Dynamical Chaos in the Wisdom-Holman Integrator: Origins and Solutions

Abstract

We examine the non-linear stability of the Wisdom-Holman (WH) mapping applied to the integration of perturbed, highly eccentric two-body orbits. We find that the method is unstable and introduces artificial chaos into the computed trajectories, unless the step size is chosen small enough to resolve pericenter. The origin of the instability is analyzed analytically using the Stark problem as a fiducial test case. We similarly examine the robustness of several alternative methods: a regularized WH map due to Mikkola (1997); the potential-splitting (PS) approach of Lee et al. (1997); and two types of Stark-based methods. Comparative simulations show the regularized WH map to be stable at high eccentricities, and an enhanced PS algorithm to perform well when close encounters are additionally present. We find Stark-based schemes to be of marginal use in N-body type integrations.