97-160 Amadeu Delshams, Rafael Ramirez-Ros

Exponentially small splitting of separatrices for standard-like maps (405K, LaTeX 2.09) Apr 2, 97

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Abstract. The splitting of separatrices for the standard-like maps $$ F(x,y) = (y,-x + 2\mu y/(1+y^{2}) + \varepsilon V'(y)), \qquad \mu=\cosh h, h>0, \varepsilon \in {\bf R}, $$ is measured. For even entire perturbative potentials $V(y) = \sum_{n\ge 2} V_{n} y^{2n}$ such that $\widehat{V}(2\pi) \neq 0$, where $\widehat{V}(\xi) = \sum_{n\ge 2} V_{n} \xi^{2n-1}/(2n-1)!$ is the Borel transform of $V(y)$, the following asymptotic formula for the area $A$ of the lobes between the perturbed separatrices is established $$ A = 8 \pi \widehat{V}(2\pi) \varepsilon e^{-\pi^{2}/h} [1 + O(h^{2})] \qquad (\varepsilon = o(h^{6} \ln^{-1}h), h \to 0^{+}). $$ This formula agrees with the one provided by the Melnikov theory, which cannot be applied directly, due to the exponentially small size of $A$ with respect to $h$.

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