Bifurcations of Clusters and Collective Oscillations in Networks of Bistable Units

TL;DR

This paper models the collective oscillations and bifurcations in networks of bistable electrochemical units, revealing how coupling induces oscillations and complex bifurcation structures, including stable limit cycles.

Contribution

It introduces a mathematical framework capturing collective oscillations in coupled bistable units and analyzes their bifurcations, including an equivariant transcritical bifurcation of limit cycles.

Findings

Coupling induces collective oscillations in the array.

Bifurcation analysis reveals complex dynamics including stable limit cycles.

System exhibits an equivariant transcritical bifurcation of limit cycles.

Abstract

We investigate dynamics and bifurcations in a mathematical model that captures electrochemical experiments on arrays of microelectrodes. In isolation, each individual microelectrode is described by a one-dimensional unit with a bistable current-potential response. When an array of such electrodes is coupled by controlling the total electric current, the common electric potential of all electrodes oscillates in some interval of the current. These coupling-induced collective oscillations of bistable one-dimensional units are captured by the model. Moreover, any equilibrium is contained in a cluster subspace, where the electrodes take at most three distinct states. We systematically analyze the dynamics and bifurcations of the model equations: We consider the dynamics on cluster subspaces of successively increasing dimension and analyze the bifurcations occurring therein. Most importantly, the system exhibits an equivariant transcritical bifurcation of limit cycles. From this bifurcation, several limit cycles branch, one of which is stable for arbitrarily many bistable units.