
Automatic
differentiation for Fourier Series and the radii
polynomial approach
J.P.
Lessard, J.D. Mireles James and J. Ransford
In this work we
develop a computerassisted technique for proving
existence of periodic solutions of nonlinear
differential equations with nonpolynomial
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 aposteriori 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
computerassisted 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 computerassisted
existence proofs for periodic orbits of the Planar
Circular Restricted ThreeBody Problem (PCRTBP).
