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Zbl 1169.37013
Gelfreich, Vassili; Simó, Carles
High-precision computations of divergent asymptotic series and homoclinic phenomena.
(English)
[J] Discrete Contin. Dyn. Syst., Ser. B 10, No. 2-3, 511-536 (2008). ISSN 1531-3492; ISSN 1553-524X/e

A method suitable for studying the exponentially small splitting of separatrices appearing in the generalizations of standard map: $$x_{1}=x+y_{1},\quad y_{1}=y+\epsilon f(x),$$ with $f$ being a polynomial, trigonometric polynomial, meromorphic or rational function is developed. The method is a combination of analytical and numerical steps, with high-precision computations. After the introduction, the analytical results on the splitting of separatrices from the generalized standard map are reviewed. Then, full details of numerical methods sketched in {\it C. Simó} [in: International conference on differential equations. Proceedings of the conference, Equadiff '99, Berlin, Germany, August 1--7, 1999. Vol. 2. Singapore: World Scientific. 967--976 (2000; Zbl 0963.65136)] are given. The numerical procedure consists of two main steps: first, the values of the homoclinic invariant are computed; then, the obtained data are used to extract coefficients of an asymptotic expansion. After that, asymptotic formulae for $f(x)$ being a polynomial of degree $2$ to $5$ are described in detail. Finally, singularities of the separatrix solutions of the ODE: $\ddot{x}_{0}=f(x_{0})$ are studied, both in the case when these solutions can be found explicitly and when this is not possible in terms of elementary functions.
MSC 2000:
*37J45 Periodic, homoclinic and heteroclinic orbits, etc.
65P10 Hamiltonian systems including symplectic integrators
37C29 Homoclinic and heteroclinic orbits
37G20 Hyperbolic singular points with homoclinic trajectories

Keywords: high-precision computation; homoclinic orbit; standard map; Hénon map; splitting of separatrices; hyperbolic fixed point; Gevrey-1 asymptotic series

Citations: Zbl 0963.65136

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