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Leonid Pavlovich Shilnikov developed the theory of Shilnikov bifurcations associated with a saddle-focus. The Shilnikov bifurcation explains a route to chaos through the return of an unstable manifold to the same equilibrium. His joint account with Andrey Shilnikov describes the role of the saddle quantity in distinguishing stable periodic behavior from more complicated nearby dynamics.^[ref]

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.

By Leonid Pavlovich Shilnikov, Andrey Shilnikov.

Scholarpedia 2(8):1891.