The periodic and chaotic regimes of motion in the exoplanet 2/1   mean-motion resonance
  We present the dynamical structure of the phase space of the planar planetary 2/1 mean-motion resonance (MMR). Inside the resonant domain, there exist two families of periodic orbits, one associated to the librational motion of the critical angle ($\sigma$-family) and the other related to the circulatory motion of the angle between the pericentres ($\Delta\varpi$-family). The well-known apsidal corotation resonances (ACR) appear at the intersections of these families. A complex web of secondary resonances exists also for low eccentricities, whose strengths and positions are dependent on the individual masses and spatial scale of the system.   Depending on initial conditions, a resonant system is found in one of the two topologically different states, referred to as \textit{internal} and \textit{external} resonances. The internal resonance is characterized by symmetric ACR and its resonant angle is $2\,\lambda_2-\lambda_1-\varpi_1$, where $\lambda_i$ and $\varpi_i$ stand for the planetary mean longitudes and longitudes of pericentre, respectively. In contrast, the external resonance is characterized by asymmetric ACR and the resonant angle is $2\,\lambda_2-\lambda_1-\varpi_2$. We show that systems with more massive outer planets always envolve inside internal resonances. The limit case is the well-known asteroidal resonances with Jupiter. At variance, systems with more massive inner planets may evolve in either internal or external resonances; the internal resonances are typical for low-to-moderate eccentricity configurations, whereas the external ones for high eccentricity configurations of the systems. In the limit case, analogous to Kuiper belt objects in resonances with Neptune, the systems are always in the external resonances characterized by asymmetric equilibria. 
Tatiana A. Michtchenko, Sylvio Ferraz-Mello, Cristian Beauge
astro-ph.EP