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High-precision computations of divergent asymptotic series and homoclinic phenomena. (English) Zbl 1169.37013

A method suitable for studying the exponentially small splitting of separatrices appearing in the generalizations of standard map:

${x}_{1}=x+{y}_{1},\phantom{\rule{1.em}{0ex}}{y}_{1}=y+ϵf\left(x\right),$

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 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 $f\left(x\right)$ being a polynomial of degree 2 to 5 are described in detail. Finally, singularities of the separatrix solutions of the ODE: ${\stackrel{¨}{x}}_{0}=f\left({x}_{0}\right)$ 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:
 37J45 Periodic, homoclinic and heteroclinic orbits; variational methods, degree-theoretic methods 65P10 Numerical methods for Hamiltonian systems including symplectic integrators 37C29 Homoclinic and heteroclinic orbits 37G20 Hyperbolic singular points with homoclinic trajectories