Deterministic Dynamical Systems and Stochastic Perturbations

2016 ◽  
pp. 25-52
1978 ◽  
Vol 15 (02) ◽  
pp. 311-320
Author(s):  
Charles J. Holland

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.


2004 ◽  
Vol 41 (03) ◽  
pp. 890-910 ◽  
Author(s):  
Katarzyna Horbacz

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.


2008 ◽  
Vol 18 (02) ◽  
pp. 509-520 ◽  
Author(s):  
TAMÁS TÉL ◽  
YING-CHENG LAI ◽  
MÁRTON GRUIZ

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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