Nonuniform relaxation oscillations near SNIPER bifurcations

Edgar Knobloch; Arik Yochelis

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arXiv · 2605.11878

Nonuniform relaxation oscillations near SNIPER bifurcations

Edgar Knobloch·Arik Yochelis

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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.

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Edgar Knobloch, Arik Yochelis. 2026-05-12. Nonuniform relaxation oscillations near SNIPER bifurcations. https://arxiv.org/abs/2605.11878

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