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