Turing–Hopf bifurcation and multi-stable spatio-temporal patterns in the Lengyel–Epstein system

In this paper, we consider the Lengyel–Epstein system of the CIMA reaction with homogeneous Neumann condition. Firstly, we derive conditions for existence of Turing/Turing–Hopf bifurcation by analysis of distribution of eigenvalues. Meanwhile, we give the concrete range of diffusion rate c preserving that spatial inhomogeneous Hopf bifurcation occurs based on the existence result in Du and Wang (2010). Secondly, existence of more complex spatio-temporal dynamical behaviors, such as spatial inhomogeneous periodic/quasi-periodic solutions and bistable/tristable phenomenon, are rigorously proved near the Turing–Hopf bifurcation point using center manifold theorem and normal form method. Finally, numerical simulations in different parameter regions not only support our analytical results but indicate the existence of tetrastable phenomenon.