Periodic Motions and their Stability in a N = ν + 1-body Regular Polygonal Configuration |
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| Authors: | Antonis D Pinotsis |
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| Institution: | 1. Department of Astrophysics, Astronomy and Mechanics, Faculty of Physics, University of Athens, Panepistimiopolis, 157 84 Zographos, Athens, Greece
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| Abstract: | We present a systematic investigation of the parametric evolution of both retrograde and direct families of periodic motions as well as their stability in the inner region of the peripheral primaries of the planar N-body regular polygonal configuration (ring model). In particular, we study the change of the bifurcation points as well as the change of the size and dynamical structure of the rings of stability for different values of the parameters ν = N?1 (number of peripheral primaries) and β (mass ratio). We find some types of bifurcations of families of periodic motions, namely period doubling pitchfork bifurcations, as well as bifurcations of symmetric and non-symmetric periodic orbits of the same period. For a given value of N ? 1, the intervals Δx and ΔC of the rings of stability (where the periodic orbits are stable) of both retrograde and direct families increase with β increasing, while for a given value of β, the interval ΔC decreases with increasing N ? 1. In general, it seems that the dynamical properties of the system depend on the ratio (N ? 1)/β. The size of each ring of stability tends to zero as the ratio (N ? 1)/β → ∞, that is, if N ? 1→∞ or β → 0, the size of each ring of stability tends to zero (Δx → 0 and ΔC → 0) and, in general, the retrograde and direct families tend to disappear. This study gives us interesting information about the evolution of these two families and the changes of the bifurcation patterns since, for example, in some cases the stability index A oscillates between ?1 ≤ Α ≤ + 1. Each time the family becomes critically stable a new dynamical structure appears. The ratios of the Jacobian constant C between the successive critical points, C i /C i+1, tend to 1. All the above depend on the parameters N ? 1, β and show changes in the topology of the phase space and in the dynamical properties of the system. |
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