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Automatic differentiation for Fourier series and the radii polynomial approach
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- article de recherche
- COAR1_1::Texte::Périodique::Revue::Contribution à un journal::Article::Article de recherche
- Rigorous numerics
- Automatic differentiation
- Fourier series
- Contraction Mapping Theorem
- periodic solutions
- Analyse numérique
- Séries de Fourier
- Équations différentielles non linéaires -- Solutions numériques
- Orbites périodiques (Mathématiques)