This paper shows that a simple measure based on how often a trajectory returns close to past states — the recurrence time entropy (RTE) — can tell regular motion, strong chaos, and the intermediate "sticky" behaviour apart in Hamiltonian flows. Sticky behaviour means chaotic trajectories spend long, but finite, times near regular parts of phase space. The authors test the idea in the well known Hénon–Heiles model and in a second driven Hamiltonian system and report consistent results.

The researchers build recurrence plots from the times when a trajectory crosses a Poincaré surface of section. A recurrence plot records pairs of times when the system revisits nearly the same point in phase space; the RTE is a summary number derived from the distribution of those recurrence times. They compare RTE values with two standard chaos diagnostics: the largest Lyapunov exponent (which measures how rapidly nearby trajectories diverge) and the smaller alignment index (SALI, a method used to classify chaotic versus regular orbits). Numerical integrations used a symplectic integrator with a timestep set so energy was conserved to better than 10^−8.

Across the Hénon–Heiles phase space the RTE reproduces known structures. Regular islands show low RTE values. The large chaotic seas show high RTE values. The thin layers around islands, where trajectories can be temporarily trapped (the sticky zones), give intermediate RTE values. When the authors count how many trajectories are chaotic, the fraction identified by RTE matches the fraction found with SALI over a range of energies below the escape threshold (they restrict attention to energy E ≤ 1/6 so orbits do not escape to infinity).

Looking at single chaotic trajectories over finite times, the RTE series shows clear low-entropy episodes when the orbit is temporarily trapped near an island and high-entropy episodes when it explores the chaotic sea. The durations of low-entropy (trapping) episodes follow an algebraic, or power-law, decay. By contrast, high-entropy episodes follow an exponential distribution. These different statistics reflect the intermittent nature of weak chaos and offer a way to separate long trapping events from ordinary chaotic wandering.