Turing-Hopf bifurcation analysis in a diffusive Gierer-Meinhardt model

The reaction-diffusion Gierer-Meinhardt system in one dimensional bounded domain is considered in the present paper. The Hopf bifurcation is investigated, which is found to be degenerate. With the aid of Maple, the normal form associated with the degenerate Hopf bifurcation is obtained to determinate the existence of Bautin bifurcation. We get the universal unfolding for the Bautin bifurcation so that we can identify the stability of periodic solutions. Then, the existence of the codimension-two Turing-Hopf bifurcation is further investigated. To research the spatiotemporal dynamics of the model near the Turing-Hopf bifurcation point, the method of the multiple time scale analysis is adopted to derive the amplitude equations. It is noted that the Gierer-Meinhardt model may show the spatial, temporal or the spatiotemporal patterns, such as the nonconstant steady state, spatially homogeneous periodic solutions and the spatially inhomogeneous periodic solutions. Finally, some numerical simulations are presented to demonstrate the applicability of the theoretical results.