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Symplectic N-body integrators are widely used to study problems in celestial mechanics. The most popular algorithms are of 2nd and 4th order, requiring 2 and 6 substeps per timestep, respectively. The number of substeps increases rapidly with order in timestep, rendering higher-order methods impractical. However, symplectic integrators are often applied to systems in which perturbations between bodies are a small factor ǫ of the force due to a dominant central mass. In this case, it is possible to create optimized symplectic algorithms that require fewer substeps per timestep. This is achieved by only considering error terms of order ǫ, and neglecting those of order ǫ 2 , ǫ 3 etc. Here we devise symplectic algorithms with 4 and 6 substeps per step which effectively behave as 4th and 6th-order integrators when ǫ is small. These algorithms are more efficient than the usual 2nd and 4th-order methods when applied to planetary systems.

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Recently, we analyzed a relation, found for chaotic orbits, between the Lyapunov time T L (the inverse of the maximum Lyapunov exponent) and the "event" time T e (the time at which an orbit becomes clearly unstable). In this paper we treat two new problems. First, we apply this T L-T e relation to numerical integrations of 25 outer-belt asteroids and show that, when viewed in the proper context of a Gaussian distribution of event time residuals, none of the 25 objects exhibits an anomalously short Lyapunov time. The current age of the solar system is approximately three standard deviations or less from the anticipated event times of all of these asteroids. We argue that the Lyapunov times of the 25 remaining bodies are each consistent with the age of the solar system, and that we are therefore seeing the remnants of a larger original distribution. The bulk of that population has been ejected by Jupiter, leaving the "tail members" as present-day survivors. This interpretation is consistent with current understanding of the behavior of trajectories near KAM tori in Hamiltonian systems. In particular, there is no need to invoke a new type of motion or class of dynamical objects to explain the short Lyapunov timescales found for solar system objects. Second, we discuss integrations of 440 fictitious outer-belt asteroids and show that the slope and offset parameters of the TL-T e relation do not change with an increase in Jupiter's mass by a factor of 10, and that the distribution of residuals in log T e is Gaussian. This allows us to sensibly and quantitatively interpret the significance of the Lyapunov timescale. However, the width of the residuals distribution is a function of mass ratio. Since knowledge of the distribution width is needed in order to interpret the significance of predicted event times, a calibration must be performed at the mass ratio of interest.