Turing Instabilities and Patterns Near a Hopf Bifurcation

TL;DR

This paper establishes a precise condition for Turing instabilities in reaction-diffusion systems near a Hopf bifurcation, analyzing their relation to pattern formation and stability in specific models.

Contribution

It derives a necessary and sufficient condition for Turing instabilities in two-component reaction-diffusion systems with Neumann boundary conditions, especially near Hopf bifurcations.

Findings

Turing instabilities are neither necessary nor sufficient for Turing patterns.

Turing patterns can exist on both sides of the Hopf bifurcation.

Time periodic and stable solutions can coexist depending on initial conditions.

Abstract

We derive a necessary and sufficient condition for Turing instabilities to occur in two-component systems of reaction-diffusion equations with Neumann boundary conditions. We apply this condition to reaction-diffusion systems built from vector fields with one fixed point and a supercritical Hopf bifurcation. For the Brusselator and the Ginzburg-Landau reaction-diffusion equations, we obtain the bifurcation diagrams associated with the transition between time periodic solutions and asymptotically stable solutions (Turing patterns). In two-component systems of reaction-diffusion equations, we show that the existence of Turing instabilities is neither necessary nor sufficient for the existence of Turing pattern type solutions. Turing patterns can exist on both sides of the Hopf bifurcation associated to the local vector field, and, depending on the initial conditions, time periodic and stable solutions can coexist.