4 research outputs found
Automatic differentiation for Fourier series and the radii polynomial approach
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