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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 ,y 1 =y+ϵf(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 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: 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:
37J45Periodic, homoclinic and heteroclinic orbits; variational methods, degree-theoretic methods
65P10Numerical methods for Hamiltonian systems including symplectic integrators
37C29Homoclinic and heteroclinic orbits
37G20Hyperbolic singular points with homoclinic trajectories