  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.
