Preprint of the project: DFG research group "Spectral analysis, asymptotic distributions and stochastic dynamics"

Numerical approximation and spectral analysis of infinite-dimensional dynamical systems

03-059 Wolf-Jürgen Beyn, Thorsten Hüls.
Error estimates for approximating non-hyperbolic heteroclinic orbits of maps

In this paper we consider heteroclinic orbits in discrete time dynamical systems that connect a hyperbolic fixed point to a non-hyperbolic fixed point with a one-dimensional center direction. A numerical method for approximating the heteroclinic orbit by a finite orbit sequence is introduced and a detailed error analysis is presented. The loss of hyperbolicity requires special tools for proving the error estimate - the polynomial dichotomy of linear difference equations and a (partial) normal form transformation near the non-hyperbolic fixed point. This situation appears, for example, when one fixed point undergoes a flip bifurcation. For this case, the approximation method and the validity of the error estimate is illustrated by an example.