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Motivated by the large number of extrasolar planetary systems that are nearmean motion resonances, this paper explores a related type of dynamicalbehavior known as "nodding". Here, the resonance angle of a planetary systemexecutes libration (oscillatory motion) for several cycles, circulates for oneor more cycles, and then enters once again into libration. This type ofcomplicated dynamics can affect our interpretation of observed planetarysystems that are in or near mean motion resonance. This work shows thatplanetary systems in (near) mean motion resonance can exhibit nodding behavior,and outlines the portion of parameter space where it occurs. This problem isaddressed using both full numerical integrations of the planetary systems andvia model equations obtained through expansions of the disturbing function. Inthe latter approach, we identify the relevant terms that allow for nodding. Thetwo approaches are in agreement, and show that nodding often occurs when asmall body is in an external mean motion resonance with a larger planet. As aresult, the nodding phenomenon can be important for interpreting observationsof transit timing variations, where the existence of smaller bodies is inferredthrough their effects on larger, observed transiting planets. For example, inactively nodding planetary systems, both the amplitude and frequency of thetransit timing variations depend on the observational time window.

Mean Motion Resonances in Exoplanet Systems: An Investigation Into Nodding Behavior