- 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
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.