Deterministic Dynamical Systems and Stochastic Perturbations

2016 ◽  
pp. 25-52
Keyword(s):  
Dynamical Systems ◽  
Stochastic Perturbations ◽  
Deterministic Dynamical Systems

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.

Random dynamical systems with jumps

2004 ◽  
Vol 41 (03) ◽  
pp. 890-910 ◽  
Author(s):  
Katarzyna Horbacz
Keyword(s):  
Dynamical Systems ◽  
Iterated Function System ◽  
Infinite Family ◽  
Learning Systems ◽  
Random Dynamical Systems ◽  
Infinite Dimensional ◽  
Markov Operators ◽  
Iterated Function ◽  
Random Evolutions ◽  
Deterministic Dynamical Systems

We consider random dynamical systems with randomly chosen jumps on infinite-dimensional spaces. The choice of deterministic dynamical systems and jumps depends on a position. The system generalizes dynamical systems corresponding to learning systems, Poisson driven stochastic differential equations, iterated function system with infinite family of transformations and random evolutions. We will show that distributions which describe the dynamics of this system converge to an invariant distribution. We use recent results concerning asymptotic stability of Markov operators on infinite-dimensional spaces obtained by T. Szarek.

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.

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