On Bifurcations of Birth of Closed Invariant Curves in the Case of Two-Dimensional Diffeomorphisms with Homoclinic Tangencies

S. V. Gonchenko ; V. S. Gonchenko

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Dynamical systems and related problems of geometry, Tome 244 (2004), pp. 87-114

Résumé

We study the bifurcations of periodic orbits in two-parameter families of two-dimensional diffeomorphisms close to a diffeomorphism with a uadratic homoclinic tangency of the manifolds of a saddle fixed point of neutral type (with multipliers $\lambda$ and $\gamma$ such that $|\lambda|1$, $|\gamma|>1$, and $\lambda\gamma =1$). In particular, we consider the question of the birth of closed invariant curves from “weak focus” periodic orbits (i.e. those with multipliers $e^{\pm i\psi}$, where $0\psi\pi $). It is shown that the first Lyapunov value of such an orbit is nonzero in general, and its sign coincides with the sign of a “separatrix value” that is a function of the coefficients of a return map near the global piece of the homoclinic orbit.