Fluctuations in the averaging scheme for differential equations with random pulsed action

Самойленко, А. М.; Stanzhitskii, A. N.

doi:10.1007/bf01060541

Cited by 2 publications

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“…Along with the continuity in t and x, the following assumptions are usually made for the right-hand side of the mapping f t x Î ( , , ) w ): For the clarity of further studies, let us briefly present the results from the paper [6], which generalizes this statement to the case of impulsive perturbations of the DE W (1) of the form…”

Section: Problem Statementmentioning

confidence: 99%

“…For clarity, let us first present the well-known results from [5,6,14]. In contrast to most studies on the averaging method of impulsive systems, we will consider impulsive systems with Markov switching.…”

Section: Introductionmentioning

confidence: 99%

“…The Bogoliubov-Mitropolsky principle of averaging is proved in the monograph [5] for impulsive systems and in [6] for the case of random perturbations. For clarity, let us first present the well-known results from [5,6,14].…”

Section: Introductionmentioning

confidence: 99%

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Stability in impulsive systems with Markov perturbations in averaging scheme. I. Averaging principle for impulsive Markov systems

2010

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The Bogoliubov-Mitropolsky small parameter method is used to study the behavior of stochastic differential systems in the analysis of the corresponding properties of solutions of averaged systems. INTRODUCTIONAveraging over explicit time [4] is one of the most popular methods for studying dynamic systems. This method also performs well in the theory of random perturbations [1, 2, 5, 6, 10-13] (see the references therein for a more detailed bibliography).This method allows not only setting up an averaged system that approximately describes the dynamics of the original model, but also writing stochastic differential equations (SDEs) for normalized deviations of the solution of the original equation from the corresponding solutions of averaged motion.The Bogoliubov-Mitropolsky principle of averaging is proved in the monograph [5] for impulsive systems and in [6] for the case of random perturbations. For clarity, let us first present the well-known results from [5,6,14]. In contrast to most studies on the averaging method of impulsive systems, we will consider impulsive systems with Markov switching.In the paper, we will prove that limit theorems can be used for stability analysis as is the case with smooth stochastic systems [10][11][12][13]. In the second and third sections, an averaged system and diffusion approximation are set up; their analysis with the use of modern computer technologies allows impruving the computational efficiency by a factor of tens of millions.

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Section: Problem Statementmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%