= Shilnikov bifurcation
{c}
A <Shilnikov bifurcation> involves a <homoclinic orbit> to a <saddle-focus equilibrium> in three dimensions. Write its <eigenvalues> as
$$
\lambda_+>0,\qquad \lambda_-\pm i\omega,\quad \lambda_-<0,\quad\omega\ne0,
$$
and 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.<Shilnikov Bifurcation by Leonid Pavlovich Shilnikov and Andrey Shilnikov>[Saddle quantity, the Shilnikov theorem and the contracting saddle-focus case]{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.
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