Shilnikov return map ($z^{\prime }=-\mu +A{z}^{\delta }\cos (\Omega \log z+\Phi )$)

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

At a saddle-focus homoclinic orbit, the leading Shilnikov return map is $f(x)=A{x}^{\delta }\cos (q\log x+\Phi )$, $x>0$, with $q>0$. If $1/2<\delta <1$, its fixed points solve $\cos (q\log x+\Phi )=x^{1-\delta }/A$. Infinitely many solutions accumulate at zero near the zeros of the cosine, with successive size ratios tending to $e^{-\pi /q}$. The derivative is $A{x}^{\delta -1}[\delta \cos (q\log x+\Phi )-q\sin (q\log x+\Phi )]$, so their expanding multiplier is unbounded. Narrow monotone strips near successive oscillations map across a common interval; inverse branches contract, supplying arbitrarily long symbolic itineraries and infinitely many saddle periodic orbits. Varying the splitting parameter shifts the graph vertically, producing accumulating fold and period-doubling thresholds and stable windows. The condition $\delta >1/2$ makes the local flow volume-contracting, whereas $\delta <1$ makes its saddle value positive.