Excitability in a Model with a Saddle-Node Homoclinic Bifurcation

TL;DR

This paper develops a two-dimensional reaction-diffusion model capturing saddle-node homoclinic bifurcation phenomena, providing insights into excitability, wave propagation, and chemical turbulence observed in experiments.

Contribution

It introduces a new theoretical framework for analyzing saddle-node homoclinic bifurcations and excitability in reaction-diffusion systems, linking mathematical bifurcation theory with chemical experiment observations.

Findings

Extended system can exhibit wave trains, solitary pulses, and pulse packets depending on initial conditions.

Chemical turbulence occurs when diffusion coefficients are equal.

The model helps design experiments to distinguish types of excitability thresholds.

Abstract

In order to describe excitable reaction-diffusion systems, we derive a two-dimensional model with a Hopf and a semilocal saddle-node homoclinic bifurcation. This model gives the theoretical framework for the analysis of the saddle-node homoclinic bifurcation as observed in chemical experiments, and for the concepts of excitability and excitability threshold. We show that if diffusion drives an extended system across the excitability threshold then, depending on the initial conditions, wave trains, propagating solitary pulses and propagating pulse packets can exist in the same extended system. The extended model shows chemical turbulence for equal diffusion coefficients and presents all the known types of topologically distinct activity waves observed in chemical experiments. In particular, the approach presented here enables to design experiments in order to decide between excitable systems with sharp and finite width thresholds.