Spatiotemporal Patterns in a Lengyel–Epstein Model Near a Turing–Hopf Singular Point

SIAM Journal on Applied Mathematics, Volume 84, Issue 2, Page 338-361, April 2024.
Abstract. In this paper, a study is carried out on the spatiotemporal dynamics of a Lengyel–Epstein model describing the chlorite-iodine-malonic-acid (CIMA) reaction with time delay and the Neumann boundary condition in a two-dimensional region. The existences for Turing, Hopf, Turing–Turing, Turing–Hopf, and Bogdanov–Takens bifurcations are derived by analyzing the dispersion relation between eigenvalues and wave numbers. In particular, to study the dynamics around a Turing–Hopf bifurcation singularity, the amplitude equations near a codimension-two bifurcation point are derived by employing the weakly nonlinear analysis method. Different spatiotemporal patterns for the system in parameter space are classified and various patterns identified, including spatially homogeneous periodic solutions, mixed mode, coexistence mode, bistable phenomenon, square, hexagon, black eye, two-phase oscillating staggered hexagon lattice, and other complex spatiotemporal patterns. The theoretical predictions are verified by numerical simulations showing an excellent agreement with many reported experiment results not only in chemistry but also in physics and biology. Results presented in this article reveal the mechanism of generating the spatiotemporal patterns of the CIMA reaction.