Chaos at the border of criticality

TL;DR

This paper introduces a new mechanism for chaotic attractors in excitable cell models near a bifurcation point, using Poincaré maps to explain complex oscillations and their statistical properties.

Contribution

It presents a novel visual and analytical approach to understanding chaos near a degenerate Andronov-Hopf bifurcation in cell membrane models.

Findings

Chaotic mixed-mode oscillations occur near the border of sub- and supercritical bifurcations.

The Poincaré map exhibits unimodality and boundary layer features.

Predicted statistical properties match numerical experiments.

Abstract

The present paper points out to a novel scenario for formation of chaotic attractors in a class of models of excitable cell membranes near an Andronov-Hopf bifurcation (AHB). The mechanism underlying chaotic dynamics admits a simple and visual description in terms of the families of one-dimensional first-return maps, which are constructed using the combination of asymptotic and numerical techniques. The bifurcation structure of the continuous system (specifically, the proximity to a degenerate AHB) endows the Poincare map with distinct qualitative features such as unimodality and the presence of the boundary layer, where the map is strongly expanding. This structure of the map in turn explains the bifurcation scenarios in the continuous system including chaotic mixed-mode oscillations near the border between the regions of sub- and supercritical AHB. The proposed mechanism yields the statistical properties of the mixed-mode oscillations in this regime. The statistics predicted by the analysis of the Poincare map and those observed in the numerical experiments of the continuous system show a very good agreement.