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**Session 69 - Instabilities in Planetary Systems.**

*Display session, Friday, January 09*

*Exhibit Hall, *

## [69.02] Dynamical Chaos in the Wisdom-Holman Mapping: Origins and Solutions

*K. P. Rauch (U. Maryland), M. J. Holman (SAO)*
The Wisdom-Holman (WH) symplectic mapping (Wisdom amp; Holman 1991)
is a widely used integration scheme for exploring the
dynamics of nearly-Keplerian N-body systems. One particular strength of the
method is its lack of long-term energy error growth in typical cases.
In a recent application by Rauch amp; Ingalls (1997), however, unstable and
apparently chaotic behavior of the mapping was found for a class of problems
involving highly eccentric orbits---in particular, systems in which
t_peri \ll \Delta t \ll t_fluc, where t_peri is the
timescale for passage through pericenter, \Delta t is the integration
timestep, and t_fluc is the minimum timescale over which the
perturbation forces fluctuate. This paper investigates in detail the dynamical
origin of this phenomenon, as well as modified schemes that are resistant to it.
To examine the non-linear stability of the WH mapping for high eccentricities
we use a resonance overlap criterion similar to that of Wisdom amp; Holman
(1992). We also test the performance of two potentially more stable
algorithms, which we call the potential-splitting method and the Stark method.
The former is based on the potential splitting approach of
Skeel amp; Biesiadecki (1994), Saha amp; Tremaine (1994), and Lee et al. (1997);
this technique recursively divides the potential into a series of
components that are evaluated with increasing frequency---effectively
producing a variable step size method that can accurately (and
efficiently) follow the trajectory of highly eccentric orbits. The latter,
a new method developed during our analysis, is based on replacement
of the Kepler stepper used in the usual WH scheme with one
appropriate to the Stark problem. We find that both schemes can significantly
outperform the WH mapping under the stated conditions. Consequences for the
integration of nearly-Keplerian systems are discussed.

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