Solutions to the Fokker-Planck-Kolmogorov equation for the van der pol system subjected to periodic and random perturbations

1983 ◽  
Vol 34 (6) ◽  
pp. 635-637
Author(s):  
Nguyen Dong Anh ◽  
Kieu Th� Dyk
Keyword(s):  
Kolmogorov Equation ◽  
Random Perturbations ◽  
Van Der Pol ◽  
Fokker Planck

A class of nonlinear oscillational systems admitting exact solution of the Fokker-Planck-Kolmogorov equation

1991 ◽  
Vol 43 (10) ◽  
pp. 1282-1286
Author(s):  
Yu. A. Mitropol'skii ◽  
Nguen Tien Kkhiem
Keyword(s):  
Exact Solution ◽  
Kolmogorov Equation ◽  
Fokker Planck

Galerkin Scheme-Based Determination of Survival Probability of Oscillators With Fractional Derivative Elements

2016 ◽  
Vol 83 (12) ◽  
Author(s):  
Pol D. Spanos ◽  
Alberto Di Matteo ◽  
Yezeng Cheng ◽  
Antonina Pirrotta ◽  
Jie Li
Keyword(s):  
Fractional Derivative ◽  
Survival Probability ◽  
Stochastic Averaging ◽  
Kolmogorov Equation ◽  
System Response ◽  
Statistical Linearization ◽  
Van Der Pol ◽  
Galerkin Scheme ◽  
Confluent Hypergeometric Functions

In this paper, an approximate semi-analytical approach is developed for determining the first-passage probability of randomly excited linear and lightly nonlinear oscillators endowed with fractional derivative elements. The amplitude of the system response is modeled as one-dimensional Markovian process by employing a combination of the stochastic averaging and the statistical linearization techniques. This leads to a backward Kolmogorov equation which governs the evolution of the survival probability of the oscillator. Next, an approximate solution of this equation is sought by resorting to a Galerkin scheme. Specifically, a convenient set of confluent hypergeometric functions, related to the corresponding linear oscillator with integer-order derivatives, is used as orthogonal basis for this scheme. Applications to the standard viscous linear and to nonlinear (Van der Pol and Duffing) oscillators are presented. Comparisons with pertinent Monte Carlo simulations demonstrate the reliability of the proposed approximate analytical solution.

On uniqueness of probability solutions to the Fokker-Planck-Kolmogorov equation

2021 ◽  
Vol 212 (6) ◽  
Author(s):  
Vladimir Igorevich Bogachev ◽  
Tikhon Il'ich Krasovitskii ◽  
Stanislav Valer'evich Shaposhnikov
Keyword(s):  
Kolmogorov Equation ◽  
Fokker Planck

Model for the Suspensions Separation Process in Hydrodynamic Filters with a Rotating Filter Element

2021 ◽  
pp. 48-56
Author(s):  
Vladimir Devisilov ◽  
V. Lvov ◽  
E. Sharay
Keyword(s):  
Theoretical Calculation ◽  
Statistical Method ◽  
Separation Process ◽  
Filter Element ◽  
Kolmogorov Equation ◽  
Diffusion Type ◽  
Self Cleaning ◽  
Fokker Planck

The possibility of applying the probabilistic-statistical method for theoretical calculation of characteristics related to suspensions separation in self-cleaning hydrodynamic filters with a rotating filter element has been demonstrated. It has been shown that the change in characteristics of the suspensions separation in hydrodynamic filters with a rotating filter element can be satisfactorily described based on equations of diffusion type, in particular, using the Fokker–Planck–Kolmogorov equation. The main parameters of the separation process in hydrodynamic filters with a rotating filter element have been determined.

A Kushner approach for small random perturbations of the Duffing-van der Pol system

Automatica ◽  
2009 ◽  
Vol 45 (4) ◽  
pp. 1097-1099 ◽  
Author(s):  
Shambhu N. Sharma
Keyword(s):  
Random Perturbations ◽  
Van Der Pol ◽  
Small Random Perturbations

Transport and evolution of classical and quantum ensembles

2020 ◽  
pp. 292-341
Author(s):  
Sandip Tiwari
Keyword(s):  
Boltzmann Transport Equation ◽  
Planck Equation ◽  
Kolmogorov Equation ◽  
Boltzmann Transport ◽  
Relaxation Time Approximation ◽  
Quantum Mechanical Description ◽  
Probabilistic Evolution ◽  
Fokker Planck Equations ◽  
And Diffusion ◽  
Fokker Planck

This chapter explores the evolution of an ensemble of electrons under stimulus, classically and quantum-mechanically. The classical Liouville description is derived, and then reformed to the quantum Liouville equation. The differences between the classical and the quantum-mechanical description are discussed, emphasizing the uncertainty-induced fuzziness in the quantum description. The Fokker-Planck equation is introduced to describe the evolution of ensembles and fluctuations in it that comprise the noise. The Liouville description makes it possible to write the Boltzmann transport equation with scattering. Limits of validity of the relaxation time approximation are discussed for the various scattering possibilities. From this description, conservation equations are derived, and drift and diffusion discussed as an approximation. Brownian motion arising in fast-and-slow events and response are related to the drift and diffusion and to the Langevin and Fokker-Planck equations as probabilistic evolution. This leads to a discussion of Markov processes and the Kolmogorov equation.

A Kolmogorov-Fokker-Planck approach for a stochastic Duffing-van der Pol system

2008 ◽  
Vol 16 (4) ◽  
pp. 351-377 ◽  
Author(s):  
Shambhu N. Sharma
Keyword(s):  
Van Der Pol ◽  
Fokker Planck
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