The reversibility error method (REM): a new, dynamical fast indicator for planetary dynamics

We describe the reversibility error method (REM) and its applications to planetary dynamics. REM is based on the time-reversibility analysis of the phase-space trajectories of conservative Hamiltonian systems. The round-off errors break the time reversibility and the displacement from the initial condition, occurring when we integrate it forward and backward for the same time interval, is related to the dynamical character of the trajectory. If the motion is chaotic, in the sense of non-zero maximal Lyapunov characteristic exponent (mLCE), then REM increases exponentially with time, as exp lambda t, while when the motion is regular (quasi-periodic), then REM increases as a power law in time, as t(alpha), where alpha and lambda are real coefficients. We compare the REM with a variant of mLCE, the mean exponential growth factor of nearby orbits. The test set includes the restricted three-body problem and five resonant planetary systems: HD 37124, Kepler-60, Kepler-36, Kepler-29 and Kepler-26. We found a very good agreement between the outcomes of these algorithms. Moreover, the numerical implementation of REM is astonishing simple, and is based on solid theoretical background. The REM requires only a symplectic and time-reversible (symmetric) integrator of the equations of motion. This method is also CPU efficient. It may be particularly useful for the dynamical analysis of multiple planetary systems in the Kepler sample, characterized by low-eccentricity orbits and relatively weak mutual interactions. As an interesting side result, we found a possible stable chaos occurrence in the Kepler-29 planetary system.