Complex oscillatory motion of multiple spikes in a three-component Schnakenberg system

TL;DR

This paper studies a three-component Schnakenberg model exhibiting complex oscillatory spike dynamics near Hopf bifurcations, including stable in-phase and out-of-phase oscillations, and chaotic behavior far from bifurcation points.

Contribution

It introduces a novel three-component Schnakenberg system with multiple spike oscillations and derives reduced equations describing their long-time dynamics near bifurcations.

Findings

Multiple spike oscillations undergo Hopf bifurcations with multiple modes.

Stable in-phase and out-of-phase oscillations coexist for two spikes.

Numerical experiments suggest the presence of chaotic oscillations away from bifurcation points.

Abstract

In this paper, we introduce a three-component Schnakenberg model. Its key feature is that it has a solution consisting of N spikes that undergoes a Hopf bifurcation with respect to N distinct modes nearly simultaneously. This results in complex oscillatory dynamics of the spikes, not seen in typical two-component models. For parameter values above the Hopf bifurcations, we derive reduced equations of motion which consist of coupled ordinary differential equations (ODEs) of order 2N for spike positions and their velocities. These ODEs fully describe the slow-time evolution of the spikes near the Hopf bifurcations. We then apply the method of multiple scales to the resulting ODEs to derive long-time dynamics. For a single spike, we find that its long-time motion consists of oscillations near the steady-state whose amplitude can be computed explicitly. For two spikes, the long-time behaviour can be either in-phase or out-of-phase oscillations. Both in and out of phase orbits are stable, coexist for the same parameter values, and the fate of orbit depends solely on the initial conditions. Further away from the Hopf bifurcation point, we offer numerical experiments indicating the existence of highly complex and chaotic oscillations.