A reaction-diffusion system coupled by two equations subject to homogeneous Neumann boundary condition on one-dimensional spatial domain

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 [

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. [

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 [

This paper is organized as follows. In the next section, following the abstract method according to [

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 (

Define the real-valued Sobolev space

Let

There exists a neighborhood

Define the second-order matrix sequence

The eigenvalues of

If

It is well known that the eigenvalue problem

From the orthogonality of the function sequence

If

Lemma

Based on the above discussion, condition (H) has the following equivalent form:

Let the linear operator

Define

For

Let

Substituting (

In addition, it can be observed from [

Assume that condition (H) (or equivalently (

From the description in the previous section we know that Hopf bifurcation of (

If condition (

From (

Let all the partial derivatives of

Notice from (

Lemma

We represent

Thus we have the following result.

Assume that condition (

Notice that the spatially nonhomogeneous periodic solutions of (

Since when

Let all the second- and third-order partial derivatives of

Let

Similarly, we can get

From (

Since

In addition, it follows from

Thus we have the following result.

Assume that condition (

The authors declare that there is no conflict of interests regarding the publication of this paper.

Cun-Hua Zhang was supported by the Natural Science Foundation of Gansu Province (1212RJZA065). Xiang-Ping Yan was supported by the National Natural Science Foundation of China (11261028), Gansu Province Natural Science Foundation (145RJZA216), and China Scholarship Council.