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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: \[ x_{1}=x+y_{1},\quad y_{1}=y+\epsilon f(x), \] 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 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:

37J45 Periodic, homoclinic and heteroclinic orbits; variational methods, degree-theoretic methods (MSC2010)
65P10 Numerical methods for Hamiltonian systems including symplectic integrators
37C29 Homoclinic and heteroclinic orbits for dynamical systems
37G20 Hyperbolic singular points with homoclinic trajectories in dynamical systems

Citations:

Zbl 0963.65136
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