Piecewise linear relaxation-oscillator period ($T\sim \log [(1+\alpha )(1-\beta )/(-\alpha \beta )]$)

For $ϵ\dot{v}=g(v)-w$, $\dot{w}=v$, let $g=-v$ below $\alpha$, $g=1-v$ above $\beta$, and join the endpoints by an increasing line, with $-1<\alpha <0<\beta <1$ and $\beta -\alpha <1$. The slow branches obey $\dot{w}=-w$ and $\dot{w}=1-w$ between $w_L=-\alpha$ and $w_R=1-\beta$. Their travel times are $\log [(1-\beta )/(-\alpha )]$ and $\log [(1+\alpha )/\beta ]$. Fast horizontal jumps contribute vanishing time as $ϵ\to 0$, giving the displayed relaxation oscillation period.