Preprint of the project: SFB 701: Spectral Structures and Topological Methods in Mathematics - Project A2

Numerical analysis of high-dimensional transfer operators

07-011 Thorsten Hüls.
Homoclinic trajectories of non-autonomous maps

For non-autonomous difference equations of the form \[x_{n+1}=f(x_n,\lambda_n),\quad n\in\mathbb{Z},\] we consider homoclinic trajectories. These are pairs of trajectories that converge in both time directions towards each other. Assuming hyperbolicity, we derive a numerical method to compute homoclinic trajectories in two steps. In the first step, one trajectory is approximated by the solution of a boundary value problem and precise error estimates are given. In particular, influences of parameters \(\lambda_n\) with \(|n|\) large are discussed in detail. A second trajectory that is homoclinic to the first one is computed in a subsequent step as follows. We transform the original system into a topologically equivalent form having \(0\) as an \(n\)-independent fixed point. Applying the boundary value ansatz to the transformed system, we obtain a non-autonomous homoclinic orbit, converging towards the origin (see [hu06]). Transforming back to the original coordinates leads to the desired homoclinic trajectories. The numerical method and the validity of the error estimates are illustrated by examples.