Abstract: This paper develops a computational method for studying
stable/unstable manifolds attached to periodic orbits of differential equations.
The method uses high order Chebyshev-Taylor series approximations in conjunction
with the parameterization methoda general functional analytic framework for invariant
manifolds. The parameterization method can follow folds in the embed- ding, recovers
the dynamics on the manifold through a simple conjugacy, and admits a natural notion
of a-posteriori error analysis. The key to the approach is the derivation of a recursive
system of linear differential equations describing the Taylor coefficients of the invariant
manifold. We find periodic solutions of these equations by solving a coupled collection of
boundary value problems with Chebyshev spectral methods. We discuss the performance of the
method for the Lorenz system, and for circular restricted three and four body problems.
We also illustrate the use of the method as a tool for computing cycle-to-cycle connecting
orbits.
Authors:
Maxime Murray and J.D. Mireles James
Preprint:
International Journal of Bifurcation and Chaos, Vol 27, No. 14 (2017).