results continue to point to the fruitful connections that exist between multiple-timescale flows and low-dimensional maps.

\[sec:intro\] Introduction
==========================

We study slow-fast dynamical systems of the form

$$\begin{aligned}
{ \varepsilon}\dot{x} &=& f(x,y,{ \varepsilon})\\
\dot{y} &=& g(x,y,{ \varepsilon}),\end{aligned}$$

where $x \in R^m$ is the [*fast*]{} variable, $y \in R^n$ is the [*slow*]{} variable, ${ \varepsilon}$ is the [*singular perturbation parameter*]{} that characterizes the ratio of the timescales, and $f,g$ are sufficiently smooth. The [*critical manifold*]{} $C = \{f  = 0\}$ is the manifold of equilibria of the fast subsystem defined by $\dot{x} = f(x,y,0)$. When ${ \varepsilon}> 0$ is sufficiently small, theorems of Fenichel[@fenichel1972] guarantee the existence of locally invariant [*slow manifolds*]{} that perturb from subsets of $C$ where the equilibria are hyperbolic. We may also project the vector field $\dot{y} = g(x,y,0)$ onto the tangent bundle $TC$. Away from folds of $C$, we may desingularize this projected vector field to define the [*slow flow*]{}. The desingularized slow flow is oriented to agree with the full vector field near stable equilibria of $C$. For sufficiently small values of ${ \varepsilon}$, trajectories of the full system can be decomposed into segments lying on the slow manifolds near $C$ together with fast jumps 