a large-amplitude passage. The first and second derivatives of the map grow very large outside of large subintervals where the map is unimodal.

We also interpret classical bifurcations of the one-dimensional map as routes to chaotic behavior in the full system. We show that a period-doubling cascade occurs in this map, which gives rise to chaotic MMOs. This cascade is reminiscent of the classical cascade in the family of quadratic maps, even though on small subsets, our return map is far from unimodal. Saddle-nodes of mixed-mode cycles, defined as fixed points of the return map with unit derivative, are also shown to occur. Finally, we identify a parameter set for which the full dynamics is close to the dynamics of a unimodal map with a critical point having dense forward orbit.

\[sec:shnf\] Three-Dimensional System of Equations
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We study the following three-dimensional flow: $$\begin{aligned}
{ \varepsilon}\dot{x} &=& y - x^2 - x^3 \nonumber\\
\dot{y} &=& z - x \label{eq:shnf}\\
\dot{z} &=& -\nu - ax -by - cz,\nonumber\end{aligned}$$

where $x$ is the fast variable, $y,z$ are the slow variables, and ${ \varepsilon},\nu,a,b,c$ are the system parameters. This system exhibits a singular Hopf bifurcation.[@braaksma1998; @guckenheimer2008siam; @guckenheimer2012dcds] The critical manifold is the S-shaped 