 Shilnikov return map

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 $T=T_g+{\lambda }_{+}^{-1}\log (z_0/z)$.