NOISE-INDUCED CHAOS: A CONSEQUENCE OF LONG DETERMINISTIC TRANSIENTS

2008 ◽  
Vol 18 (02) ◽  
pp. 509-520 ◽  
Author(s):  
TAMÁS TÉL ◽  
YING-CHENG LAI ◽  
MÁRTON GRUIZ
Keyword(s):  
Dynamical Systems ◽  
Dynamical Model ◽  
Invariant Sets ◽  
Transient Chaos ◽  
Weak Noise ◽  
Fractal Properties ◽  
Deterministic Dynamical Systems ◽  
Chaotic Transients ◽  
Noise Phenomenon

We argue that transient chaos in deterministic dynamical systems is a major source of noise-induced chaos. The line of arguments is based on the fractal properties of the dynamical invariant sets responsible for transient chaos, which were not taken into account in previous works. We point out that noise-induced chaos is a weak noise phenomenon since intermediate noise strengths destroy fractality. The existence of a deterministic nonattracting chaotic set, and of chaotic transients, underlying noise-induced chaos is illustrated by examples, among others by a population dynamical model.

scholarly journals Characteristic exponents of dynamical systems in metric spaces

1983 ◽  
Vol 3 (1) ◽  
pp. 119-127 ◽  
Author(s):  
Yuri Kifer
Keyword(s):  
Dynamical Systems ◽  
Metric Spaces ◽  
Invariant Sets ◽  
Characteristic Exponents ◽  
Smooth Case

AbstractWe introduce for dynamical systems in metric spaces some numbers which in the case of smooth dynamical systems turn out to be the maximal and the minimal characteristic exponents. These numbers have some properties similar to the smooth case. Analogous quantities are defined also for invariant sets.

scholarly journals Invariant Sets in Quasiperiodically Forced Dynamical Systems

2020 ◽  
Vol 19 (1) ◽  
pp. 329-351
Author(s):  
Yoshihiko Susuki ◽  
Igor Mezić
Keyword(s):  
Dynamical Systems ◽  
Invariant Sets

Stochastically perturbed limit cycles

1978 ◽  
Vol 15 (02) ◽  
pp. 311-320
Author(s):  
Charles J. Holland
Keyword(s):  
Dynamical Systems ◽  
Invariant Measure ◽  
Probability Measure ◽  
Limit Cycles ◽  
Noise Intensity ◽  
Stable Limit Cycle ◽  
Stable Limit ◽  
Additive White Noise ◽  
Small Intensity ◽  
Deterministic Dynamical Systems

In this paper we examine the effects of perturbing certain deterministic dynamical systems possessing a stable limit cycle by an additive white noise term with small intensity. We place assumptions on the system guaranteeing that when noise is present the corresponding random process generates an ergodic probability measure. We then determine the behavior of the invariant measure when the noise intensity is small.

Certain new robust properties of invariant sets and attractors of dynamical systems

1999 ◽  
Vol 33 (2) ◽  
pp. 95-105 ◽  
Author(s):  
A. S. Gorodetski ◽  
Yu. S. Ilyashenko
Keyword(s):  
Dynamical Systems ◽  
Invariant Sets

TRACKING SUSTAINED CHAOS

2000 ◽  
Vol 10 (03) ◽  
pp. 571-578 ◽  
Author(s):  
IRA B. SCHWARTZ ◽  
IOANA TRIANDAF
Keyword(s):  
Dynamical Systems ◽  
Simple Method ◽  
Periodically Driven ◽  
Unstable Solutions ◽  
Chaotic Transients ◽  
Parameter Values

Tracking unstable periodic states first introduced in [Schwartz & Triandaf, 1992] is the process of continuing unstable solutions as a systems parameter is varied in experiments. The tracked dynamical objects have been periodic saddles of well-defined finite periods. However, other saddles, such as chaotic saddles, have not been successfully "tracked," or continued. In this paper, we introduce a new yet simple method which can be used to track chaotic saddles in dynamical systems, which allows an experimentalist to sustain chaotic transients far away from crisis parameter values. The method is illustrated on a periodically driven CO 2 laser.

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