Spatiotemporal Patterns Induced by Turing-Hopf Interaction and Symmetry on a Disk

TL;DR

This paper investigates the complex spatiotemporal patterns arising from Turing-Hopf interactions on a disk, providing normal forms and numerical examples of various wave-like solutions.

Contribution

It introduces normal forms for three types of Turing-Hopf bifurcations on a disk and demonstrates the resulting patterns through numerical simulations.

Findings

Identification of breathing, standing wave-like, and rotating wave-like patterns.

Derivation of normal forms for Turing-Hopf bifurcations.

Numerical examples illustrating complex spatiotemporal behaviors.

Abstract

Turing bifurcation and Hopf bifurcation are two important kinds of transitions giving birth to inhomogeneous solutions, in spatial or temporal ways. On a disk, these two bifurcations may lead to equivariant Turing-Hopf bifurcations. In this paper, normal forms for three kinds of Turing-Hopf bifurcations are given and the breathing, standing wave-like, and rotating wave-like patterns are found in numerical examples.