We examine the non-linear stability of the Wisdom-Holman ( WH ) symplectic mapping applied to the integration of perturbed , highly eccentric ( e \gtrsim 0.9 ) two-body orbits .
We find that the method is unstable and introduces artificial chaos into the computed trajectories for this class of problems , unless the step size is chosen small enough to always resolve periapse , in which case the method is generically stable .
This ‘ radial orbit instability ’ persists even for weakly perturbed systems .
Using the Stark problem as a fiducial test case , we investigate the dynamical origin of this instability and show that the numerical chaos results from the overlap of step size resonances ( cf .
Wisdom & Holman 1992 ) ; interestingly , for the Stark problem many of these resonances appear to be absolutely stable .

We similarly examine the robustness of several alternative integration methods : a regularized version of the WH mapping suggested by Mikkola ( 1997 ) ; the potential-splitting ( PS ) method of Lee et al .
( 1997 ) ; and two methods incorporating approximations based on Stark motion instead of Kepler motion ( cf . [ Newman et al .
1997 ] ) .
The two fixed point problem and a related , more general problem are used to comparatively test the various methods for several types of motion .
Among the tested algorithms , the regularized WH mapping is clearly the most efficient and stable method of integrating eccentric , nearly-Keplerian orbits in the absence of close encounters .
For test particles subject to both high eccentricities and very close encounters , we find an enhanced version of the PS method—incorporating time regularization , force-center switching , and an improved kernel function—to be both economical and highly versatile .
We conclude that Stark-based methods are of marginal utility in N -body type integrations .
Additional implications for the symplectic integration of N -body systems are discussed .