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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