scholarly journals Nonlinear stochastic perturbations of dynamical systems and quasi-linear parabolic PDE’s with a small parameter

2009 ◽  
Vol 147 (1-2) ◽  
pp. 273-301 ◽  
Author(s):  
M. Freidlin ◽  
L. Koralov
Keyword(s):  
Dynamical Systems ◽  
Small Parameter ◽  
Stochastic Perturbations

Symplectic analysis of dynamical systems with a small parameter. A new criterion for stabilization of homoclinic separatrices and its application

1992 ◽  
Vol 44 (1) ◽  
pp. 41-58
Author(s):  
Yu. O. Mitropol'skii ◽  
I. O. Antonishin ◽  
A. K. Prikarpats'kyy ◽  
V. G. Samoilenko
Keyword(s):  
Dynamical Systems ◽  
Small Parameter ◽  
New Criterion

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.

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