Parametric continuity of solutions to stochastic functional differential equations with poisson perturbations

Yasinsky, V. K.; Malyka, I. V.

doi:10.1007/s10559-012-9464-1

Cited by 2 publications

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References 4 publications

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“…Such parametric systems are considered in [3,4]. A stochastic model of PVTODC operation taking into account external random jump-like actions is a second-order stochastic differential equation [6,7] specified on probability basis ( , , , ) W F P ¹ [8] with regard for the Poisson random component with the initial condition…”

Section: Problem Statementmentioning

confidence: 99%

Analysis of Fluctuations of a Parametric Vacuum Tube Oscillator with Delayed Feedback

Yasynskyy

Malyk

2015

Cybern Syst Anal

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The generating equation and equations for amplitude and phase fluctuations of the parametric tube oscillator with delayed feedback are analyzed in the paper. The steady-state oscillation modes and the influence of fluctuations in the natural frequency of the oscillator on the operation of the self-oscillator and parametric "pumping" in the presence of interference are investigated. The domains for the parameters of the original equation corresponding to unstable nodes, stable nodes, and focal points are identified.

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Section: Problem Statementmentioning

confidence: 99%

Analysis of Fluctuations of a Parametric Vacuum Tube Oscillator with Delayed Feedback

Yasynskyy

Malyk

2015

Cybern Syst Anal

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Analysis of oscillations in quasilinear stochastic dynamic hereditary systems

Yasinsky

Malyk

2013

Cybern Syst Anal

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Oscillations in quasilinear stochastic hereditary systems and their application to specific problems are investigated. The algorithm of transition to a standard system with the help of the averaging method with small parameter in the presence of fast time is justified.In [1], the strong solution of a stochastic functional differential equation with Poisson perturbations (SFDEPP) is proved to be continuous with respect to a small parameter e > 0. This allows replacing the solutions of the SFDEPP with some Markov process [2][3][4]. As a consequence from the continuous dependence of the solution of the SFDEPP from parameter e > 0 , we will use the averaging method [5, 6] to establish the proximity of these processes on an asymptotically large time interval [ , / ] 0 T e . Moreover, we will present the method of reducing a second-order quasilinear SFDEPP to the standard form and prove the averaging theorem for this system. We will describe the splitting method, where the generating operator of the deterministic part of the stochastic equation on the chosen space C([ , ]) -t 0 of continuous functions is represented as a contour integral of a complex plane [7, 8]. On the probability basis ( , , , ) W F Ã ¹ [9], a random process R R +´® W 1 is defined as a strong solution [10-12] of the quasilinear SFDEPP dx t m x dt t dw t t u dt du f t dt t U ( ) ( ) ( ) ( ) ( , )~( , ) ( ) = + + + ò s l n (1)with the initial condition-® t 0 1 ; S is a Skorokhod space [13,14]; the random processes f R R : +´® W 1 and s : R R +´® W 1 and the random function l:Moreover, w t ( ) is a scalar Wiener process;~( , ): ( , ) ( , ) n n n t A t A t A = -¸is a centered Poisson measure [9, 10]; n( , ) ( ) t A t A = P ; and w t ( ) and~( , ) n t A are independent random processes. Denote by H t ( ) the solution of the deterministic equation [7, 15] dH t m H dt t ( ) ( ) = (5) with respect to the initial condition H t t t ( ) , , , . = = < ì í î 1 0 0 0 Following [7], we will call this solution the fundamental solution of (5).

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Parametric continuity of solutions to stochastic functional differential equations with poisson perturbations

Cited by 2 publications

References 4 publications

Analysis of Fluctuations of a Parametric Vacuum Tube Oscillator with Delayed Feedback

Analysis of Fluctuations of a Parametric Vacuum Tube Oscillator with Delayed Feedback

Analysis of oscillations in quasilinear stochastic dynamic hereditary systems

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