Nonautonomous systems with transversal homoclinic structures under discretization

We consider homoclinic orbits in continuous time nonautonomous dynamical systems. Unlike the autonomous case, stable and unstable fiber bundles that generalize stable and unstable manifolds, typically intersect transversally in isolated points. In the first part, we establish persistence and error estimates for one-step discretizations of transversal homoclinic orbits. Secondly, we extend an algorithm by England, Krauskopf, Osinga to nonautonomous systems and illustrate transversally intersecting fibers along homoclinic orbits for three examples. The first one is constructed artificially in order to study numerical errors, while the second one is a periodically forced model that reveals the influence of underlying autonomous dynamics. The third example originates from mathematical biology.