Mapping for Gravitational Few - Body Problem with Dissipation 13

tion the evolution of a system is, in general, extremely sensitive to initial conditions and to small changes in system parameters. This sensitivity is an intrinsic physical property of the system, and not merely of numerical solutions. Sensitivity to initial conditions { aka chaos { may be found in the conservative system also, but not necessarily in all regions of phase space. In contrast, in the presence of weak dissipation, a nearly-Hamiltonian system is forced to sweep across large regions of phase space, including chaotic regions near resonances. Consequently, sensitivity of the evolution is always present in this case, for any initial conditions. The scatter in Figure 3 is indirect evidence of this. It is related to the complicated dynamics of resonance passage phenomena. A discussion of this phenomenon is beyond the scope of this paper. However, it is worth pointing out that resonance passage can produce \anomalous" behavior, i.e. evolution that is characterized by singularly large changes of orbital parameters in relatively short times. This phenomenon is well-known in the literature and is an active area of current research (see, for example, Malhotra 1994, for a recent review). The meaning of numerical solutions in the presence of such dynamical behavior is also an interesting question (see, for example, Quinlan & Tremaine 1992). Acknowledgements I thank Jack Lissauer and Martin Duncan for pertinent comments. I am grateful to the organizers of the Planet Formation program at the Institute for Theoretical Physics, Santa Barbara, CA, for their hospitality during the early stages of this work. In this paper, we have introduced a new numerical integration method for Solar system problems with small dissipation. This method is a modiication of the second-order \N-body map" introduced recently by Wisdom & Holman (1991) (also independently by Kinoshita et al. (1991)) for nearly-integrable Hamiltonian systems of the form H = H kepler + H 0 where H kepler and H 0 are separately integrable. H kepler describes the unperturbed Keplerian motion and H 0 describes the small mutual gravitational perturbations of the planets (or satellites, in a planetary satellite system). The new method is summarized as follows. Recall that for the conservative problem, a single step of the N-body map with size requires three operations. These are: (i) evolve the system according to H kepler for time 1 2 ; (ii) evolve the system according to H 0 for time ; (iii) evolve the system …