Spatiotemporal attractors generated by the Turing-Hopf bifurcation in a time-delayed reaction-diffusion system

TL;DR

This paper derives explicit normal form formulas for Turing-Hopf bifurcations in two-component reaction-diffusion systems with delays, enabling analysis of complex spatiotemporal attractors.

Contribution

It extends the normal forms method to Hopf-zero singularity and provides an automated Matlab implementation for analyzing bifurcations.

Findings

Explicit formulas for normal forms of Turing-Hopf bifurcation.

Identification of possible attractors including steady-states, periodic, and quasi-periodic solutions.

Extension of normal forms method to systems with time delays.

Abstract

We study the Turing-Hopf bifurcation and give a simple and explicit calculation formula of the normal forms for a general two-components system of reaction-diffusion equation with time delays. We declare that our formula can be automated by Matlab. At first, we extend the normal forms method given by Faria in 2000 to Hopf-zero singularity. Then, an explicit formula of the normal forms for Turing-Hopf bifurcation are given. Finally, we obtain the possible attractors of the original system near the Turing-Hopf singularity by the further analysis of the normal forms, which mainly include, the spatially non-homogeneous steady-state solutions, periodic solutions and quasi-periodic solutions.