00-402 Vassili Gelfreich

Splitting of separatrices near resonant periodic orbit (1267K, LaTeX 2e) Oct 12, 00

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Abstract. We consider an analytic family of area-preserving maps $F_\varepsilon$ with an elliptic fixed point. We assume that for $\varepsilon=0$ the fixed point is resonant of an order $n=1,2$ or $3$. In each of these cases the fixed point can be unstable at the exact resonance, and close to the exact resonance there is a hyperbolic periodic orbit. The resonant normal form is integrable and its separatrices form a small loop. Separatrices of the map $F_\varepsilon$ are close to the separatrices of the normal form but can intersect transversally. Asymptotic formulae for the splitting of separatrices are provided. The splitting is exponentially small compared to $\varepsilon$ and can not be detected by Melnikov method. This problem is equivalent to studying a generic family of close-to-resonant elliptic periodic orbits in an analytic Hamiltonian system with two degrees of freedom.

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