Parameterization of Invariant Manifolds for Periodic Orbits I: Efficient Numerics via the Floquet Normal Form

Abstract

We present an efficient numerical method for computing Fourier--Taylor expansions of (un)stable manifolds associated with hyperbolic periodic orbits. Three features of the method are that (1) we obtain accurate representation of the invariant manifold as well as the dynamics on the manifold, (2) it admits natural a posteriori error analysis, and (3) it does not require numerically integrating the vector field. Our approach is based on the parameterization method for invariant manifolds, and studies a certain partial differential equation which characterizes a chart map of the manifold. The method requires only that some mild nonresonance conditions hold. The novelty of the present work is that we exploit the Floquet normal form in order to efficiently compute the Fourier--Taylor expansion. A number of example computations are given including manifolds in phase space dimension as high as ten and manifolds which are two and three dimensional. We also discuss computations of cycle-to-cycle connecting orbits which exploit these manifolds.