Stickiness in generic low-dimensional Hamiltonian systems: A recurrence-time statistics approach

Abstract

We analyze the structure and stickiness in the chaotic components of generic Hamiltonian systems with divided phase space. Following the method proposed recently in Lozej and Robnik [Phys. Rev. E 98, 022220 (2018)], the sticky regions are identified using the statistics of recurrence times of a single chaotic orbit into cells dividing the phase space into a grid. We perform extensive numerical studies of three example systems: the Chirikov standard map, the family of Robnik billiards, and the family of lemon billiards. The filling of the cells is compared to the random model of chaotic diffusion, introduced in Robnik et al. [J. Phys. A: Math. Gen. 30, L803 (1997)] for the description of transport in the phase spaces of ergodic systems. The model is based on the assumption of completely uncorrelated cell visits because of the strongly chaotic dynamics of the orbit and the distribution of recurrence times is exponential. In generic systems the stickiness induces correlations in the cell visits. The distribution of recurrence times exhibits a separation of timescales because of the dynamical trapping. We model the recurrence time distributions to cells inside sticky areas as a mixture of exponential distributions with different decay times. We introduce the variable S, which is the ratio between the standard deviation and the mean of the recurrence times as a measure of stickiness. We use S to globally assess the distributions of recurrence times. We find that in the bulk of the chaotic sea S=1, while S>1 in areas of stickiness. We present the results in the form of animated grayscale plots of the variable S in the largest chaotic component for the three example systems, included as supplemental material to this paper.

arXiv comment15 pages, 14 figures

References in corpus (19)

• Power-law distributions in empirical data
• Long-Time Correlations in the Stochastic Regime
• Thirty Years of Turnstiles and Transport
• Stickiness in Hamiltonian systems: from sharply divided to hierarchical phase space
• Asymptotic Statistics of Poincaré Recurrences in Hamiltonian Systems with Divided Phase Space
• On universality of algebraic decays in Hamiltonian systems
• Average exit times in volume preserving maps
• Stickiness in mushroom billiards
• Universality of algebraic laws in Hamiltonian systems
• Recurrence Time Statistics for Finite Size Intervals
• Survey on the role of accelerator modes for anomalous diffusion: The case of the standard map
• Statistical properties of the localization measure of chaotic eigenstates and the spectral statistics in a mixed-type billiard
• New Universal Aspects of Diffusion in Strongly Chaotic Systems
• Global and Local diffusion in the Standard Map
• On another edge of defocusing: hyperbolicity of asymmetric lemon billiards
• Fluctuations in the relaxation dynamics of mixed chaotic systems
• Structure, size, and statistical properties of chaotic components in a mixed-type Hamiltonian system
• Characteristic times in the standard map
• Few Islands Approximation of Hamiltonian System with divided Phase Space

Cited by in corpus (8)

• Stickiness and recurrence plots: an entropy-based approach
• Effects of stickiness in the classical and quantum ergodic lemon billiard
• Phenomenology of quantum eigenstates in mixed-type systems: lemon billiards with complex phase space structure
• Classical and quantum mixed-type lemon billiards without stickiness
• Spectral form factors and dynamical localization
• Spacing ratios in mixed-type systems
• Superexponential Interactions and the Dynamical Unfolding of Confined Degrees of Freedom
• Recurrence and Stickiness in the Noisy Harper Map