= Shilnikov return map
{c}
{title2=$z'=-\mu+Az^\delta\cos(\Omega\log z+\Phi)$}
Near a <saddle-focus equilibrium>, linearized coordinates satisfy $\dot z=\lambda_+z$, $\dot r=\lambda_-r$ and $\dot\theta=\omega$. A trajectory entering at a small positive coordinate $z$ takes
$$
T_{\mathrm{loc}}=\frac1{\lambda_+}\log\frac{z_0}{z}
$$
to reach a fixed outgoing section. Hence $r_{\mathrm{out}}\propto z^\delta$ with $\delta=-\lambda_-/\lambda_+$, while the accumulated angle is logarithmic in $z$. Composing this passage with a smooth global return gives the leading scalar form
$$
P_\mu(z)=-\mu+Az^\delta\cos(\Omega\log z+\Phi),\qquad z>0,
$$
where $|\Omega|=|\omega|/\lambda_+$ and the phase convention absorbs constant angular shifts. The map applies only on branches whose trajectories return to the section.<Shilnikov Bifurcation by Leonid Pavlovich Shilnikov and Andrey Shilnikov>[Local saddle-focus passage and the oscillatory Poincaré return map]{ref}
Differentiation gives
$$
P_\mu'(z)=Az^{\delta-1}\left[\delta\cos(\Omega\log z+\Phi)-\Omega\sin(\Omega\log z+\Phi)\right].
$$
For $\delta<1$, the envelope becomes unbounded as $z\downarrow0$, but the derivative also has infinitely many zeros. The map therefore has strongly expanding branches, rather than uniform expansion everywhere.
The full two-dimensional return has <Jacobian determinant> of order $z^{2\delta-1}$ in regular section coordinates: radial scaling contributes $z^{\delta-1}$ and conversion of polar coordinates contributes $z^\delta$. For $\delta>1/2$ this contracts area near the singular limit, but area contraction alone does not prove that a scalar reduction is valid. A one-passage periodic trajectory has period $T_g+T_{\mathrm{loc}}$, explaining the logarithmically diverging time near the homoclinic connection.
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