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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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