Normal Forms of Hopf Bifurcation for a Reaction-Diffusion System Subject to Neumann Boundary Condition

A reaction-diffusion system coupled by two equations subject to homogeneous Neumann boundary condition on one-dimensional spatial domain (0, lπ) with l > 0 is considered. According to the normal form method and the center manifold theorem for reaction-diffusion equations, the explicit formulas determining the properties of Hopf bifurcation of spatially homogeneous and nonhomogeneous periodic solutions of system near the constant steady state (0, 0) are obtained.


Introduction
As an important dynamic bifurcation phenomenon in dynamical systems, Hopf bifurcation of periodic solutions has attracted great interest of many authors in the last several decades [1][2][3][4][5][6][7][8]. In general, the study of Hopf bifurcation includes the existence and the properties such as the direction of bifurcation and the stability of bifurcating periodic solutions. In application, however, it is more difficult to determine the properties of Hopf bifurcation than to find the existence of a Hopf bifurcation. An approach applied to determine the properties of Hopf bifurcation is to derive the projected equation of original equations on the associated center manifold, that is, the so-called normal form. Then one may explore the local dynamical behaviors of a higher dimensional or even infinitely dimensional dynamical system near a certain nonhyperbolic steady state according to the normal form obtained. The normal form of Hopf bifurcation in ordinary differential equations (ODEs) with or without delays has been established well [1,3,5] since in this case the equilibrium is always constant and there are also no effects of spatial diffusion.
Under some certain conditions, the reaction-diffusion equations under the homogeneous Neumann boundary condition may have the constant steady state and thus one can study the Hopf bifurcation of system at this constant steady state. Compared with the ODEs, it is more difficult to derive the normal form of Hopf bifurcation for reaction-diffusion equations at the constant steady state. Although Hassard et al. [3] established the method computing the normal form of Hopf bifurcation in reaction-diffusion equations with the homogeneous Neumann boundary condition and also considered the Hopf bifurcation of spatially homogeneous periodic solutions in Brusselator system, using the same method, Jin et al. [9] and Ruan [10] as well as Yi et al. [11,12] considered the Hopf bifurcation of spatially homogeneous periodic solutions for Gierer-Meinhardt system and CIMA reaction, respectively. There are few results regarding Hopf bifurcation of spatially nonhomogeneous periodic solutions for spatially homogeneous reaction-diffusion equations [7].
Based on the reason mentioned above, in this paper we consider the normal form of Hopf bifurcation of reactiondiffusion equations at the constant steady state following the idea in [3]. In order to have a clearer structure, we are concerned with the following general reaction-diffusion system coupled by two equations defined on one-dimensional spatial domain (0, ℓ ) with ℓ > 0 and subject to Neumann boundary conditions; that is,

Algorithm Determining the Properties of Hopf Bifurcation
In this section, we will describe the explicit algorithm determining the direction of Hopf bifurcation and the stability of the bifurcating periodic solutions of system (1) at (0, 0). Define the real-valued Sobolev space by In terms of , the complex-valued Sobolev space C is given by and the inner product ⟨⋅, ⋅⟩ on C is defined by Assume that, for some 0 ∈ R, the following condition holds: (H) There exists a neighborhood of 0 such that, for ∈ , ( ) has a pair of simple and continuously differentiable eigenvalues ( )± ( ) with ( 0 ) = 0, ( 0 ) = 0 > 0, and ( 0 ) ̸ = 0. In addition, all other eigenvalues of ( ) have nonzero real parts for ∈ .
Define the second-order matrix sequence ( ) by Then the characteristic equation of ( ) is where The eigenvalues of ( ) can be determined by the eigenvalues of ( ) ( ∈ N 0 ) and we have the following conclusion. Lemma 1. If ( ) ∈ C is an eigenvalue of the operator ( ), then there exists some ∈ N 0 such that ( ) is the eigenvalue of ( ) and vice versa.
If ( ) is the eigenvalue of some matrix , then there exists a nonzero vector ( , ) ∈ C × C such that (13) holds. Let Then ( , ) ̸ = 0 and This demonstrates that ( ) is an eigenvalue of ( ) and thus the proof is complete.

Lemma 1 shows that, under assumption (H), there is a unique
∈ N 0 such that ± 0 are purely imaginary eigenvalues of ( 0 ); that is, ( 0 ) = 0 and ( 0 ) > 0. Furthermore, it is easy to see that ( 0 ) ̸ = 0 for any ̸ = . Therefore, ( 0 ) ( ̸ = ) has eigenvalues with zero real parts if and only if ( 0 ) = 0. Assume that ( ) = ( ) + ( ) is the eigenvalue of ( ) for sufficiently approaching 0 . Then by the smoothness of ( = 1, 2) we know that ( ) is also the eigenvalue of ( ); namely, ( ) satisfies the following equation: Under the assumption (H), differentiating the above equation with respect to at 0 yields Based on the above discussion, condition (H) has the following equivalent form: Then we know that 0 = √ ( 0 ) and ( 0 ), ( 0 ) cannot be equal to zero simultaneously when the hypothesis (H) is satisfied. Therefore, the eigenvector of ( 0 ) corresponding to the eigenvalue 0 can be chosen as and thus the eigenfunction of ( 0 ) corresponding to the eigenvalue 0 has the form Let the linear operator ) . (21) Similar to the choice of the eigenfunction of the operator ( 0 ) corresponding to the eigenvalue 0 , we can choose * = ( * * ) cos ℓ such that Define and by = { + | ∈ C} and = { ∈ | ⟨ * , ⟩ = 0}, respectively. Then can be decomposed as the direct sum of and ; that is, = ⊕ . Thus, for any = ( , V) ∈ , there exists ∈ C and = ( 1 , 2 ) ∈ such that = + + , Journal of Applied Mathematics Define ( , ) by Then system (1) can be rewritten into the following abstract form: When = 0 , system (26) is reduced to where 0 ( ) = ( , )| = 0 . In terms of (23) and decomposition (24), system (27) can be transformed into the following system in ( , ) coordinates: where For = ( 1 , 2 ), = ( 1 , 2 ), and = ( 1 , 2 ) ∈ C , define the symmetric multilinear forms ( , ) and ( , , ), respectively, by ) . (31) For the simplicity of notations, we will use and to denote ( , ) and ( , , ), respectively. Let Then from (29) and (32), one can get From the center manifold theorem in [3], we can rewrite in the form The second equation of (28), (33), and (35) yields Substituting (35) into the first equation of (28) gives the equation of reaction-diffusion system (1) restricted on the center manifold at ( 0 , 0, 0) as where 20 = ⟨ * , ⟩, 11 = ⟨ * , ⟩, 02 = ⟨ * , ⟩, and The dynamics of (28) can be determined by the dynamics of (37). In addition, it can be observed from [3] that when approaches sufficiently 0 , the Poincaré normal form of (26) has the forṁ= where is a complex variable, ≥ 1, and ( ) are complexvalued coefficients with The direction of Hopf bifurcation and the stability of the bifurcating periodic solutions of (1) at ( 0 , 0, 0) can be determined by the sign of Re 1 ( 0 ) and we have the following conclusion.
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