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A Shilnikov bifurcation involves a homoclinic orbit to a saddle-focus equilibrium in three dimensions. Write its eigenvalues asand put $\delta =-{\lambda }_{-}/{\lambda }_{+}$. The Shilnikov return map combines contraction toward the equilibrium with arbitrarily many rotations before departure.

The sign of the saddle value ${\lambda }_{+}+{\lambda }_{-}={\lambda }_{+}(1-\delta )$ separates two regimes. At a nondegenerate homoclinic connection with $0<\delta <1$, the Shilnikov theorem gives countably many nearby saddle periodic orbits. When $\delta >1$, the usual generic unfolding instead creates a locally attracting periodic orbit on the appropriate side. The boundary $\delta =1$ requires additional terms.^[ref]

The spectral and nondegeneracy assumptions matter: a planar saddle loop has no rotating stable plane and does not supply this mechanism. The result is associated with Leonid Pavlovich Shilnikov; Andrey Shilnikov coauthored the cited account.

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 $z$ takesto 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 formwhere $|\Omega |=|\omega |/{\lambda }_{+}$ and the phase convention absorbs constant angular shifts. The map applies only on branches whose trajectories return to the section.^[ref]

Differentiation givesFor $\delta <1$, the envelope becomes unbounded as $z\downarrow 0$, 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.