Turing Instability of Periodic Solutions in the Fitzhugh–nagumo Model With Cross-Diffusion

We investigate spatiotemporal pattern formation in a FitzHugh–Nagumo reaction-diffusion system with density-dependent cross-diffusion. In particular, we analyze how diffusion perturbs Hopf-bifurcating periodic orbits and derive conditions under which these oscillations lose stability via a Turing mechanism. Moreover, we obtain an explicit criterion, in terms of the diffusion coefficients, that predicts the onset of Turing instability for Hopf-bifurcating periodic solutions in the FitzHugh–Nagumo reaction-diffusion system with cross-diffusion. Numerical simulations validate the theoretical analysis and demonstrate the existence of spatiotemporal pattern formation.