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We apply a generalized Kepler map theory to describe the qualitative chaotic dynamics around cometary nuclei, based on accessible observational data for five comets whose nuclei are well-documented to resemble dumb-bells. The sizes of chaotic zones around the nuclei and the Lyapunov times of the motion inside these zones are estimated. In the case of Comet 1P/Halley, the circumnuclear chaotic zone seems to engulf an essential part of the Hill sphere, at least for orbits of moderate to high eccentricity. [ABSTRACT FROM AUTHOR]

Modern theoretical methods for estimating the width of the chaotic layer in the presence of prominent marginal resonances are considered in the perturbed pendulum model of nonlinear resonance. The fields of applicability of these methods are explicitly and precisely formulated. The comparative accuracy is investigated in massive and long-run numerical experiments. It is shown that the methods are naturally subdivided in classes applicable for adiabatic and nonadiabatic cases of perturbation. It is explicitly shown that the pendulum approximation of marginal resonance works well in the nonadiabatic case. In this case, the role of marginal resonances in determining the total layer width is demonstrated to diminish with increasing main parameter &lgr; (equal to the ratio of the perturbation frequency to the frequency of small-amplitude phase oscillations on the resonance). Solely the "bending effect" is important in determining the total amplitude of the energy deviations of the near-separatrix motion at &lgr; ≳ 7. In the adiabatic case, it is demonstrated that the geometrical form of the separatrix cell can be described analytically quite easily by means of using a specific representation of the separatrix map. It is shown that the nonadiabatic (and, to some extent, intermediary) case is most actual, in comparison with the adiabatic one, for the physical or technical applications that concern the energy jumps in the near-separatrix chaotic motion [ABSTRACT FROM AUTHOR]

Abstract: The Kepler map was derived by as a tool for description of the long-term chaotic orbital behaviour of the comets in nearly parabolic motion. It is a two-dimensional area-preserving map, describing the motion of a comet in terms of energy and time. Its second equation is based on Kepler’s third law, hence the title of the map. Since 1980s the Kepler map has become paradigmatic in a number of applications in celestial mechanics and atomic physics. It represents an important kind of general separatrix maps. used refined methods of mathematical physics to derive analytical expressions for its single parameter. These methods became available only in the second half of the 20th century, and it may seem that the map is inherently a very modern mathematical tool. With the help of the Jacobi integral I show that the Kepler map, including analytical formulae for its parameter, can be derived by quite elementary methods. The prehistory and applications of the Kepler map are considered and discussed. [Copyright &y& Elsevier]

The chaotic orbital motion of Prometheus and Pandora, the 16th and 17th satellites of Saturn, is studied. Chaos in their orbital motion, as found by Goldreich & Rappaport and Renner & Sicardy, is due to interaction of resonances in the resonance multiplet corresponding to the 121:118 commensurability of the mean motions of the satellites. It is shown rigorously that the system moves in adiabatic regime. The Lyapunov time (the ‘time horizon of predictability’ of the motion) is calculated analytically and compared to the available numerical–experimental estimates. For this purpose, a method of analytical estimation of the maximum Lyapunov exponent in the perturbed pendulum model of non-linear resonance is applied. The method is based on the separatrix map theory. An analytical estimate of the width of the chaotic layer is made as well, based on the same theory. The ranges of chaotic diffusion in the mean motion are shown to be almost twice as big compared to previous estimates for both satellites. [ABSTRACT FROM AUTHOR]