3 ) = A ( x 1 2 + x 2 2 ) 2 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_Equt_HTML.gif

      according to Theorems 2.1, 2.2 and Remark 2.1, we can conclude that if λ > μ + α http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_IEq80_HTML.gif, equation (1.1) bifurcates from u = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_IEq13_HTML.gif to an attractor Σ λ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_IEq17_HTML.gif, which is homeomorphic to S 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_IEq18_HTML.gif.

      Step 4. In the last step, we show that the bifurcated attractor of (3.2) consists of a singularity cycle.

      Since the even function space is an invariant subspace of L λ + G http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_IEq86_HTML.gif defined by (3.1), we shall consider the bifurcation problem in the even function space and prove that system (1.1) bifurcates from ( u , λ ) = ( 0 , μ + α ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_IEq105_HTML.gif to two steady solutions. For any function v in the even function space can be expressed as follows:
      v = k 1 x 2 k e 2 k , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_Equu_HTML.gif
      by the Lyapunov-Schmidt reduction method used in Step 3, we can deduce that the reduction equation of (1.1) is as follows:
      d x 2 d t = ( λ μ α ) x 2 + A x 2 3 + O ( | λ μ α | 2 | x 2 | 3 ) + o ( | x 2 | 3 ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_Equ17_HTML.gif
      (4.6)

      which implies that (1.1) bifurcates from ( u , λ ) = ( 0 , μ + α ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_IEq105_HTML.gif to two steady solutions V λ ± ( x , t ) = ± 4 ( 16 μ + 4 α λ ) ( μ + α λ ) 3 a 3 ( 16 μ + 4 α λ ) + 2 a 2 2 cos x + h.o.t. http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_IEq106_HTML.gif in the space of even functions.

      Since the solutions of (2.1) are translation invariant,
      V λ + ( x , t ) V λ + ( x + θ , t ) , θ R , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_Equv_HTML.gif
      the set
      T = { V λ + ( x + θ , t ) | θ R } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_Equw_HTML.gif

      represents S 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_IEq18_HTML.gif in H 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_IEq19_HTML.gif, which implies that λ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_IEq107_HTML.gif consists of exactly one circle of steady solutions of (1.1). This completes the proof of Theorem 4.1. □

      Remark 4.1 Suppose that { e i } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_IEq108_HTML.gif, the generalized eigenvectors of L λ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_IEq24_HTML.gif, form a basis of H with the dual basis { e i } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_IEq109_HTML.gif such that
      e i , e j H { = 0 , if  i j , 0 , if  i = j . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_Equx_HTML.gif
      We have
      v = x + y E 1 λ E 2 λ , x = i = 1 m x i e i E 1 λ , y = i = m + 1 x i e i E 2 λ . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_Equy_HTML.gif
      Then near λ = λ 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_IEq47_HTML.gif, the center manifold function ϕ ( x , λ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_IEq110_HTML.gif in Theorem 3.8 in [15] can be expressed as follows:
      ϕ ( x , λ ) = j = m + 1 ϕ j ( x , λ ) e j + O ( | Re β ( λ ) | x k ) + o ( x k ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_Equ18_HTML.gif
      (4.7)
      where
      ϕ j ( x , λ ) = 1 β j ( λ ) G k ( x , , x ) , e j H . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_Equz_HTML.gif

      Remark 4.2 If g ( s ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_IEq5_HTML.gif in (1.1) is not a polynomial but a C ω http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_IEq111_HTML.gif with Taylor’s expansion in s = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_IEq112_HTML.gif as g ( s ) = k = 2 a k s k http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_IEq113_HTML.gif; if 2 a 2 2 + 45 μ a 3 + 9 α a 3 < 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_IEq79_HTML.gif is satisfied, then the conclusions of Theorem 4.1 also hold true.

      Remark 4.3 If the higher order terms k = 4 p a k u k http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_IEq114_HTML.gif in g ( u ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_IEq115_HTML.gif are omitted, from the proof of Theorem 4.1, it is easy to see that the conclusions of Theorem 4.1 also hold true.

      5 Applications

      In this section, we apply Theorem 4.1 to a model of the population density for single-species as follows:
      { v t = μ 4 x 4 v + α 2 x 2 v + b 1 v + b 2 v 2 + a 3 v 3 + b 0 , ( x , t ) R × ( 0 , ) , 0 2 π v ( x , t ) d x = a 2 2 a 3 π , t 0 , v ( x , t ) = v ( x + 2 k π , t ) , k Z , v ( x , 0 ) = u 0 + v 0 , x R , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_Equ19_HTML.gif
      (5.1)

      where μ, α are the diffusion coefficients, v is the population density for single-species, and a 2 < 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_IEq116_HTML.gif, a 3 < 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_IEq117_HTML.gif, b 0 = λ a 2 4 a 3 + 3 64 a 2 3 a 3 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_IEq118_HTML.gif, b 1 = λ 5 16 a 2 2 a 3 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_IEq119_HTML.gif, b 2 = a 2 4 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_IEq120_HTML.gif. It is easy to see that b 0 < 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_IEq121_HTML.gif, b 1 > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_IEq122_HTML.gif and b 2 < 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_IEq123_HTML.gif. Inspired by the work of Murray [20], b 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_IEq124_HTML.gif represents the birth rate, b 2 v 2 + a 3 v 3 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_IEq125_HTML.gif describes the intra specific competition, and b 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_IEq126_HTML.gif stands for the emigration which arises from disease.

      It is not difficult to verify that v 0 = a 2 4 a 3 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_IEq127_HTML.gif is a positive steady solution of system (5.1). From the translation
      u ( x , t ) = v ( x , t ) v 0 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_Equ20_HTML.gif
      (5.2)
      we derive the following system:
      { u t = μ 4 x 4 u + α 2 x 2 u + λ u + a 2 u 2 + a 3 u 3 , ( x , t ) R × ( 0 , ) , 0 2 π u ( x , t ) d x = 0 , t 0 , u ( x , t ) = u ( x + 2 k π , t ) , k Z , u ( x , 0 ) = u 0 , x R . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_Equ21_HTML.gif
      (5.3)

      According to Remark 4.3, if the condition 2 a 2 2 + 45 μ a 3 + 9 α a 3 < 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_IEq79_HTML.gif is satisfied, the conclusions of Theorem 4.1 for system (5.3) also hold true. Consequently, from the translation (5.2), we have the following results for (5.1).

      Theorem 5.1 For problem (5.1), if 2 a 2 2 + 45 μ a 3 + 9 α a 3 < 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_IEq79_HTML.gif is satisfied, then the following assertions hold true:
      1. (1)

        If b 1 μ + α 5 16 a 2 2 a 3 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_IEq128_HTML.gif, the steady state v 0 = a 2 4 a 3 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_IEq127_HTML.gif is locally asymptotically stable (Figure 1).

         
      2. (2)

        If b 1 > μ + α 5 16 a 2 2 a 3 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_IEq129_HTML.gif, system (5.1) bifurcates from the solution v 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_IEq130_HTML.gif to an attractor Σ b 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_IEq131_HTML.gif. This implies that the stability will switch from the original state (i.e., v 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_IEq130_HTML.gif) to a new one (i.e., Σ b 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_IEq131_HTML.gif) (Figure 1).

         
      3. (3)

        Σ b 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_IEq131_HTML.gif is homeomorphic to S 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_IEq18_HTML.gif and consists of exactly one cycle of steady solutions of (5.1) (Figure 1).

         
      4. (4)
        Σ b 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_IEq131_HTML.gif can be expressed as
        Σ b 1 = { v 0 + x ˜ cos ( x + θ ) + o ( | x ˜ | ) | θ R } , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_Equaa_HTML.gif
         

      where x ˜ = 4 ( 16 μ + 4 α λ ) ( μ + α λ ) 3 a 3 ( 16 μ + 4 α λ ) + 2 a 2 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_IEq81_HTML.gif, and μ + α < λ < μ + α + ϵ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_IEq85_HTML.gif, ϵ is sufficiently small.

      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_Fig1_HTML.jpg
      Figure 1

      Bifurcation diagram for the model of the population density for single-species. (1) Bifurcation appears at λ 0 = μ + α 5 16 a 2 2 a 3 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_IEq132_HTML.gif. (2) Bifurcated attractor Σ b 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_IEq131_HTML.gif is the boundary of the shaded region. (3) The first horizontal solid line from above denotes that the solution v = v 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_IEq133_HTML.gif is stable, and the horizontal dotted line means this solution is unstable.

      Furthermore, Theorem 5.1 and the equality
      0 2 π v ( x , t ) d x = a 2 2 a 3 π , t 0 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_Equab_HTML.gif

      yield the following biological results.

      Biological results For the model (5.1), if 2 a 2 2 + 45 μ a 3 + 9 α a 3 < 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_IEq79_HTML.gif is satisfied, we have the following assertions:
      1. (1)

        The population of this single-species is a conservative quantity.

         
      2. (2)

        If the birth rate is low, then the population density will keep a uniform spatial distribution (Figure 2(A)).

         
      3. (3)

        If the birth rate becomes high enough, then the spatial distribution of the population density will not keep uniform but change periodically with space (Figure 2(B)).

         
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_Fig2_HTML.jpg
      Figure 2

      The spatial distribution of the population density. (1) Figure 2(A) shows that the population density keeps a uniform spatial distribution when the birth rate is low. (2) Figure 2(B) shows that the population density changes periodically with space when the birth rate becomes high enough. (3) The area of the shaded regions stands for the population of this single-species. And the area of the shaded region in Figure 2(A) is equal to the area of the shaded region in Figure 2(B).

      6 Discussion

      Taking α = 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_IEq134_HTML.gif, λ = 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_IEq135_HTML.gif, g ( u ) = u 3 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_IEq136_HTML.gif in (1.1), Peletier and Troy [6] analyzed stationary antisymmetric single-bump periodic solutions. They found that the coefficient of the fourth-order derivative term μ played a role of system parameter. If μ 1 8 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_IEq137_HTML.gif, the family of periodic solutions is still very similar to that of the Fisher-Kolmogorov equations. However, if μ > 1 8 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_IEq138_HTML.gif, different families of periodic solutions emerged.

      Taking μ = 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_IEq139_HTML.gif, λ = 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_IEq135_HTML.gif in (1.1), and under hypothesis that g ( 1 ) = 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_IEq140_HTML.gif, g ( 1 ) < 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_IEq141_HTML.gif, g ( u ) < 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_IEq142_HTML.gif for 0 < u < 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_IEq143_HTML.gif, Rottschäer and Wayne [8] showed that for every positive wavespeed, there exists a traveling wave. And they also found that there exists a critical wavespeed c http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_IEq144_HTML.gif. If c c http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_IEq145_HTML.gif, the solution is monotonic; otherwise, the solution is oscillatory.

      Unlike the work mentioned above, which focuses on the structure of solutions varying with the system parameter (μ or c), the manuscript presented here investigates the topological structure and the stability of solutions varying with the system parameter, i.e.λ. Firstly, if λ μ + α http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_IEq12_HTML.gif, the bifurcated attractor consists of the trivial solution; if λ > μ + α http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_IEq80_HTML.gif, the bifurcated attractor consists of only one cycle of steady state solutions and is homeomorphic to S 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_IEq18_HTML.gif. Secondly, if λ μ + α http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_IEq12_HTML.gif, the trivial solution is locally asymptotically stable. However, if λ > μ + α http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_IEq80_HTML.gif, the stability switches from the trivial solution to the bifurcated attractor.

      Since the increment of dimension of spatial domain may lead to much richer bifurcated behavior, further investigation on higher dimension of spatial domain is necessary in the future.

      7 Conclusions

      In this article, we first prove the existence of attractor bifurcation when the system parameter crosses critical number μ + α http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_IEq14_HTML.gif, which is the first eigenvalue of the eigenvalue problem of the linearized equation of (1.1). Second, we show that the stability of solutions varies with the system parameter λ. If λ μ + α http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_IEq12_HTML.gif, the trivial solution u = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_IEq13_HTML.gif is locally asymptotically stable. However, if λ > μ + α http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_IEq146_HTML.gif, the stability switches from u = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_IEq13_HTML.gif to Σ λ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_IEq17_HTML.gif. Third, the topological structure of the attractor is investigated. We prove that the attractor Σ λ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_IEq17_HTML.gif consists of only one cycle of steady state solutions and is homeomorphic to S 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-169/MediaObjects/13661_2012_Article_444_IEq18_HTML.gif. At last, the expression of bifurcated solution is also obtained.

      Declarations

      Acknowledgements

      The authors are grateful to the anonymous referees whose careful reading of the manuscript and valuable comments were very helpful for revising and improving our work.

      Authors’ Affiliations

      (1)
      College of Computer Science, Civil Aviation Flight University of China
      (2)
      College of Mathematics and Software Science, Sichuan Normal University

      References

      1. Coullet P, Elphick C, Repaux D: Nature of spatial chaos. Phys. Rev. Lett. 1987, 58: 431–434. 10.1103/PhysRevLett.58.431MathSciNetView Article
      2. Dee G, Saarloose W: Bistable systems with propagating fronts leading to pattern formation. Phys. Rev. Lett. 1988, 60: 2641–2644. 10.1103/PhysRevLett.60.2641View Article
      3. Zimmermann W: Propagating fronts near a Lipschitz point. Phys. Rev. Lett. 1991, 66: 1546. 10.1103/PhysRevLett.66.1546View Article
      4. Kalies WD, van der Vorst RCAM: Multitransition homoclinic and heteroclinic solutions of the extended Fisher-Kolmogorov equation. J. Differ. Equ. 1996, 131: 209–228. 10.1006/jdeq.1996.0161MathSciNetView Article
      5. Tersian S, Chaparova J: Periodic and homoclinic solutions of extended Fisher-Kolmogorov equations. J. Math. Anal. Appl. 2001, 260: 490–506. 10.1006/jmaa.2001.7470MathSciNetView Article
      6. Peletier LA, Troy WC: Spatial patterns described by the extended Fisher-Kolmogorov equation: periodic solutions. SIAM J. Math. Anal. 1997, 28: 1317–1353. 10.1137/S0036141095280955MathSciNetView Article
      7. van den Berg JB: The phase-plane picture for a class of fourth-order conservative differential equations. J. Differ. Equ. 2000, 161: 110–153. 10.1006/jdeq.1999.3698MathSciNetView Article
      8. Rottschäer V, Wayne CE: Existence and stability of traveling fronts in the extended Fisher-Kolmogorov equation. J. Differ. Equ. 2001, 176: 532–560. 10.1006/jdeq.2000.3984View Article
      9. Luo H, Zhang Q: Regularity of global attractor for the fourth-order reaction-diffusion equation. Commun. Nonlinear Sci. Numer. Simul. 2012, 17: 3824–3831. 10.1016/j.cnsns.2012.02.007MathSciNetView Article
      10. Kwapisz J: Uniqueness of the stationary wave for the extended Fisher-Kolmogorov equation. J. Differ. Equ. 2000, 165: 235–253. 10.1006/jdeq.1999.3750MathSciNetView Article
      11. Bartuccelli MV: On the asymptotic positivity of solutions for the extended Fisher-Kolmogorov equation with nonlinear diffusion. Math. Methods Appl. Sci. 2002, 25: 701–708. 10.1002/mma.309MathSciNetView Article
      12. Benguria RD, Depassier MC: On the transition from pulled to pushed monotonic fronts of the extended Fisher-Kolmogorov equation. Physica A 2005, 356: 61–65. 10.1016/j.physa.2005.05.013MathSciNetView Article
      13. Peletier LA, Troy WC: A topological shooting method and the existence of kinks of the extended Fisher-Kolmogorov equation: periodic solutions. Topol. Methods Nonlinear Anal. 1995, 6: 331–355.MathSciNet
      14. Ma T, Wang SH: Attractor bifurcation theory and its applications to Rayleigh-Bénard convection. Commun. Pure Appl. Anal. 2003, 2: 591–599.MathSciNetView Article
      15. Ma T, Wang SH Nonlinear Science Series A 53. In Bifurcation Theory and Applications. World Scientific, Singapore; 2005.
      16. Park J: Bifurcation and stability of the generalized complex Ginzburg-Landau equation. Commun. Pure Appl. Anal. 2008, 7(5):1237–1253.MathSciNetView Article
      17. Zhang YD, Song LY, Axia W: Dynamical bifurcation for the Kuramoto-Sivashinsky equation. Nonlinear Anal. 2011, 74(4):1155–1163. 10.1016/j.na.2010.09.052MathSciNetView Article
      18. Temam R Appl. Math. Sci. 68. In Infinite-Dimensional Dynamical Systems in Mechanics and Physics. Springer, New York; 1997.View Article
      19. Pazy A Appl. Math. Sci. 44. In Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer, New York; 2006.
      20. Murray JD: Mathematical Biology. Springer, New York; 1989.View Article

      Copyright

      © Zhang and Luo; licensee Springer. 2013

      This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://​creativecommons.​org/​licenses/​by/​2.​0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.