Dynamical aspects of Hamiltonian systems: Chaos, stickiness, and recurrence plots

Abstract

Hamiltonian systems represent a vast class of dynamical systems that have the special fea- ture of preserving volume in phase space. The phase space of a typical Hamiltonian system is

neither integrable nor uniformly hyperbolic. It exhibits both regular and chaotic components. For two-dimensional quasi-integrable systems with a hierarchical phase space, chaotic orbits can spend an arbitrarily long time around islands of stability, in which they behave similarly

to quasiperiodic orbits. This phenomenon is called stickiness and is one of the main conse- quences of the complex hierarchical structure of islands-around-islands embeded in the chaotic

sea. Stickiness affects the global transport properties of the system and the convergence of the

Lyapunov exponents. In this thesis, we analyze nonstandard dynamical measures for the quan- tification of chaotic motion and the detection of the stickiness effect in Hamiltonian systems.

Initially, we consider the standard map, which is a simple, paradigmatic system that displays all the features of quasi-integrable Hamiltonian systems. First, we introduce a recently proposed dynamical measure based on ergodic theory and a weighted Birkhoff average. By using this measure, we successfully distinguish chaos and regularity for different values of the standard map’s non-linearity parameter k, and we apply it together with the uncertainty fraction method to determine the fractal dimension of the islands’ boundary for a special value of k, namely, k = 6.908745. For this value, the standard map’s phase space is composed of a self-similar hierarchical structure of islands within the chaotic sea, and we show that the deeper we go into this structure, the longer it takes for the orbits to escape the trapping region, and the higher the boundary dimension becomes. Additionally, the dimension depends on the position in phase space as well as on the scale of the initial condition uncertainty, which implies the existence of an effective fractal dimension. As a further measure, we propose the use of an entropy-based measure of the recurrence plots (RPs). We estimate the recurrence times of an orbit from the RP and calculate the Shannon entropy of its distribution, known as the recurrence time entropy (RTE). We find that the RTE is an alternative way of detecting chaotic orbits and sticky regions. We show that the largest Lyapunov exponent and the RTE exhibit a high correlation coefficient even when considering relatively small time series (5000 data points). By computing the RTE in smaller time windows along the evolution of a single chaotic orbit, we find the finite-time RTE distribution to be multi-modal when sticky regions are present in phase space, and we successfully identify the specific areas in phase space that correspond to each mode. We also

quantify the duration of each stickiness regime and find that the cumulative distribution of trap- ping times in the sticky regions follows a power law tail, while the distribution when the orbit

wanders in the chaotic sea displays an exponential decay. Seeking a more robust analysis of the aforementioned dynamical measures, we consider another two-dimensional Hamiltonian system

with a hierarchical divided phase space: the billiard system. We demonstrate that these mea- sures characterize all dynamical behavior of such a system, with the advantage of not relying on

the Jacobian matrix for their calculation, unlike the Lyapunov exponents.