Relaxation Oscillations and the Entry-Exit Function in Multi-Dimensional Slow-Fast Systems

Abstract

For a slow-fast system of the form $\dot{p}=\epsilon f(p,z,\epsilon )+h(p,z,\epsilon )$, $\dot{z}=g(p,z,\epsilon )$ for $(p,z)\in {\mathbb{R}}^n\times {\mathbb{R}}^m$, we consider the scenario that the system has invariant sets $M_i=\{(p,z):z=z_i\}$, $1\le i\le N$, linked by a singular closed orbit formed by trajectories of the limiting slow and fast systems. Assuming that the stability of $M_i$ 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 multi-dimensional 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.

32 pages

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