= Shilnikov return map
{c}
{title2=$z'=-\mu+Az^\delta\cos(\Omega\log z+\Phi)$}
A passage near a saddle-focus contracts a transverse radius by $z^\delta$ while rotating it by an angle proportional to $\log z$. A smooth global reinjection converts these transverse coordinates into a sinusoidal return in $z$. The map is defined only for positive passage coordinates that return on the chosen branch. The full two-dimensional map has determinant of order $z^{2\delta-1}$; for $\delta>1/2$ its thin transverse geometry supports a leading one-dimensional approximation. A one-passage orbit has period $T=T_g+\lambda_+^{-1}\log(z_0/z)$.
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