Physics Reports, 587, 1–39p. (2015) DOI:10.1016/j.physrep.2015.05.004

Local and global approaches to the problem of Poincaré recurrences. Applications in nonlinear dynamics

V. S. Anishchenko, Y. Boev, N. I. Semenova, G. I. Strelkova

We review rigorous and numerical results on the statistics of Poincaré recurrences which are related to the modern development of the Poincaré recurrence problem. We analyze and describe the rigorous results which are achieved both in the classical (local) approach and in the recently developed global approach. These results are illustrated by numerical simulation data for simple chaotic and ergodic systems. It is shown that the basic theoretical laws can be applied to noisy systems if the probability measure is ergodic and stationary. Poincaré recurrences are studied numerically in nonautonomous systems.

Statistical characteristics of recurrences are analyzed in the framework of the global approach for the cases of positive and zero topological entropy. We show that for the positive entropy, there is a relationship between the Afraimovich-Pesin dimension, Lyapunov exponents and the Kolmogorov-Sinai entropy either without and in the presence of external noise. The case of zero topological entropy is exemplified by numerical results for the Poincare recurrence statistics in the circle map. We show and prove that the dependence of minimal recurrence times on the return region size demonstrates universal properties for the golden and the silver ratio. The behavior of Poincaré recurrences is analyzed at the critical point of Feigenbaum attractor birth. We explore Poincaré recurrences for an ergodic set which is generated in the stroboscopic section of a nonautonomous oscillator and is similar to a circle shift.

Based on the obtained results we show how the Poincaré recurrence statistics can be applied for solving a number of nonlinear dynamics issues. We propose and illustrate alternative methods for diagnosing effects of external and mutual synchronization of chaotic systems in the context of the local and global approaches. The properties of the recurrence time probability density can be used to detect the stochastic resonance phenomenon. We also discuss how the fractal dimension of chaotic attractors can be estimated using the Poincaré recurrence statistics.

back


Creative Commons License © 2017 SOME RIGHTS RESERVED
The content of this web site is licensed under a Creative Commons Attribution-NonCommercial-NoDerivs 2.0 Germany License.

Please note: The abstracts of the bibliography database may underly other copyrights.

Ihr Browser versucht gerade eine Seite aus dem sogenannten Internet auszudrucken. Das Internet ist ein weltweites Netzwerk von Computern, das den Menschen ganz neue Möglichkeiten der Kommunikation bietet.

Da Politiker im Regelfall von neuen Dingen nichts verstehen, halten wir es für notwendig, sie davor zu schützen. Dies ist im beidseitigen Interesse, da unnötige Angstzustände bei Ihnen verhindert werden, ebenso wie es uns vor profilierungs- und machtsüchtigen Politikern schützt.

Sollten Sie der Meinung sein, dass Sie diese Internetseite dennoch sehen sollten, so können Sie jederzeit durch normalen Gebrauch eines Internetbrowsers darauf zugreifen. Dazu sind aber minimale Computerkenntnisse erforderlich. Sollten Sie diese nicht haben, vergessen Sie einfach dieses Internet und lassen uns in Ruhe.

Die Umgehung dieser Ausdrucksperre ist nach §95a UrhG verboten.

Mehr Informationen unter www.politiker-stopp.de.