Homoclinic trajectories of non-autonomous maps

Abstract

For non-autonomous difference equations of the form

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 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 (T. Hüls, J. Difference Equ. Appl. 12(11) (2006), pp. 1103–1126). 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.

Keywords:

• non-autonomous discrete time dynamical systems
• homoclinic trajectories
• numerical approximation
• error analysis

AMS Subject Classification::

• 70K44
• 34C37
• 37B55

Acknowledgements

The author wishes to thank Wolf-Jürgen Beyn for stimulating discussions about this paper. The author also thanks an anonymous referee for several helpful suggestions. Supported by CRC 701 ‘Spectral Structures and Topological Methods in Mathematics’.