Automatic differentiation for Fourier series and the radii polynomial approach

作者：Lessard Jean Philippe; James J D Mireles; Ransford Julian

来源：Physica D: Nonlinear Phenomena , 2016, 334: 174-186.

DOI：10.1016/j.physd.2016.02.007

• Rigorous numerics
• Automatic differentiation
• Fourier series
• Contraction Mapping Theorem
• Periodic-solutions

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).

• 出版日期2016-11-1