** Authors: **

Roberto Castelli

Jean-Philippe Lessard

J. D. Mireles-James

** 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 non-resonance 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.

SIAM Journal on Applied Dynamical Systems, Vol. 14, No. 1, pp. 132-167.