Relaxation oscillations and the entry-exit function in multidimensional slow-fast systems

Abstract

For a slow-fast system of the form p = ∊ f(p, z, ∊) + h(p, z, ∊), z = g(p, z, ∊) for (p, z) ∊ Rn × Rm, we consider the scenario that the system has invariant sets Mi = {(p, z): z = zi} , 1 ≤ i ≤ N, linked by a singular closed orbit formed by trajectories of the limiting slow and fast systems. Assuming that the stability of Mi changes along the slow trajectories at certain turning points, we derive criteria for the existence and stability of relaxation oscillations for the slow-fast system. Our approach is based on a generalization of the entry-exit relation to systems with multidimensional fast variables. We then apply our criteria to several predator-prey systems with rapid ecological evolutionary dynamics to show the existence of relaxation oscillations in these models.