Turing–Hopf bifurcation in the reaction–diffusion equations and its applications

• Normal forms of the Turing–Hopf bifurcation for a general reaction–diffusion system are calculated.
• The spatiotemporal dynamics near the Turing–Hopf bifurcation point can be exactly studied by the unfolding parameters.
• The stable spatially inhomogeneous periodic solution in an autocatalysis reaction–diffusion model is found.

In this paper, we consider the Turing–Hopf bifurcation arising from the reaction–diffusion equations. It is a degenerate case and where the characteristic equation has a pair of simple purely imaginary roots and a simple zero root. First, the normal form theory for partial differential equations (PDEs) with delays developed by Faria is adopted to this degenerate case so that it can be easily applied to Turing–Hopf bifurcation. Then, we present a rigorous procedure for calculating the normal form associated with the Turing–Hopf bifurcation of PDEs. We show that the reduced dynamics associated with Turing–Hopf bifurcation is exactly the dynamics of codimension–two ordinary differential equations (ODE), which implies the ODE techniques can be employed to classify the reduced dynamics by the unfolding parameters. Finally, we apply our theoretical results to an autocatalysis model governed by reaction–diffusion equations; for such model, the dynamics in the neighbourhood of this bifurcation point can be divided into six categories, each of which is exactly demonstrated by the numerical simulations; and then according to this dynamical classification, a stable spatially inhomogeneous periodic solution has been found.