= Three-time-scale cubic relaxation oscillator
{title2=$\dot x=\alpha-y,\ \epsilon\dot y=x-z,\ \delta\dot z=f(y)-z$}
For $f(y)=y-y^2+\gamma y^3/3$ with $0<\gamma<1$, the <fold points of a critical manifold> are $y_\pm=(1\pm\sqrt{1-\gamma})/\gamma$. When $\delta\ll\epsilon\ll1$, the fastest variable satisfies $z\simeq f(y)$ even during jumps of the intermediate variable. At fixed $x$, the outer branches of $x=f(y)$ attract the fast $y$ dynamics because $f'(y)>0$; its middle branch repels. If $y_-<\alpha<y_+$, the slow equation $\dot x=\alpha-y$ drives the low branch to its <fold point of a critical manifold> and the high branch back to the other <fold point of a critical manifold>. Fast jumps at almost fixed $x$ close a relaxation cycle. This explains why a reduction justified on the slow arcs must also be checked on the jump time scale. For $\gamma>1$, the cubic is strictly increasing; the reduced <energy> $V=[x-f(\alpha)]^2/2+\epsilon(y-\alpha)^2/2$ has $\dot V=-(y-\alpha)[f(y)-f(\alpha)]\leq0$, precluding this oscillation mechanism.
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