Characteristic times in the standard map

Mirella Harsoula, Kostas Karamanos, and George Contopoulos
Phys. Rev. E 99, 032203 – Published 5 March 2019

Abstract

We study and compare three characteristic times of the standard map: the Lyapunov time tL, the Poincaré recurrence time tr, and the stickiness (or escape) time tst. The Lyapunov time is the inverse of the Lyapunov characteristic number (L) and in general is quite small. We find empirical relations for the L as a function of the nonlinearity parameter K and of the chaotic area A. We also find empirical relations for the Poincaré recurrence time tr as a function of the nonlinearity parameter K, of the chaotic area A, and of the size of the box of initial conditions ε. As a consequence, we find relations between tr and L. We compare the distributions of the stickiness time and the Poincaré recurrence time. The stickiness time inside the sticky regions at the boundary of the islands of stability is orders of magnitude smaller than the Poincaré recurrence time tr and this affects the diffusion exponent μ, which converges always to the value μ=1. This is shown in an extreme stickiness case. The diffusion is anomalous (ballistic motion) inside the accelerator mode islands of stability with μ=2 but it is normal everywhere outside the islands with μ=1. In a particular case of extreme stickiness, we find the hierarchy of islands around islands.

  • Figure
  • Figure
  • Figure
  • Figure
  • Figure
  • Figure
  • Figure
9 More
  • Received 26 October 2018

DOI:https://doi.org/10.1103/PhysRevE.99.032203

©2019 American Physical Society

Physics Subject Headings (PhySH)

Nonlinear Dynamics

Authors & Affiliations

Mirella Harsoula1, Kostas Karamanos2, and George Contopoulos1

  • 1Research Center for Astronomy, Academy of Athens Soranou Efesiou 4, GR-115 27 Athens, Greece
  • 2University of West Attica, Department of Mechanical Engineering, 250 Thivon and P. Ralli Str. GR-12244-Egaleo, Athens, Greece

Article Text (Subscription Required)

Click to Expand

References (Subscription Required)

Click to Expand
Issue

Vol. 99, Iss. 3 — March 2019

Reuse & Permissions
Access Options
Author publication services for translation and copyediting assistance advertisement

Authorization Required


×
×

Images

×

Sign up to receive regular email alerts from Physical Review E

Log In

Cancel
×

Search


Article Lookup

Paste a citation or DOI

Enter a citation
×