Stochastic Perturbations to Conservative Dynamical Systems on the Plane. I: Convergence of Invariant Distributions

1988 ◽  
Vol 309 (2) ◽  
pp. 621
Author(s):  
G. Wolansky
Keyword(s):  
Dynamical Systems ◽  
Stochastic Perturbations ◽  
Invariant Distributions

Averaging Principle for Stochastic Perturbations of Multifrequency Systems

2003 ◽  
Vol 03 (03) ◽  
pp. 393-408 ◽  
Author(s):  
M. I. Freidlin ◽  
A. D. Wentzell
Keyword(s):  
Dynamical Systems ◽  
White Noise ◽  
Lebesgue Measure ◽  
Hamiltonian Systems ◽  
Measure Zero ◽  
Lebesgue Measure Zero ◽  
Averaging Principle ◽  
Stochastic Perturbations ◽  
Main Assumption ◽  
Noise Type

We consider the averaging principle for deterministic and stochastic perturbations of multidimensional dynamical systems for which coordinates can be introduced in such a way that the "fast" coordinates change in a torus (for Hamiltonian systems, "action-angle coordinates"). Stochastic perturbations of the white-noise type are considered. Our main assumption is that the set of action values for which the frequencies of the motion on corresponding tori are rationally dependent (and so the motion reduces to a torus of smaller dimension) has Lebesgue measure zero. Our results about stochastic perturbations imply some new results for averaging of purely deterministic perturbations.

scholarly journals ON STOCHASTIC PERTURBATIONS OF DYNAMICAL SYSTEMS WITH FAST AND SLOW COMPONENTS

2001 ◽  
Vol 01 (02) ◽  
pp. 261-281 ◽  
Author(s):  
MARK FREIDLIN
Keyword(s):  
Dynamical Systems ◽  
Large Deviations ◽  
Stochastic Resonance ◽  
Slow Motion ◽  
Point Of View ◽  
Random Perturbations ◽  
Stochastic Perturbations ◽  
Large Deviations Theory ◽  
Stable Equilibria ◽  
Small Random Perturbations

Dynamical systems with fast and slow components are considered. We show that small random perturbations of the fast component can lead to essential changes in the limiting slow motion. For example, new stable equilibria or deterministic oscillations with amplitude and frequency of order 1 can be introduced by the perturbations. These are stochastic resonance type effects, and they are considered from the point of view of large deviations theory.

Invariant distributions for shapes in sequences of randomly-divided rectangles

1999 ◽  
Vol 31 (1) ◽  
pp. 1-14 ◽  
Author(s):  
Francis K. C. Chen ◽  
Richard Cowan
Keyword(s):  
Dynamical Systems ◽  
Markov Processes ◽  
Shape Descriptor ◽  
Shape Space ◽  
Probability Measures ◽  
Shape Theory ◽  
Stochastic Dynamical Systems ◽  
One Dimensional ◽  
Invariant Distributions ◽  
Dimensional Shape

Interest has been shown in Markovian sequences of geometric shapes. Mostly the equations for invariant probability measures over shape space are extremely complicated and multidimensional. This paper deals with rectangles which have a simple one-dimensional shape descriptor. We explore the invariant distributions of shape under a variety of randomised rules for splitting the rectangle into two sub-rectangles, with numerous methods for selecting the next shape in sequence. Many explicit results emerge. These help to fill a vacant niche in shape theory, whilst contributing at the same time, new distributions on [0,1] and interesting examples of Markov processes or, in the language of another discipline, of stochastic dynamical systems.

scholarly journals Integrability of dominated decompositions on three-dimensional manifolds

2016 ◽  
Vol 37 (2) ◽  
pp. 606-620
Author(s):  
STEFANO LUZZATTO ◽  
SİNA TÜRELİ ◽  
KHADIM WAR
Keyword(s):  
Dynamical Systems ◽  
Three Dimensional ◽  
Regularity Conditions ◽  
Two Dimensional ◽  
Lipschitz Continuous ◽  
Invariant Distributions

We investigate the integrability of two-dimensional invariant distributions (tangent sub-bundles) which arise naturally in the context of dynamical systems on 3-manifolds. In particular, we prove unique integrability of dynamically dominated and volume-dominated Lipschitz continuous invariant decompositions as well as distributions with some other regularity conditions.

Invariant distributions for shapes in sequences of randomly-divided rectangles

1999 ◽  
Vol 31 (01) ◽  
pp. 1-14 ◽  
Author(s):  
Francis K. C. Chen ◽  
Richard Cowan
Keyword(s):  
Dynamical Systems ◽  
Markov Processes ◽  
Shape Descriptor ◽  
Shape Space ◽  
Probability Measures ◽  
Shape Theory ◽  
Stochastic Dynamical Systems ◽  
One Dimensional ◽  
Invariant Distributions ◽  
Dimensional Shape

Interest has been shown in Markovian sequences of geometric shapes. Mostly the equations for invariant probability measures over shape space are extremely complicated and multidimensional. This paper deals with rectangles which have a simple one-dimensional shape descriptor. We explore the invariant distributions of shape under a variety of randomised rules for splitting the rectangle into two sub-rectangles, with numerous methods for selecting the next shape in sequence. Many explicit results emerge. These help to fill a vacant niche in shape theory, whilst contributing at the same time, new distributions on [0,1] and interesting examples of Markov processes or, in the language of another discipline, of stochastic dynamical systems.

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