Automatic differentiation for Fourier series and the radii polynomial approach

PHYSICA D-NONLINEAR PHENOMENA

卷:334

Article

Lessard, Jean-Philippe¹  James, J. D. Mireles²  Ransford, Julian¹ 

[1] Univ Laval, Dept Math & Stat, 1045 Ave Med, Quebec City, PQ G1V 0A6, Canada

[2] Florida Atlantic Univ, Dept Math Sci, Sci Bldg,Room 234,777 Glades Rd, Boca Raton, FL 33431 USA

关键词: Rigorous numerics;    Automatic differentiation;    Fourier series;    Contraction Mapping Theorem;    Periodic-solutions;   

DOI  :  10.1016/j.physd.2016.02.007

来源: Elsevier

【 摘 要 】

In this work we develop a computer-assisted technique for proving existence of periodic solutions of nonlinear differential equations with non-polynomial nonlinearities. We exploit ideas from the theory of automatic differentiation in order to formulate an augmented polynomial system. We compute a numerical Fourier expansion of the periodic orbit for the augmented system, and prove the existence of a true solution nearby using an a-posteriori validation scheme (the radii polynomial approach). The problems considered here are given in terms of locally analytic vector fields (i.e. the field is analytic in a neighborhood of the periodic orbit) hence the computer-assisted proofs are formulated in a Banach space of sequences satisfying a geometric decay condition. In order to illustrate the use and utility of these ideas we implement a number of computer-assisted existence proofs for periodic orbits of the Planar Circular Restricted Three-Body Problem (PCRTBP). (C) 2016 Elsevier B.V. All rights reserved.

【 授权许可】

Free