= Local passage map near a dissipative saddle-node
{title2=$Y=h\left[\frac{(h+k)(x-k)}{(h-k)(x+k)}\right]^{\lambda/(2k)}$}
For $\dot x=x^2-\mu$, $\dot y=-\lambda y$, use entry $(x,h)$ and exit $(h,Y)$. If $\mu=k^2>0$, $x>k$, integration gives $Y=h[(h+k)(x-k)/((h-k)(x+k))]^{\lambda/(2k)}$. A smooth global return has $P(x)=\nu+cY+O(Y^2)$. If $\lambda>2k$, its derivative vanishes as $x\downarrow k$, and a small $\nu-k>0$ yields an attracting fixed point. If $\mu=-k^2<0$, the passage is $Y=h\exp[-(\lambda/k)(\arctan(h/k)-\arctan(x/k))]$. For small parameters the resulting global return is strongly contracting, with an attracting fixed point near $\nu$ on either side of zero.
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