scholarly journals Singular perturbations in noisy dynamical systems - CORRIGENDUM

2018 ◽  
Vol 29 (4) ◽  
pp. 594-594
Author(s):  
B. J. MATKOWSKY
Keyword(s):  
Dynamical Systems ◽  
Singular Perturbations ◽  
Kolmogorov Equation ◽  
Backward Kolmogorov Equation

On the seventh line of the fifth page of the article the Backward Kolmogorov equation should read pt = Lp, and not ps = Lp.The author apologises for the error.

Reliability Estimation of Stochastic Dynamical Systems with Fractional Order PID Controller

2018 ◽  
Vol 18 (06) ◽  
pp. 1850083 ◽  
Author(s):  
Wei Li ◽  
Lincong Chen ◽  
Junfeng Zhao ◽  
Natasa Trisovic
Keyword(s):  
Dynamical Systems ◽  
Fractional Order ◽  
First Passage Time ◽  
Gaussian White Noise ◽  
Passage Time ◽  
Reliability Estimation ◽  
Kolmogorov Equation ◽  
Critical Parameters ◽  
Stochastic Dynamical Systems ◽  
Stochastic Dynamical System

In this paper, the reliability of stochastic dynamical systems under Gaussian white noise excitations with fractional order proportional–inegral–derivative (FOPID) controller is estimated. First, the FOPID controller is approximated by a set of combination of displacement and velocity based on the generalized van der Pol transformation. Then, the stochastic averaging method of energy envelope is applied to obtain a diffusive differential equation, from which the Backward Kolmogorov equation, governing the conditional reliability function, and the Generalized Pontryagin equation, governing the statistical moments of first-passage time, are derived from the averaged equation and solved numerically. Finally, in the two examples, the critical parameters in the FOPID controller are shown to be capable of improving the reliability of the stochastic dynamical system apparently, and all numerical results are verified to be efficient and correct by the Monte Carlo simulation.

scholarly journals The nonlocal boundary value problem for one-dimensional backward Kolmogorov equation and associated semigroup

2019 ◽  
Vol 11 (2) ◽  
pp. 463-474
Author(s):  
R.V. Shevchuk ◽  
I.Ya. Savka ◽  
Z.M. Nytrebych
Keyword(s):  
Boundary Condition ◽  
Boundary Value Problem ◽  
Boundary Value ◽  
Nonlocal Boundary ◽  
Boundary Integral ◽  
Kolmogorov Equation ◽  
Linear Parabolic Equation ◽  
One Dimensional ◽  
Wentzell Boundary Condition ◽  
Backward Kolmogorov Equation

This paper is devoted to a partial differential equation approach to the problem of construction of Feller semigroups associated with one-dimensional diffusion processes with boundary conditions in theory of stochastic processes. In this paper we investigate the boundary-value problem for a one-dimensional linear parabolic equation of the second order (backward Kolmogorov equation) in curvilinear bounded domain with one of the variants of nonlocal Feller-Wentzell boundary condition. We restrict our attention to the case when the boundary condition has only one term and it is of the integral type. The classical solution of the last problem is obtained by the boundary integral equation method with the use of the fundamental solution of backward Kolmogorov equation and the associated parabolic potentials. This solution is used to construct the Feller semigroup corresponding to such a diffusion phenomenon that a Markovian particle leaves the boundary of the domain by jumps.

scholarly journals On Deterministic and Stochastic Multiple Pathogen Epidemic Models

Epidemiologia ◽  
2021 ◽  
Vol 2 (3) ◽  
pp. 325-337
Author(s):  
Fernando Vadillo
Keyword(s):  
Variable Coefficients ◽  
Epidemic Models ◽  
Kolmogorov Equation ◽  
The Finite Element Method ◽  
Persistence Time ◽  
Main Challenge ◽  
Stochastic Epidemic ◽  
The Mean ◽  
Backward Kolmogorov Equation ◽  
Multiple Pathogen

In this paper, we consider a stochastic epidemic model with two pathogens. In order to analyze the coexistence of two pathogens, we compute numerically the expectation time until extinction (the mean persistence time), which satisfies a stationary partial differential equation with degenerate variable coefficients, related to backward Kolmogorov equation. I use the finite element method in order to solve this equation, and we implement it in FreeFem++. The main conclusion of this paper is that the deterministic and stochastic epidemic models differ considerably in predicting coexistence of the two diseases and in the extinction outcome of one of them. Now, the main challenge would be to find an explanation for this result.

scholarly journals MARKOV MODELS FOR THE ANALYSIS OF DYNAMICAL SYSTEMS

2017 ◽  
Vol 16 (1) ◽  
pp. 35-49
Author(s):  
A. A. AKINTUNDE ◽  
S. O.N AGWUEGBO ◽  
O. M. OLAYIWOLA
Keyword(s):  
Exchange Rate ◽  
Dynamical Systems ◽  
Markov Models ◽  
Effective Means ◽  
Kolmogorov Equation ◽  
State Variables ◽  
Stationary Probability Distribution ◽  
Signal Model ◽  
State Space Approach ◽  
Space Approach

Most real world situations involve modelling of physical processes that evolve with time and space, especially those exhibiting high variability. Such events that have to flow with time or space are called dynamical systems. The mathematical notions of a dynamical system serves to depict the flow of causation from past into future (Kalman 1960). In this study, Markov model which is a signal model based on the Markovian property with state space approach was adopted for the analysis of dynamical systems. The Nigerian monetary exchange rate data was used in the application with the use of R statistical software package. The study incorporated the Chapman-Kolmogorov equation in the construction of absolute limiting distribution of the system via the state variables. The procedure gives an easy and effective means of analysing complex and time varying dynamical systems. The study showed that the Nigerian monetary exchange rate is ergodic with stationary probability distribution.  

The stationary moments of Poisson-driven non-linear dynamical systems

1982 ◽  
Vol 19 (3) ◽  
pp. 702-706
Author(s):  
Charles E. Smith ◽  
Loren Cobb
Keyword(s):  
Dynamical Systems ◽  
Probability Density ◽  
Stochastic Systems ◽  
Transition Probability ◽  
Kolmogorov Equation ◽  
Transition Probability Density ◽  
Linear Dynamical Systems ◽  
Stationary Density ◽  
Non Linear ◽  
Noise Input

Moment recursion relations have previously been derived for the stationary probability density functions of continuous-time stochastic systems with Wiener (white noise) input. These results are extended in this paper to the case of Poisson (shot noise) input. The non-linear dynamical systems are expressed, in general, as stochastic differential equations, with an independent increment input. The transition probability density function evolves according to the appropriate Kolmogorov equation. Moments of the stationary density are obtained from the Fourier transform of the stationary density. The moment relations can be used to estimate the parameters of linear and non-linear stochastic systems from empirical moments, given either Wiener or Poisson input.

Solution to the Backward-Kolmogorov Equation for a Nonstationary Oscillation Problem

1982 ◽  
Vol 49 (4) ◽  
pp. 923-925 ◽  
Author(s):  
G. P. Solomos ◽  
P-T. D. Spanos
Keyword(s):  
Random Excitation ◽  
Response Amplitude ◽  
Kolmogorov Equation ◽  
Linear Oscillator ◽  
Markov Approximation ◽  
Nonstationary Oscillation ◽  
Oscillation Problem ◽  
Backward Kolmogorov Equation

The solution of a backward-Kolmogorov equation is presented. This equation is associated with a Markov approximation of the response amplitude of a lightly damped linear oscillator driven by an evolutionary random excitation.

DISPERSIVE GROUNDWATER FLOW AND POLLUTION

1991 ◽  
Vol 01 (01) ◽  
pp. 61-81 ◽  
Author(s):  
ONNO A. VAN HERWAARDEN ◽  
JOHAN GRASMAN
Keyword(s):  
Random Walk ◽  
Dirichlet Problem ◽  
Groundwater Flow ◽  
Arrival Time ◽  
Kolmogorov Equation ◽  
Dispersive Flow ◽  
Backward Kolmogorov Equation ◽  
Protected Zone

By solving asymptotically the Dirichlet problem for the backward Kolmogorov equation describing the random walk of a particle in a dispersive flow, it is computed at what rate contaminated particles cross the boundary of a protected zone. The method also yields an estimate of the expected arrival time.

Sign in / Sign up

Export Citation Format

Share Document