Discretization of Homoclinic Orbits, Rapid Forcing and Invisible Chaos

One-step discretizations of order $p$ and step size $e$ of autonomous ordinary differential equations can be viewed as time-$e$ maps of a certain first order ordinary differential equation that is a rapidly forced nonautonomous system. Fiedler and Scheurle study the behavior of a homoclinic orbit for $e = 0$, under discretization. Under generic assumptions they show that this orbit becomes transverse for positive $e$. Likewise, the region where complicated, "chaotic" dynamics prevail is under certain conditions estimated to be exponentially small. These results are illustrated by high precision numerical experiments. The experiments show that, due to exponential smallness, homoclinic transversality is already practically invisible under normal circumstances, for only moderately small discretization steps.