Nonuniform relaxation oscillations near SNIPER bifurcations

TL;DR

This paper investigates spatially dependent relaxation oscillations near SNIPER bifurcations, revealing long-wavelength instabilities and diverse oscillation patterns like chimeras and chaos across different systems.

Contribution

It demonstrates how SNIPER bifurcations can lead to spatially modulated oscillations and instabilities in extended media, with implications for various physical systems.

Findings

SNIPER bifurcation creates large-amplitude, long-period orbits.

Long-wavelength instabilities cause spatially modulated oscillations.

Different systems exhibit chimera states and chaotic spiking.

Abstract

Properties of spatially dependent relaxation oscillations near a SNIPER bifurcation are described. A SNIPER bifurcation creates a large-amplitude long-period periodic orbit via the annihilation of a pair of fixed points in a saddle-node bifurcation. We show that in spatially extended media, this orbit may undergo a long-wavelength instability, leading to spatially modulated oscillations that persist on both sides of the SNIPER. The oscillations take different forms depending on the system: a chimera state in a theta-reaction-diffusion model, and chaotic spiking in an activator-inhibitor-substrate model. The results are expected to have applications in a number of physical systems exhibiting SNIPER bifurcations, ranging from models of the nervous system through chemical reactions to nonlinear optics.