Localized spatiotemporal reaction-diffusion patterns on a line and a disk arising from a subcritical finite wavenumber Hopf instability

TL;DR

This paper investigates localized and extended spatiotemporal patterns arising from a subcritical Hopf bifurcation in a reaction-diffusion model, revealing complex snaking behavior and structures on lines and disks with implications for biological wave phenomena.

Contribution

It provides a detailed numerical and theoretical analysis of localized structures and snaking behavior in a reaction-diffusion system near a subcritical Hopf bifurcation, including two-dimensional pattern organization.

Findings

Identification of wall-attached traveling and oscillating spots on disks

Demonstration of domain-filling and disordered structures in 2D

Correlation of 1D results with 2D pattern formation

Abstract

Spatiotemporal localized and extended structures associated with a subcritical finite wavenumber Hopf bifurcation are studied in the Purwins model (a three-variable FitzHugh-Nagumo version). Steady and time-dependent numerical continuation procedures are used to investigate snaking behavior of localized standing and traveling waves on the real line, and the results are corroborated using weakly nonlinear theory. The results shed light on the origin of so-called jumping oscillons and the organization of a nontypical homoclinic snaking structure of traveling pulses. The computations are extended to moderate size disks and used to identify wall-attached spots that travel along the disk boundary as well as wall-attached spots that oscillate in place and wall-attached jumping oscillons. The one-dimensional results are shown to be useful in interpreting the two-dimensional results. Domain-filling and mixed structures are also studied, demonstrating the variety of extended and localized states that emerge in two-space dimensions, ranging from periodic to disordered. The latter are potentially important for observations of waves in far-from-equilibrium media, such as those often observed in cell biology.