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

Gelfreich, Vassili; Simó, Carles

Discrete Contin. Dyn. Syst., Ser. B 10, No. 2-3, 511-536 (2008).

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.

Reviewer: Milena Radnovic (Beograd)

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

Keywords:

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

Citations:

Zbl 0963.65136