** Authors: **

J.L. Gonzalez and J.D. Mireles James

** Abstract:**
We consider the problem of computing stable/unstable manifolds
attached to periodic orbits of maps, and develop seminumerical methods
for polynomial approximation of the manifolds to any desired order.
Our approach avoids function compositions by exploiting an idea inspired
by multiple shooting schemes for periodic orbits. We consider a system of
conjugacy equations which characterize chart maps for the local stable/unstable
manifold segments attached to the points of the periodic orbit. We develop a
formal series solution for the system of conjugacy equations, and show that
the coefficients of the series are determined by recursively solving certain
linear systems of equations. We derive the recursive equations for some example
problems in dimension two and three, and for examples with both polynomial and
transcendental nonlinearities. Finally we present some numerical results which
illustrate the utility of the method and highlight some technical numerical issues
such as controlling the decay rate of the coefficients and managing truncation
errors via a-posteriori indicators.

SIAM Journal on Applied Dynamical Systems. Vol. 16, No. 3, pp. 1748--1795 (2017).