Hyperbolic Chaos of Turing Patterns

TL;DR

This paper demonstrates that Turing patterns under periodic modulation exhibit hyperbolic chaos, characterized by an expanding circle map and a Smale-Williams attractor, which is robust to parameter and boundary changes.

Contribution

It introduces a novel theoretical and numerical analysis showing hyperbolic chaos in Turing patterns governed by a modified Swift-Hohenberg equation.

Findings

Turing patterns exhibit hyperbolic chaos with a Smale-Williams attractor.

The chaos is robust against parameter and boundary condition variations.

Spatial phases follow an expanding circle map leading to hyperbolic dynamics.

Abstract

We consider time evolution of Turing patterns in an extended system governed by an equation of the Swift-Hohenberg type, where due to an external periodic parameter modulation long-wave and short-wave patterns with length scales related as 1:3 emerge in succession. We show theoretically and demonstrate numerically that the spatial phases of the patterns, being observed stroboscopically, are governed by an expanding circle map, so that the corresponding chaos of Turing patterns is hyperbolic, associated with a strange attractor of the Smale-Williams solenoid type. This chaos is shown to be robust with respect to variations of parameters and boundary conditions.