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September  2008, 10(2&3, September): 511-536. doi: 10.3934/dcdsb.2008.10.511

High-precision computations of divergent asymptotic series and homoclinic phenomena

Vassili Gelfreich 1, and  Carles Simó 2,

1. 

Mathematics Institute, University of Warwick, Coventry CV4 7AL, United Kingdom
2. 

Departament de Matemàtica Aplicada i Anàlisi, Universitat de Barcelona, Gran Via 585, 08007 Barcelona, Spain

Received  November 2006 Revised  May 2007 Published  June 2008

We study asymptotic expansions for the exponentially small splitting of separatrices of area preserving maps combining analytical and numerical points of view. Using analytic information, we conjecture the basis of functions of an asymptotic expansion and then extract actual values of the coefficients of the asymptotic series numerically. The computations are performed with high-precision arithmetic, which involves up to several thousands of decimal digits. This approach allows us to obtain information which is usually considered to be out of reach of numerical methods. In particular, we use our results to test that the asymptotic series are Gevrey-1 and to study positions and types of singularities of their Borel transform. Our examples are based on generalisations of the standard and Hénon maps.
Keywords: standard map., homoclinic orbit, High-precision computation.
Mathematics Subject Classification: Primary: 37J45; Secondary: 65P1.
Citation: Vassili Gelfreich, Carles Simó. High-precision computations of divergent asymptotic series and homoclinic phenomena. Discrete and Continuous Dynamical Systems - B, 2008, 10 (2&3, September) : 511-536. doi: 10.3934/dcdsb.2008.10.511
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