Diffusion-driven destabilization of spatially homogeneous limit cycles in reaction-diffusion systems

We study the diffusion-driven destabilization of a spatially homogeneous limit cycle with large amplitude in a reaction-diffusion system on an interval of finite size under the periodic boundary condition. Numerical bifurcation analysis and simulations show that the spatially homogeneous limit cycle becomes unstable and changes to a stable spatially nonhomogeneous limit cycle for appropriate diffusion coefficients. This is analogous to the diffusion-driven destabilization (Turing instability) of a spatially homogeneous equilibrium. Our approach is based on a reaction-diffusion system with mass conservation and its perturbed system considered as an infinite dimensional slow-fast system (relaxation oscillator).