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The existence of a chaotic region due to the overlap of secular resonances ν5 and ν6

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Abstract

A nonlinear theory of secular resonances is developed. Both terms corresponding to secular resonances ν5 and ν6 are taken into account in the Hamiltonian. The simple overlap criterion is applied and the condition for the overlap of these resonances is found. It is shown that in given approximation the value p = (1 - e2)1/2(1 - cosI) is an integral of motion, where the mean eccentricity e and mean inclination I are obtained by eliminating short-period perturbations as well as the nonresonant terms from the planets. The overlap criterion yields a critical value of parameter p depending on the semi-major axis a of the asteroid. For p greater than the critical value, resonance overlap occurs and chaotic motion has to be expected. A mapping is presented for fast calculation of the trajectories. The results are illustrated by level curves in surfaces of section method.

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  1. Astronomical Institute of the Czechoslovak Academy of Sciences, 6, 120 23 Praha 2, Budečská, Czechoslovakia

    M. Šidlichovský

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  1. M. Šidlichovský
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Šidlichovský, M. The existence of a chaotic region due to the overlap of secular resonances ν5 and ν6 . Celestial Mech Dyn Astr 49, 177–196 (1990). https://doi.org/10.1007/BF00050713

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  • DOI: https://doi.org/10.1007/BF00050713

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