Dynamics of the SNARE‐SM System With Weak Diffusion

This paper is devoted to the dynamics of the SNARE‐SM system with weak diffusion, including stability and bifurcation. Firstly, Turing instability in the SNARE‐SM reaction–diffusion system around the equilibrium is discussed. Then, using traveling wave transformation, we transform our problem into a multiscale slow‐fast system with two small parameters. Based on the normal hyperbolic property of the critical manifold, the 4‐dimensional system is very ingeniously reduced to a 2‐dimensional fast‐slow system with a small parameter. Then, making use of the geometric singular perturbation theory again, which includes the slow‐fast normal form and entry‐exit function, the existence of relaxation oscillation and canard explosion phenomenon on the critical manifold are investigated. Eventually, sufficient conditions for the existence of traveling waves and periodic solutions of the original reaction–diffusion system are obtained.