Mean Motion Resonances in Exoplanet Systems: An Investigation Into   Nodding Behavior
  Motivated by the large number of extrasolar planetary systems that are near mean motion resonances, this paper explores a related type of dynamical behavior known as "nodding". Here, the resonance angle of a planetary system executes libration (oscillatory motion) for several cycles, circulates for one or more cycles, and then enters once again into libration. This type of complicated dynamics can affect our interpretation of observed planetary systems that are in or near mean motion resonance. This work shows that planetary systems in (near) mean motion resonance can exhibit nodding behavior, and outlines the portion of parameter space where it occurs. This problem is addressed using both full numerical integrations of the planetary systems and via model equations obtained through expansions of the disturbing function. In the latter approach, we identify the relevant terms that allow for nodding. The two approaches are in agreement, and show that nodding often occurs when a small body is in an external mean motion resonance with a larger planet. As a result, the nodding phenomenon can be important for interpreting observations of transit timing variations, where the existence of smaller bodies is inferred through their effects on larger, observed transiting planets. For example, in actively nodding planetary systems, both the amplitude and frequency of the transit timing variations depend on the observational time window. 
Jacob A. Ketchum, Fred C. Adams and Anthony M. Bloch
astro-ph.EP