Spatiotemporal rotating waves near equivariant Hopf singularity in the delayed Lengyel–Epstein model with cross-diffusion on a 2D square domain

Cross-diffusion and time delay synergistically induce complex 2D patterns (e.g., rotating waves, standing waves). Based on the 2D delayed Lengyel–Epstein model with cross-diffusion, we investigated pattern formation, structure and stability via equivariant Hopf bifurcation. For the delay-free model, we derived equivariant Turing/Hopf bifurcation conditions and stability region of the coexistence equilibrium. We characterized equivariant Hopf bifurcations co-induced by time delay and cross-diffusion (including rare mode-(1,0)(0,1)/(1,1)/(2,0)(0,2) ones), derived the third-order normal form via normal form method and center manifold theory, and theoretically predicted and numerically validated bistable rotating waves and monostable standing waves from these bifurcations. This work reveals the core spatiotemporal dynamic and pattern selection mechanisms of the CIMA reaction-diffusion system under symmetric domain constraints.