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Abstract

The stickiness effect is a fundamental feature of quasi-integrable Hamiltonian systems. We propose the use of an entropy-based measure of the recurrence plots (RPs), namely, the entropy of the distribution of the recurrence times (estimated from the RP), to characterize the dynamics of a typical quasi-integrable Hamiltonian system with coexisting regular and chaotic regions. We show that the recurrence time entropy (RTE) is positively correlated to the largest Lyapunov exponent, with a high correlation coefficient. We obtain a multi-modal distribution of the finite-time RTE and find that each mode corresponds to the motion around islands of different hierarchical levels. In two-dimensional quasi-integrable Hamiltonian systems with hierarchical phase space, chaotic orbits can spend an arbitrarily long time around islands, in which they behave similarly as quasiperiodic orbits. This phenomenon is called stickiness, and it is due to the presence of partial barriers to the transport around the hierarchical levels of islands-around-islands. The stickiness affects the convergence of the Lyapunov exponents, making the task of characterizing the dynamics more difficult, especially when only short time series are known. Due to the intrinsic property of dynamical systems that quasiperiodic orbits can have at most three different return times (Slater’s theorem1,2 ), which is the time needed to the orbit return to a given region at the curve, in this paper, we propose the use of the recurrence time entropy (RTE) (estimated from the recurrence plots) to characterize the dynamics of nonlinear systems. We find that the RTE is an alternative way of detecting chaotic orbits and sticky regions. Furthermore, the finite-time RTE distribution is multi-modal when sticky regions are present in the phase space, and each mode corresponds to a different hierarchical level in the islands-around-islands structure embedded in the chaotic sea.