** Authors: **

Jason D. Mireles-James

Konstantin Mischaikow

** Abstract:**This work is concerned with high order polynomial approximation of stable and unstable
manifolds for analytic discrete time dynamical systems. We develop a-posteriori theorems for these
polynomial approximations which allow us to obtain rigorous bounds on the truncation errors via a
computer assisted argument. Moreover we represent the truncation error as an an- alytic function, so
that that the derivatives of the truncation error can be bound using classical estimates of complex
analysis. As an application of these ideas we combine the approximate mani- folds and rigorous bounds
with a standard Newton-Kantorovich argument in order to obtain a kind of analytic-shadowing result
for connecting orbits between fixed points of discrete time dynamical systems. Examples of the manifold
computation are given for invariant manifolds which have dimen- sion between two and ten. Examples of
the a-posteriori error bounds and the analytic shadowing argument for connecting orbits are given for
dynamical systems in dimension three and six.

SIAM Journal on Applied Dynamical Systems, Vol. 12, No. 2, pp. 957-1006 (2013).