 Cubic-nullcline relaxation period

For $ϵ\dot{x}=x-x^3/3+y$, $\dot{y}=-x$ and $ϵ\to 0^{+}$, 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 ${\int }_2^1(1/x-x)dx=3/2-\log 2$. Reflection supplies the other half; fast-jump and fold times vanish at leading order. This proves the displayed asymptotic period. Negative $ϵ$ reverses fast stability and does not preserve this attracting cycle.