Nonlinear dynamics of finite perturbation: collapse and revival of spatial patterns

Abstract

A full-scale nonlinear stability analysis is performed on a reaction-diffusion system that includes a cubic polynomial source term and Cattaneo's modification of Fick's law of diffusion. This modification incorporates the effect of a small, finite relaxation time of flux at the macroscopic level of the description of the process. While linear stability analysis predicts the decay of small wavelength perturbations on a homogeneous steady state for large reaction time, consideration of finite perturbations leads to a spatiotemporal instability, resulting in an interesting phenomenon of periodic collapse and revival of spatial patterns. This instability is relaxation (time) driven, and the time period is determined by self-sustaining oscillations due to the limit cycle of the underlying dynamics. The nonlinear dynamics of finite perturbations may thus be generically different from what is expected from a linear stability analysis.