Chaotic dynamics

The chaotic movement of small celestial bodies within the Solar System may result from factors such as close encounters, collisions, or resonance overlapping. Various methods can be employed to identify chaotic motion, including those that gauge the separation rate of trajectories starting infinitesimally close or assess frequencies of time series. In this chapter, we illustrate a novel approach utilizing the autocorrelation function of time series, referred to as the ACF index (ACFI). Autocorrelation coefficients measure the correlation between a time series and a lagged copy of itself. By evaluating the fraction of autocorrelation coefficients exceeding the 5% null hypothesis threshold after a specific time lag, we can ascertain how well the time series autocorrelates with itself. This aids in pinpointing unpredictable time series, characterized by low ACFI values. When applied to orbital regions affected by both resonance overlapping and chaos induced by close encounters, ACFI demonstrates effectiveness in correctly identifying motion stemming from resonance overlapping, albeit showing limited sensitivity to chaos induced by close encounters. We apply this to the Henon-Heiles system, asteroids families, and to the circular restricted three body problem. ACFI holds potential for discerning chaotic effects acting as global dynamical thermometer and a tool for pre-processing data for further time series analysis using machine learning.