The Stability Analysis of the Extrasolar Planetary Systems
  To date, more than 100 giant Jupiter-like planets have been discovered in Doppler surveys of solar-type stars. In this paper, we perform simulations to investigate three systems: GJ 876, HD 82943 and 55 Cnc. The former two systems both have a pair of planets in the 2:1 Mean Motion Resonance (MMR), while the inner two companions of the later is close to 3:1 MMR. By integrating hundreds of the planetary orbits of three systems for million years, we find that for GJ 876 and HD 82943, the critical argument $\lambda_{1} - 2\lambda_{2} + \varpi_{1}$ and $\lambda_{1} - 2\lambda_{2} + \varpi_{2}$ librate about $0^{\circ}$ or $180^{\circ}$, indicating 2:1 MMR can play an important role in stabilizing the motion of the planets so that they are protected from frequent close encounters. As for 55 Cnc, we further show the three resonant arguments for 3:1 MMR execute librations for millions of years respectively, which reveals the evidence of the resonance for this system. Additionally, we should emphasize another vital mechanism is the apsidal phase-locking between a couple of planets for a certain system. For GJ 876 and HD 82943, we discover the relative apsidal longitudes $\varpi_{1} - \varpi_{2}$ move about $0^{\circ}$ or $180^{\circ}$, respectively; but for 55 Cnc, we find that there exists an asymmetric apsidal libration between two inner planets. Finally, we made a brief discussion about the Habitable Zones in the exoplanetary systems. 
Jianghui Ji (1, 2), Lin Liu (3), H. Kinoshita (4), Guangyu Li (1,2),   H. Nakai (4) ((1) Purple Mountain Observatory, (2) NA...