  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.
