Oscillating Turing patterns, chaos and strange attractors in a reaction-diffusion system augmented with self- and cross-diffusion terms

TL;DR

This paper introduces a reaction-diffusion model with self- and cross-diffusion, revealing complex behaviors like oscillating Turing patterns, chaos, and strange attractors through bifurcation analysis and spectral methods.

Contribution

The study presents a novel reaction-diffusion model incorporating self- and cross-diffusion, along with a spectral method for analyzing equilibria and chaotic dynamics.

Findings

Development of new Turing patterns

Identification of Hopf bifurcations leading to oscillations

Observation of strange attractors in phase space

Abstract

In this article we introduce an original model in order to study the emergence of chaos in a reaction diffusion system in the presence of self- and cross-diffusion terms. A Fourier Spectral Method is derived to approximate equilibria and orbits of the latter. Special attention is paid to accuracy, a necessary condition when one wants to catch periodic orbits and to perform their linear stability analysis via Floquet multipliers. Bifurcations with respect to a single control parameter are studied in four different regimes of diffusion: linear diffusion, self-diffusion for each of the two species, and cross-diffusion. Key observations are made: development of original Turing patterns, Hopf bifurcations leading to oscillating patterns and period doubling cascades leading to chaos. Eventually, original strange attractors are reported in phase space.