Homoclinic trajectories of non-autonomous maps

Thorsten Hüls
2011 Journal of difference equations and applications (Print)  
For non-autonomous difference equations of the form x n+1 = f (x n , λ n ), n ∈ we consider homoclinic trajectories. These are pairs of trajectories that converge in both time directions towards each other. We derive a numerical method to approximate such homoclinic trajectories in two steps. In the first step one of the infinite trajectories is approximated by a finite segment and precise error estimates are given. In a subsequent step, a second trajectory that is homoclinic to the first one
more » ... computed as follows. We transform the original system into a topologically equivalent form, having zero as an n-independent fixed point. Then, the techniques, developed in Hüls (2006) apply and we gain an approximation of a non-autonomous homoclinic orbit, converging towards the origin. Transforming back to the original coordinates leads to the desired homoclinic trajectories. The approximation method and the validity of the error estimates are illustrated by an example.
doi:10.1080/10236190902932742 fatcat:idlk4iwmzndjjiub2uz3ho2o6q