Turing instabilities from a limit cycle

TL;DR

This paper generalizes the concept of Turing instability to systems with periodic solutions, including oscillation death in coupled oscillators, providing conditions for instability in both spatial and networked systems.

Contribution

It introduces a new framework extending Turing instability analysis to systems with limit cycle solutions, applicable to continuum and network structures.

Findings

Derived closed conditions for Turing instability in periodic systems

Validated the theory with numerical simulations on various reaction schemes

Unified understanding of oscillation death as a Turing instability

Abstract

The Turing instability is a paradigmatic route to patterns formation in reaction-diffusion systems. Following a diffusion-driven instability, homogeneous fixed points can become unstable when subject to external perturbation. As a consequence, the system evolves towards a stationary, nonhomogeneous attractor. Stable patterns can be also obtained via oscillation quenching of an initially synchronous state of diffusively coupled oscillators. In the literature this is known as the oscillation death phenomenon. Here we show that oscillation death is nothing but a Turing instability for the first return map associated to the excitable system in its synchronous periodic state. In particular we obtain a set of closed conditions for identifying the domain in the parameters space that yields the instability. This is a natural generalisation of the original Turing relations, to the case where the homogeneous solution of the examined system is a periodic function of time. The obtained framework applies to systems embedded in continuum space, as well as those defined on a network-like support. The predictive ability of the theory is tested numerically, using different reaction schemes.