Chaos, Order, and Periodic Orbits in 3 : 1 Resonant Planetary Dynamics

Abstract

This study addresses the long-term evolution of possible two-planet extrasolar systems that are initially trapped in 3:1 mean motion resonance. A planar general three-body problem model is used, and its resonant dynamics are examined by computing periodic orbits in a rotating frame, Poincare maps, and maps of dynamical stability. We computed the families of symmetric resonant periodic orbits that obey bifurcations, giving rise to families of asymmetric periodic orbits. The linear stability of such orbits has also been computed, and their relation to the long-term stability of their nearby phase-space domain has been studied. The maps of dynamical stability reveal a complicated structure in the phase space, where chaos and order coexist and alternate as the initial eccentricities or the phases of the planets change. The regular orbits are classified into various types according to the librating or rotating evolution of the resonant angles. Apsidal symmetric librations are common in the domain of resonant motion, but asymmetric ones are associated exclusively with the existence of asymmetric periodic orbits. Such a stable asymmetric configuration seems to correspond to the companions b and c of the 55 Cnc extrasolar system, which are trapped in the 3:1 mean motion resonance according to the study of McArthur et al. However, a recent study by Fischer et al. shows the existence of a new planet (the companion f) in the system, and that the planets b and c are not in mean motion resonance.