Homoclinic orbits of non-autonomous maps and their approximation

We consider homoclinic orbits in non-autonomous discrete time dynamical systems of the form Xn+l = f(n)(x(n)), n is an element of Z, where it is assumed that an n independent fixed point exists. A numerical method for computing finite approximations of transversal homoclinic orbits is introduced and a detailed error analysis is presented. The non-autonomous setup requires special tools. We prove that the analytic condition of transversality of the orbit corresponds to a transversal intersection of the corresponding invariant fiber bundles. The approximation method and the validity of the error estimate is illustrated by an example.