V. Gelfreich
Splitting of a small separatrix loop near the saddle-center bifurcation
in area-preserving maps
(183K, LaTeX 2.09)
ABSTRACT. When the saddle-center bifurcation occurs in an analytic family of
area-preserving maps, first a parabolic fixed point appears at the origin
and then this point bifurcates, creating an elliptic and hyperbolic fixed
point. Separatrices of the hyperbolic fixed point form a small loop around
the elliptic point. In general the separatrices intersect transversaly and
the splitting is exponentially small with respect to the perturbation
parameter. We derive an asymptotic formula, which describes the
splitting, and study the properties of the preexponential factor.