Automatic differentiation for Fourier series and the radii polynomial approach

Authors: Lessard, Jean-Philippe; Mireles James, Jason D.; Ransford, Julian
Abstract: 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)
Document Type: Article de recherche
Issue Date: 2 March 2016
Open Access Date: 2 March 2018
Document version: AM
Permalink: http://hdl.handle.net/20.500.11794/1345
This document was published in: Physica D. Nonlinear phenomena, Vol. 317, 1-24 (2016)
https://doi.org/10.1016/j.physd.2016.02.007
Amsterdam : North-Holland
Alternative version: 10.1016/j.physd.2016.02.007
Collection:Articles publiés dans des revues avec comité de lecture

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