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

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1 Zhang and Luo Boundary Value Problems 213, 213:169 R E S E A R C H Open Access Attractor bifurcation for the extended Fisher-Kolmogorov equation with periodic boundary condition Qiang Zhang 1* andhongluo 2 * Correspondence: zqcs7@163.com 1 College of Computer Science, Civil Aviation Flight University of China, Guanghan, Sichuan 61837, P.R. China Full list of author information is available at the end of the article 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 4 = μ u + α 2 u + λu + gu, t x 4 x 2 2π ux, t dx =, t, ux, t=ux +2kπ, t, k Z, ux,=u, x R, x, t R,, 1.1 where u = ux, t :R [, R is an unknown function, μ >,α > are constants, λ R + is the system parameter. gs is a polynomial on s R,whichisgivenby gs= p a k s k, k=2 where 2 p N and a k are given constants. The extended Fisher-Kolmogorov EFK equation has been proposed as a model for phase transitions and other bistable phenomena [1 3]. 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 213 Zhang and Luo; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

2 Zhangand Luo Boundary Value Problems 213, 213:169 Page 2 of 13 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] provedthatequation1.1 possesses a global attractor in the Sobolev space H k for all k > provided that a p <andpisanoddnumber. We refer the interested readers to the references in [4 9] for other results on the EFK equation; see also, among others, [1 13]. Returning to problem 1.1, our main interest in the present paper is the bifurcationand 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 existence of attractor bifurcation when the system parameter crosses some critical number, 2 dynamic stability of bifurcated solutions, and 3 the topological structure of the bifurcated attractor. Ourmainresultscanbesummarizedasfollows. 1. If λ μ + α, thesteady state u = is locally asymptotically stable. 2. As λ crosses μ + α, i.e.,thereexistsanɛ >such that for any μ + α < λ < λ + ɛ, system 1.1 bifurcates from the trivial solution to an attractor λ. 3. λ is homeomorphic to S 1 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.Themathematical 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 singlespecies 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 be two Hilbert spaces, and let H 1 H be a dense and compact inclusion. We consider the following nonlinear evolution equations du dt = L λu + Gu, u = u, 2.1 where u :[, H is the unknown function, λ R is the system parameter, and L λ : H 1 H are parameterized linear completely continuous fields depending continuously

3 Zhangand Luo Boundary Value Problems 213, 213:169 Page 3 of 13 on λ,whichsatisfy L λ = A + B λ A : H 1 H B λ : H 1 H a sectorial operator, alinearhomeomorphism, parameterized linear compact operators. 2.2 Since L λ is a sectorial operator which generates an analytic semigroup S λ t={e tl λ} t for any λ R, we can define fractional power operators L λ μ for μ 1 with domain H μ = D L λ μ suchthath μ1 H μ2 if μ 1 > μ 2,andH = H see [18, 19]. In addition, we assume that the nonlinear terms G : H α H for some α <1area family of parameterized C r bounded operators r 1 such that Gu=o u Hα. 2.3 Definition 2.1 [15] Aset H is called an invariant set of 2.1 ifst = for any t. An invariant set H of 2.1 is said to be an attractor if is compact, and there exists a neighborhood of W H of such that for any ϕ W we have lim dist H ut, ϕ, =, t where dist H ut, ϕ, =inf v ut, ϕ v H, t. Definition 2.2 [15] 1 We say that the solution to equation 2.1bifurcatesfromu, λ=, λ toaninvariantset λ if there exists a sequence of invariant sets { λn } of 2.1such that / λn,and lim λ n = λ, n lim max v H =. n v λn 2 If the invariant sets λ 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 be given by β k λ C k 1, and the principle of exchange of stabilities holds true: <, ifλ < λ, Re β i λ =, ifλ = λ 1 i m, >, ifλ > λ, 2.4 Re β j λ <, j m

4 Zhangand Luo Boundary Value Problems 213, 213:169 Page 4 of 13 Let the eigenspace of L λ at λ = λ be E = 1 j m k=1 { u, v H1 L λ β j λ k } w =,w = u + iv. It is known that dim E = m. The following attractor bifurcation theorem can be found in [15]. Theorem 2.1 Let H 1 = H = R n, conditions 2.4, 2.5 hold true, and u =is a locally asymptotically stable equilibrium point of 2.1 at λ = λ. Then the following assertions hold true: 1 Equation 2.1 bifurcates from u, λ=,λ to attractors λ for λ > λ, with dimension m 1 dim λ m, which is connected as m >1. 2 The attractor λ is a limit of a sequence of m-dimensional annuli A k with A k+1 A k ; especially, if λ is a finite simplicial complex, then λ has the homology type of the m 1-dimensional sphere S m 1. 3 For any u λ λ, u λ can be expressed as u λ = v λ + o v λ H1, vλ E. 4 If u =is globally asymptotically stable for 2.1 at λ = λ, then for any bounded open set U H with U, there is an ɛ >such that λ < λ < λ + ɛ, the attractor λ attracts U\Ɣ in H, where Ɣ is the stable manifold of u =with codimension m. In particular, if 2.1 has a global attractor for all λ near λ, then U = H. Remark 2.1 As H 1 and H are infinite dimensional Hilbert spaces, if 2.1 satisfies conditions andu = is a locally global asymptotically stable equilibrium point of 2.1atλ = λ, 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 r 1 vector field given by v λ x=λx Fx 2.6 for x R 2.Here Fx=F k x+o x k, 2.7 where F k is a k-multilinear field, which satisfies the inequality C 1 x k+1 F k x, x C 2 x k for some constants < C 1 < C 2 and k =2m +1,m 1. Theorem 2.2 Theorem 5.1 in [15] Under conditions 2.7, 2.8, the vector field 2.6 bifurcates from x, λ =,to an attractor λ for λ >,which is homeomorphic to S 1. Moreover, one and only one of the following conclusions is true:

5 Zhangand Luo Boundary Value Problems 213, 213:169 Page 5 of 13 1 λ is a period orbit. 2 λ consistsofinfinitelymanysingularpoints. 3 λ contains at most 2k +1=4m +1singular 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. 3 Mathematical setting Let H = L 2, 2π and { H 1 = u H 4, 2π ux +2π=ux, 2π } udx=. We define L λ = A + B λ : H 1 H and G : H 1 H by Au = μ d4 u + α d2 u, dx 4 dx 2 B λ u = λu, Gu=gu. 3.1 Consequently, we have an operator equation which is equivalent to problem 1.1asfollows: du dt = L λu + Gu, 3.2 u = u. 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 2a μa 3 +9αa 3 <is satisfied, then the following assertions hold true: 1 If λ μ + α, the steady state u = is locally asymptotically stable. 2 If λ > μ + α, system 1.1 bifurcates from the trivial solution u =to an attractor λ. 3 λ is homeomorphic to S 1 and consists of exactly one cycle of steady solutions of λ can be expressed as λ = { x cosx + θ+o x θ R }, where x = 416μ+4α λμ+α λ a 3a 3 16μ+4α λ+2a 2 2, or x = 4μ+α λ 3a 2 3 a 2 =, and μ + α < λ < μ + α + ɛ, ɛ 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.

6 Zhangand Luo Boundary Value Problems 213, 213:169 Page 6 of 13 Consider the eigenvalue problem of the linear equation, L λ u = βu. 4.1 It is not difficult to find that the eigenvalues and the normalized eigenvectors of 4.1are β 2k 1 = β 2k = λ μk 4 αk 2, k =1,2,..., 4.2 sin kx e 2k 1 = cos kx π, e 2k = π, under condition that we get the principle of exchange of stabilities <, λ < μ + α, β 1 λ=β 2 λ =, λ = μ + α, >, λ > μ + α, β j μ + α<, j 3. Step 2. We verify that for any λ R,operatorL λ + G satisfies conditions 2.2and2.3. Thanks to the results in [9, 18, 19], we know that the operator L λ : H 1 H is a sectorial operator which implies that condition 2.2holds true. It is easy to get the following inequality: 2π Gu 2 H = gu 2 dx 2π p C u 2k dx C C p k=2 p k=2 k=2 u 2k L 2k,2π u 2k H 12, which implies that Gu=o u H 12, where { H 1 = u H 2, 2π 2 2π } udx =,ux +2π=ux, then condition 2.3holdstrue. Step 3. In this part, we prove the existence of attractor bifurcation and analyze the topological structure of the attractor λ. Let E1 λ = E = span{e 1, e 2 }, E2 λ = E.Let be the center manifold function, in the neighborhood of u, λ=,μ + α, we have u = y + y, where y = x 1 e 1 + x 2 e 2.

7 Zhangand Luo Boundary Value Problems 213, 213:169 Page 7 of 13 Then the reduction equations of 3.2 are as follows: dx 1 =λ μ αx dt 1 + Gu, e 1, dx 2 =λ μ αx dt 2 + Gu, e To get the exact form of the reduction equations, we need to obtain the expression of Gu, e 1 and Gu, e 2. Let G 2 : H 1 H 1 H and G 3 : H 1 H 1 H 1 H 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. Since G2 y, y, e 1 = G2 y, y, e 2 =, 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 a 2 π x 1 x 2, k =3, G2 y, y, e k = a 2 2 π x2 2 a 2 2 π x2 1, k =4,, k 3,4. According to the formula of Theorem 3.8 in [15] or Remark 4.1, the center manifold function, in the neighborhood of u, λ=,μ + α, can be expressed as y= k=3 β k 1 G2 y, y, e k ek + O β β 2 2 y 2 + o y 2 = λ 16μ 4α 1 a 2 2 π 2x1 x 2 e 3 + x 2 2 e 4 x 2 1 e 4 + O λ μ α 2 x x o x x 2 2. In the following, we calculate Gu, e j, j =1,2. Gu, ej = G2 y, y, ej + G2 y, y, ej + G3 y, y, y, e j + O λ μ α 2 x x o x x 2 3, j =1,2. By direct calculation, we have G2 y, y, e1 = G 2 y, y, e1

8 Zhangand Luo Boundary Value Problems 213, 213:169 Page 8 of 13 = λ 16μ 4α 1 a2 2 4π x3 1 λ 16μ 4α 1 a2 2 4π x 1x O λ μ α 2 x x o x x 2 3, G2 y, y, e2 = G 2 y, y, e2 = λ 16μ 4α 1 a2 2 4π x3 2 λ 16μ a2 4α 1 2 4π x2 1 x 2 + O λ μ α 2 x x o x x 2 3, 3a 3 G3 y, y, y, e 1 = 4π x a 3 4π x 1x 2 2, 3a 3 G3 y, y, y, e 2 = 4π x a 3 4π x2 1 x 2, then we obtain the expression of Gu, e j, j =1,2. Gu, e1 = Ax Ax 1 x O λ μ α 2 x x o x x 2 3, Gu, e2 = Ax 2 1 x 2 + Ax O λ μ α 2 x x o x x 2 3, 4.4 where A = λ 16μ 4α 1 a2 2 2π + 3a 3 4π. Putting 4.4into4.3, we havethe reductionequations dx 1 dt =λ μ αx 1 + Ax Ax 1x O λ μ α 2 x x o x x 2 3, dx 2 dt =λ μ αx 2 + Ax 2 1 x 2 + Ax O λ μ α 2 x x o x x For the case of λ < μ + α,itisobviousthatu = is locally asymptotically stable. For the case of λ = μ + α, if2a μ +9αa 3 <, which implies that A <,thenu =isalso locally asymptotically stable. Assertion 1 of Theorem 4.1 is proved. Since the following equality holds true: x 1 Ax Ax 1 x x2 Ax 2 1 x 2 + Ax 3 2 = A x x 2 2, 2 according to Theorems 2.1, 2.2 and Remark 2.1, we can conclude that if λ > μ + α,equation 1.1bifurcatesfromu = to an attractor λ, which is homeomorphic to S 1. 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 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=,μ + α to two steady solutions. For any function v in the even function space can be expressed as follows: v = x 2k e 2k, k 1

9 Zhangand Luo Boundary Value Problems 213, 213:169 Page 9 of 13 by the Lyapunov-Schmidt reduction method used in Step 3, we can deduce that the reduction equation of 1.1 is as follows: dx 2 dt =λ μ αx 2 + Ax O λ μ α 2 x o x 2 3, 4.6 which implies that 1.1bifurcatesfromu, λ=,μ+αtotwosteadysolutionsv ± λ x, t= ± 416μ+4α λμ+α λ cos x + h.o.t. in the space of even functions. 3a 3 16μ+4α λ+2a 2 2 Since the solutions of 2.1 are translation invariant, V λ + x, t V λ + x + θ, t, θ R, the set T = { V λ + x + θ, t θ R} represents S 1 in H 1, which implies that λ 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 }, the generalized eigenvectors of L λ,formabasisofh with the dual basis {e i } such that ei, e j We have H =, ifi j,, ifi = j. v = x + y E λ 1 Eλ 2, x = y = m x i e i E1 λ, i=1 x i e i E2 λ. i=m+1 Then near λ = λ, the center manifold function φx, λ intheorem3.8in[15] canbe expressed as follows: φx, λ= φ j x, λe j + O Re βλ x k + o x k, 4.7 j=m+1 where φ j x, λ= 1 Gk x,...,x, e j β j λ H. Remark 4.2 If gsin1.1 is not a polynomial but a C ω with Taylor s expansion in s =as gs= k=2 a ks k ;if2a μa 3 +9αa 3 < is satisfied, then the conclusions of Theorem 4.1 also hold true.

10 Zhangand Luo Boundary Value Problems 213, 213:169 Page 1 of 13 Remark 4.3 If the higher order terms p k=4 a ku k in gu 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 singlespeciesasfollows: v 4 = μ v + α 2 v + b t x 4 x 2 1 v + b 2 v 2 + a 3 v 3 + b, 2π vx, t dx = a 2 2a 3 π, t, vx, t=vx +2kπ, t, k Z, vx,=u + v, x R, x, t R,, 5.1 where μ, α are the diffusion coefficients, v is the population density for single-species, and a 2 <,a 3 <,b = λ a 2 4a a 3 2, b 64 a 2 1 = λ 5 a a 3 3, b 2 = a 2 4.Itiseasytoseethatb <,b 1 > and b 2 <. Inspired by the work of Murray [2], b 1 represents the birth rate, b 2 v 2 + a 3 v 3 describes the intra specific competition, and b stands for the emigration which arises from disease. It is not difficult to verify that v = a 2 4a 3 is a positive steady solution of system 5.1. From the translation ux, t=vx, t v, 5.2 we derive the following system: u 4 = μ u + α 2 u + λu + a t x 4 x 2 2 u 2 + a 3 u 3, 2π ux, t dx =, t, ux, t=ux +2kπ, t, k Z, ux,=u, x R. x, t R,, 5.3 According to Remark 4.3, if the condition 2a μa 3 +9αa 3 < is satisfied, the conclusions of Theorem 4.1 for system 5.3 alsoholdtrue.consequently,fromthetranslation 5.2, we have the following results for 5.1. Theorem 5.1 For problem 5.1, if 2a μa 3 +9αa 3 <is satisfied, then the following assertions hold true: a If b 1 μ + α 5 16 a 3, the steady state v = a 2 4a 3 is locally asymptotically stable Figure 1. a If b 1 > μ + α 5 16 a 3, system 5.1 bifurcates from the solution v to an attractor b1. This implies that the stability will switch from the original state i.e., v to a new one i.e., b1 Figure 1. 3 b1 is homeomorphic to S 1 and consists of exactly one cycle of steady solutions of 5.1 Figure 1.

11 Zhangand Luo Boundary Value Problems 213, 213:169 Page 11 of 13 Figure 1 Bifurcation diagram for the model of the population density for single-species. 1 Bifurcation appears at λ = μ + α 5 a a. 2 Bifurcated attractor b1 is the boundary of the shaded 3 region. 3 The first horizontal solid line from above denotes that the solution v = v is stable, and the horizontal dotted line means this solution is unstable. Figure 2 The spatial distribution of the population density. 1 Figure 2A shows that the population densitykeeps a uniformspatial distributionwhenthe birthrateis low. 2 Figure2B 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 2A is equal to the area of the shaded region in Figure 2B. 4 b1 can be expressed as b1 = { v + x cosx + θ+o x θ R }, where x = 416μ+4α λμ+α λ 3a 3 16μ+4α λ+2a 2 2 Furthermore, Theorem 5.1 and the equality, and μ + α < λ < μ + α + ɛ, ɛ is sufficiently small. 2π vx, t dx = a 2 2a 3 π, t, yield the following biological results. Biological results For the model 5.1, if 2a μa 3 +9αa 3 <issatisfied,wehavethe following assertions: 1 The population of this single-species is a conservative quantity. 2 If the birth rate is low, then the population density will keep a uniform spatial distribution Figure 2A. 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 2B. 6 Discussion Taking α =1,λ =1,gu= u 3 in 1.1, Peletier and Troy [6] analyzed stationary antisymmetric single-bump periodic solutions. They found that the coefficient of the fourth-order

12 Zhangand Luo Boundary Value Problems 213, 213:169 Page 12 of 13 derivative term μ played a role of system parameter. If μ 1, the family of periodic solutions is still very similar to that of the Fisher-Kolmogorov equations. However, if μ > 1 8, 8 different families of periodic solutions emerged. Taking μ =1,λ =1in1.1, and under hypothesis that g1 = 1, g 1 < 1, g u<for <u < 1, 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.ifc c, 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, the bifurcated attractor consists of the trivial solution; if λ > μ + α, the bifurcated attractor consists of only one cycle of steady state solutions and is homeomorphic to S 1. Secondly, if λ μ + α, the trivial solution is locally asymptotically stable. However, if λ > μ + α, 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, 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.ifλ μ + α, the trivial solution u = is locally asymptotically stable. However, if λ > μ + α, the stability switches from u =to λ. Third, the topological structure of the attractor is investigated. We prove that the attractor λ consists of only one cycle of steady state solutions and is homeomorphic to S 1. At last, the expression of bifurcated solution is also obtained. Competing interests The authors declare that they have no competing interests. Authors contributions Both authors read and approved the final manuscript. Author details 1 College of Computer Science, Civil Aviation Flight University of China, Guanghan, Sichuan 61837, P.R. China. 2 College of Mathematics and Software Science, Sichuan Normal University, Chengdu, Sichuan 6166, P.R. China. 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. Received: 24 October 212 Accepted: 5 July 213 Published: 19 July 213 References 1. Coullet, P, Elphick, C, Repaux, D: Nature of spatial chaos. Phys. Rev. Lett. 58, Dee, G, Saarloose, W: Bistable systems with propagating fronts leading to pattern formation. Phys. Rev. Lett. 6, Zimmermann, W: Propagating fronts near a Lipschitz point. Phys. Rev. Lett. 66, Kalies, WD, van der Vorst, RCAM: Multitransition homoclinic and heteroclinic solutions of the extended Fisher-Kolmogorov equation. J. Differ. Equ. 131, Tersian, S, Chaparova, J: Periodic and homoclinic solutions of extended Fisher-Kolmogorov equations. J. Math. Anal. Appl. 26,

13 Zhangand Luo Boundary Value Problems 213, 213:169 Page 13 of Peletier, LA, Troy, WC: Spatial patterns described by the extended Fisher-Kolmogorov equation: periodic solutions. SIAM J. Math. Anal. 28, van den Berg, JB: The phase-plane picture for a class of fourth-order conservative differential equations. J. Differ. Equ. 161, Rottschäer, V, Wayne, CE: Existence and stability of traveling fronts in the extended Fisher-Kolmogorov equation. J. Differ. Equ. 176, Luo, H, Zhang, Q: Regularity of global attractor for the fourth-order reaction-diffusion equation. Commun. Nonlinear Sci. Numer. Simul. 17, Kwapisz, J: Uniqueness of the stationary wave for the extended Fisher-Kolmogorov equation. J. Differ. Equ. 165, Bartuccelli, MV: On the asymptotic positivity of solutions for the extended Fisher-Kolmogorov equation with nonlinear diffusion. Math. Methods Appl. Sci. 25, Benguria, RD, Depassier, MC: On the transition from pulled to pushed monotonic fronts of the extended Fisher-Kolmogorov equation. Physica A 356, 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. 6, Ma, T, Wang, SH: Attractor bifurcation theory and its applications to Rayleigh-Bénard convection. Commun. Pure Appl. Anal. 2, Ma, T, Wang, SH: Bifurcation Theory and Applications. Nonlinear Science Series A, vol. 53. World Scientific, Singapore Park, J: Bifurcation and stability of the generalized complex Ginzburg-Landau equation. Commun. Pure Appl. Anal. 75, Zhang, YD, Song, LY, Axia, W: Dynamical bifurcation for the Kuramoto-Sivashinsky equation. Nonlinear Anal. 744, Temam, R: Infinite-Dimensional Dynamical Systems in Mechanics and Physics. Appl. Math. Sci., vol. 68. Springer, New York Pazy, A: Semigroups of Linear Operators and Applications to Partial Differential Equations. Appl. Math. Sci., vol. 44. Springer, New York Murray, JD: Mathematical Biology. Springer, New York 1989 doi:1.1186/ Cite this article as: Zhang and Luo: Attractor bifurcation for the extended Fisher-Kolmogorov equation with periodic boundary condition. Boundary Value Problems :169.