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

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.