= Cubic-nullcline relaxation period
{title2=$T\sim3-2\log2$}
For $\epsilon\dot x=x-x^3/3+y$, $\dot y=-x$ and $\epsilon\to0^+$, the attracting branches of the cubic <nullcline> $y=x^3/3-x$ have $|x|>1$. The limiting cycle follows $x=2$ to $x=1$, jumps to $x=-2$ at fixed $y=-2/3$, follows the other outer branch to $x=-1$, and jumps back at $y=2/3$. On a slow branch, $(x^2-1)\dot x=-x$, so one branch takes $\int_2^1(1/x-x)\,dx=3/2-\log2$. Reflection supplies the other half; fast-jump and fold times vanish at leading order. This proves the displayed asymptotic <period>. Negative $\epsilon$ reverses fast stability and does not preserve this attracting cycle.
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