The results of an extensive numerical study of the periodic orbits of planar, elliptic restricted three-body planetary systems consisting of a star, an inner massive planet and an outer mass-less body in the external 1:2 mean-motion resonance are presented. Using the method of differential continuation, the locations of the resonant periodic orbits of such systems are identified and through an extensive study of their phase-parameter space, it is found that the majority of the resonant periodic orbits are unstable. For certain values of the mass and the orbital eccentricity of the inner planet, however, stable periodic orbits can be found. The applicability of such studies to the 1:2 resonance of the extrasolar planetary system GJ876 is also discussed.
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