The Averaging Principle. Fluctuations in Dynamical Systems with Averaging

2012 ◽  
pp. 192-257
Author(s):  
Mark I. Freidlin ◽  
Alexander D. Wentzell
Keyword(s):  
Dynamical Systems ◽  
Averaging Principle

scholarly journals Stochastic averaging principle for dynamical systems with fractional Brownian motion

2014 ◽  
Vol 19 (4) ◽  
pp. 1197-1212 ◽  
Author(s):  
Yong Xu ◽  
◽  
Rong Guo ◽  
Di Liu ◽  
Huiqing Zhang ◽  
...  
Keyword(s):  
Brownian Motion ◽  
Dynamical Systems ◽  
Fractional Brownian Motion ◽  
Stochastic Averaging ◽  
Averaging Principle

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.

An averaging principle for stochastic dynamical systems with Lévy noise

2011 ◽  
Vol 240 (17) ◽  
pp. 1395-1401 ◽  
Author(s):  
Yong Xu ◽  
Jinqiao Duan ◽  
Wei Xu
Keyword(s):  
Dynamical Systems ◽  
Lévy Noise ◽  
Averaging Principle ◽  
Stochastic Dynamical Systems ◽  
Levy Noise

scholarly journals Periodic averaging method for impulsive stochastic dynamical systems driven by fractional Brownian motion under non-Lipschitz condition

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Anas Dheyab Khalaf ◽  
Mahmoud Abouagwa ◽  
Xiangjun Wang
Keyword(s):  
Brownian Motion ◽  
Dynamical Systems ◽  
Differential Equations ◽  
Fractional Brownian Motion ◽  
Lipschitz Condition ◽  
Averaging Method ◽  
Averaging Principle ◽  
Mean Square ◽  
Stochastic Dynamical Systems ◽  
Theoretical Results

AbstractThis paper presents the periodic averaging principle for impulsive stochastic dynamical systems driven by fractional Brownian motion (fBm). Under non-Lipschitz condition, we prove that the solutions to impulsive stochastic differential equations (ISDEs) with fBm can be approximated by the solutions to averaged SDEs without impulses both in the sense of mean square and probability. Finally, an example is provided to illustrate the theoretical results.

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