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Andrey Shilnikov studies bifurcation theory and dynamical systems. With Leonid Pavlovich Shilnikov, he authored an account of the Shilnikov bifurcation, explaining how a homoclinic orbit to a saddle-focus can produce complicated dynamics. The construction connects local contraction and expansion near an equilibrium with a global return of its unstable manifold.^[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.