Fourier-Taylor Approximation of Unstable Manifolds for Compact Maps: Numerical Implementation and Computer-Assisted Error Bounds

We develop and implement a semi-numerical method for computing high order Taylor approximations of the unstable manifold at a hyperbolic fixed point of a compact infinite dimensional analytic map. Even though the method involves several layers of truncation our goal is to obtain a representation of the invariant manifold which is accurate in a large region about the fixed point. In order to insure the accuracy of our computations we develop a-posteriori error bounds for the approximations. Numerical implementation of the a-posteriori theory, combined with deliberate control of floating point round-off errors (or interval arithmetic), leads to mathematically rigorous computer assisted theorems describing precisely the truncation errors for the approximation of the invariant manifold. The method is illstrated for the Kot-Schaffer model of population dispersion.