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 with 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 are obtained.
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 [1–8]. 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.  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.  and Ruan  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 .
Based on the reason mentioned above, in this paper we consider the normal form of Hopf bifurcation of reaction-diffusion equations at the constant steady state following the idea in . 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 with and subject to Neumann boundary conditions; that is,in which are the diffusion coefficients, is the parameter, and are functions with for any . Although Yi et al.  described the algorithm determining the properties of Hopf bifurcation of spatially homogeneous and nonhomogeneous periodic solutions for (1) at and also considered the Hopf bifurcation of a diffusive predator-prey system with Holling type-II functional response and subject to the homogeneous Neumann boundary condition, they did not give the normal form of Hopf bifurcation of spatially homogeneous and nonhomogeneous periodic solutions of the general reaction-diffusion system (1) at .
This paper is organized as follows. In the next section, following the abstract method according to , we describe the algorithm determining the properties of Hopf bifurcation of spatially homogeneous and nonhomogeneous periodic solutions for system (1) at the constant steady state . In Section 3, the explicit formulas determining the properties of Hopf bifurcation of spatially homogeneous periodic solutions for system (1) at are obtained. The explicit formulas determining the properties of Hopf bifurcation of spatially nonhomogeneous periodic solutions for (1) at are also derived in Section 4.
2. 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 .
Define the real-valued Sobolev space by In terms of , the complex-valued Sobolev space is given by and the inner product on is defined by
Let , , , and and define the linear operator with the domain byAssume that, for some , the following condition holds: (H)There exists a neighborhood of such that, for , has a pair of simple and continuously differentiable eigenvalues with , , and . In addition, all other eigenvalues of have nonzero real parts for .Then from [3, 7] we know that system (1) undergoes a Hopf bifurcation at when crosses through .
Define the second-order matrix sequence byThen the characteristic equation of iswhere
The eigenvalues of can be determined by the eigenvalues of and we have the following conclusion.
Lemma 1. If is an eigenvalue of the operator , then there exists some such that is the eigenvalue of and vice versa.
Proof. It is well known that the eigenvalue problemhas eigenvalues with eigenfunctions . Assume that is an eigenvalue of the operator and the corresponding eigenfunction is ; that is,Notice that can be represented as where . Then (10) can be written intoFrom the orthogonality of the function sequence , one can get from (12) that, for each ,Since is the eigenfunction of corresponding to the eigenvalue , and so there must be some such that . Therefore, is the eigenvalue of the matrix .
If is the eigenvalue of some matrix , then there exists a nonzero vector such that (13) holds. Let Then andThis demonstrates that is an eigenvalue of and thus the proof is complete.
Lemma 1 shows that, under assumption (H), there is a unique such that are purely imaginary eigenvalues of ; that is, and . Furthermore, it is easy to see that for any . Therefore, has eigenvalues with zero real parts if and only if . Assume that is the eigenvalue of for sufficiently approaching . Then by the smoothness of 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 yields
Based on the above discussion, condition (H) has the following equivalent form:Then we know that and cannot be equal to zero simultaneously when the hypothesis (H) is satisfied. Therefore, the eigenvector of corresponding to the eigenvalue can be chosen asand thus the eigenfunction of corresponding to the eigenvalue has the form
Let the linear operator with the domain be defined by Then is the adjoint operator of the operator such that with . Similar to the choice of the eigenfunction of the operator corresponding to the eigenvalue , we can choosesuch thatDefine and by and , respectively. Then can be decomposed as the direct sum of and ; that is, . Thus, for any , there exists and such that
Define byThen system (1) can be rewritten into the following abstract form:When , system (26) is reduced towhere . In terms of (23) and decomposition (24), system (27) can be transformed into the following system in coordinates:where
For , , and , define the symmetric multilinear forms and , respectively, byThen, for , we haveFor the simplicity of notations, we will use and to denote and , respectively.
Substituting (35) into the first equation of (28) gives the equation of reaction-diffusion system (1) restricted on the center manifold at aswhere , , , andThe dynamics of (28) can be determined by the dynamics of (37).
In addition, it can be observed from  that when approaches sufficiently , the Poincaré normal form of (26) has the form where is a complex variable, , and are complex-valued coefficients withThe direction of Hopf bifurcation and the stability of the bifurcating periodic solutions of (1) at can be determined by the sign of and we have the following conclusion.
Theorem 2. Assume that condition (H) (or equivalently (18)) holds. Then system (1) undergoes a supercritical (or subcritical) Hopf bifurcation at when if In addition, if all other eigenvalues of have negative real parts, then the bifurcating periodic solutions are stable (resp., unstable) when ().
3. Spatially Homogeneous Hopf Bifurcation
From the description in the previous section we know that Hopf bifurcation of (1) at is spatially homogeneous if condition (18) holds when . In the present section, we compute in (40) in order to determine the direction of spatially homogeneous Hopf bifurcation and the stability of bifurcating periodic solutions of (1) at following the algorithm described in this pervious section.
Lemma 3. If condition (18) is satisfied when , then .
Proof. From (20) and (22) one can seewhereLet all the partial derivatives of be evaluated at , and let , , and be defined, respectively, byThen from (30) and (31) we can getTherefore,From (34) and (46), one can obtainNotice from (42) thatThe conclusion follows by substituting (48) into (47).
We represent , , and by , and , respectively, for the simplicity of notations and under assumption (18) with , substituting in (42) into (44) yields that, for ,From (42), (50), and (51) one can derive where
Thus we have the following result.
Theorem 4. Assume that condition (18) is satisfied when and is defined by (53). Then the spatially homogeneous Hopf bifurcation of system (1) at is supercritical (resp., subcritical) ifMoreover, if each eigenvalue of has negative real parts for all , then the above spatially homogeneous bifurcating periodic solutions are stable (resp., unstable) when
4. Spatially Nonhomogeneous Hopf Bifurcation
Notice that the spatially nonhomogeneous periodic solutions of (1) at from Hopf bifurcation are unstable. Accordingly, in this section we will calculate in (40) in order to determine the direction of Hopf bifurcation of spatially nonhomogeneous periodic solutions of system (1) at . To this end, we always assume that in (18) throughout this section and still represent , , and by , and , respectively. Thus defined in (22) has the formwhere
Since when , one can obtainThus, in order to calculate , it remains to compute
Let all the second- and third-order partial derivatives of with respect to and be evaluated at and letThen from (30) and (31), one can observeIn view of (34), (59), and (62), we haveEqualities (63) show that the calculation of and will be restricted on the subspaces spanned by eigenmodes and .
Similarly, we can getwhere