Basin of Attraction of Solutions with Pattern Formation in Slow-Fast Reaction-Diffusion Systems

Abstract

This article is devoted to the characterization of the basin of attraction of pattern solutions for some slow-fast reaction-diffusion systems with a symmetric property and an underlying oscillatory reaction part. We characterize some subsets of initial conditions that prevent the dynamical system to evolve asymptotically toward solutions which are homogeneous in space. We also perform numerical simulations that illustrate theoretical results and give rise to symmetric and non-symmetric pattern solutions. We obtain these last solutions by choosing particular random initial conditions.

Keywords: FitzHugh–Nagumo; Limit-cycles; Pattern formation; Reaction diffusion systems; Slow–fast analysis.