Stabilization of Stochastic Diffusive Dynamical Systems with Impulse Markov Switchings and Parameters. Part I. Stability of Impulse Stochastic Systems with Markov Parameters

2009 ◽  
Vol 41 (2) ◽  
pp. 1-24 ◽  
Author(s):  
Taras O. Lukashiv ◽  
Vladimir K. Yasinskiy ◽  
Evgeniy V. Yasinskiy
Keyword(s):  
Dynamical Systems ◽  
Stochastic Systems ◽  
Markov Parameters

scholarly journals Bifurcation in Mean Phase Portraits for Stochastic Dynamical Systems with Multiplicative Gaussian Noise

2020 ◽  
Vol 30 (11) ◽  
pp. 2050216
Author(s):  
Hui Wang ◽  
Athanasios Tsiairis ◽  
Jinqiao Duan
Keyword(s):  
Dynamical Systems ◽  
Gaussian Noise ◽  
Stochastic Systems ◽  
Planck Equation ◽  
Phase Portraits ◽  
Stochastic Bifurcation ◽  
Stochastic Dynamical Systems ◽  
Solution Processes ◽  
Bifurcation Phenomena ◽  
Stability Type

We investigate the bifurcation phenomena for stochastic systems with multiplicative Gaussian noise, by examining qualitative changes in mean phase portraits. Starting from the Fokker–Planck equation for the probability density function of solution processes, we compute the mean orbits and mean equilibrium states. A change in the number or stability type, when a parameter varies, indicates a stochastic bifurcation. Specifically, we study stochastic bifurcation for three prototypical dynamical systems (i.e. saddle-node, transcritical, and pitchfork systems) under multiplicative Gaussian noise, and have found some interesting phenomena in contrast to the corresponding deterministic counterparts.

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.

scholarly journals Implications of Noise on Neural Correlates of Consciousness: A Computational Analysis of Stochastic Systems of Mutually Connected Processes

Entropy ◽  
2021 ◽  
Vol 23 (5) ◽  
pp. 583
Author(s):  
Pavel Kraikivski
Keyword(s):  
Dynamical Systems ◽  
Stochastic Systems ◽  
System Size ◽  
Neural Correlates ◽  
Spectral Entropy ◽  
Modeling And Analysis ◽  
Neural Correlates Of Consciousness ◽  
Power Spectral ◽  
Neuronal Processes ◽  
Random Fluctuations

Random fluctuations in neuronal processes may contribute to variability in perception and increase the information capacity of neuronal networks. Various sources of random processes have been characterized in the nervous system on different levels. However, in the context of neural correlates of consciousness, the robustness of mechanisms of conscious perception against inherent noise in neural dynamical systems is poorly understood. In this paper, a stochastic model is developed to study the implications of noise on dynamical systems that mimic neural correlates of consciousness. We computed power spectral densities and spectral entropy values for dynamical systems that contain a number of mutually connected processes. Interestingly, we found that spectral entropy decreases linearly as the number of processes within the system doubles. Further, power spectral density frequencies shift to higher values as system size increases, revealing an increasing impact of negative feedback loops and regulations on the dynamics of larger systems. Overall, our stochastic modeling and analysis results reveal that large dynamical systems of mutually connected and negatively regulated processes are more robust against inherent noise than small systems.

scholarly journals Controlled Discrete-Time Semi-Markov Random Evolutions and Their Applications

Mathematics ◽  
2021 ◽  
Vol 9 (2) ◽  
pp. 158
Author(s):  
Anatoliy Swishchuk ◽  
Nikolaos Limnios
Keyword(s):  
Dynamical Systems ◽  
Discrete Time ◽  
Diffusion Approximation ◽  
Limit Theorems ◽  
Stochastic Systems ◽  
Renewal Processes ◽  
Additive Functionals ◽  
Markov Random ◽  
Markov Renewal Processes ◽  
Random Evolutions

In this paper, we introduced controlled discrete-time semi-Markov random evolutions. These processes are random evolutions of discrete-time semi-Markov processes where we consider a control. applied to the values of random evolution. The main results concern time-rescaled weak convergence limit theorems in a Banach space of the above stochastic systems as averaging and diffusion approximation. The applications are given to the controlled additive functionals, controlled geometric Markov renewal processes, and controlled dynamical systems. We provide dynamical principles for discrete-time dynamical systems such as controlled additive functionals and controlled geometric Markov renewal processes. We also produce dynamic programming equations (Hamilton–Jacobi–Bellman equations) for the limiting processes in diffusion approximation such as controlled additive functionals, controlled geometric Markov renewal processes and controlled dynamical systems. As an example, we consider the solution of portfolio optimization problem by Merton for the limiting controlled geometric Markov renewal processes in diffusion approximation scheme. The rates of convergence in the limit theorems are also presented.

On the Theory of a Nonlinear Dynamic Circuit Filtering

2020 ◽  
Vol 19 (03) ◽  
pp. 2050022
Author(s):  
Dhruvi S. Bhatt ◽  
Shaival H. Nagarsheth ◽  
Shambhu N. Sharma
Keyword(s):  
Dynamical Systems ◽  
Stochastic Systems ◽  
Observation Equation ◽  
Time Varying ◽  
Physical Systems ◽  
Random Forcing ◽  
Potential Case ◽  
Set Up ◽  
Nonlinear Sampling ◽  
Nonlinear Observation

Stochastic Differential Equations (SDEs) describe physical systems to account for random forcing terms in the evolution of the state trajectory. The noisy sampling mixer, a component of digital wireless communications, can be regarded as a potential case from the dynamical systems’ viewpoint. The universality of the noisy sampling mixer is attributed to the fact that it adopts the structure of a nonlinear SDE and its linearized version becomes a time-varying bilinear SDE. This paper develops a mathematical theory for the nonlinear noisy sampling mixer from the filtering viewpoint. Since the filtering of stochastic systems hinges on the structure of dynamical systems and observation equation set up, we consider three ‘filtering models’. The first model, accounts for a nonlinear SDE coupled with a nonlinear observation equation. In the second model, we consider a bilinear SDE with a linear observation equation to achieve the nonlinear sampling filtering. Note that the bilinear SDE coupled with the linear observation is a consequence of the Carleman linearization to the nonlinear SDE and the nonlinear observation equation. In the third model, we consider a Stratonovich SDE coupled with a nonlinear observation equation. The filtering equation of this paper can be further utilized to guide the design process of the noisy sampling mixer.

Universal Feedback Controllers and Inverse Optimality for Nonlinear Stochastic Systems

2019 ◽  
Vol 142 (2) ◽  
Author(s):  
Wassim M. Haddad ◽  
Xu Jin
Keyword(s):  
Dynamical Systems ◽  
Feedback Control ◽  
Stochastic Control ◽  
Stochastic Systems ◽  
Sufficient Conditions ◽  
Optimal Feedback Control ◽  
Control Law ◽  
Stochastic Dynamical Systems ◽  
Control Lyapunov Function ◽  
Inverse Optimality

Abstract In this paper, we develop a constructive finite time stabilizing feedback control law for stochastic dynamical systems driven by Wiener processes based on the existence of a stochastic control Lyapunov function. In addition, we present necessary and sufficient conditions for continuity of such controllers. Moreover, using stochastic control Lyapunov functions, we construct a universal inverse optimal feedback control law for nonlinear stochastic dynamical systems that possess guaranteed gain and sector margins. An illustrative numerical example involving the control of thermoacoustic instabilities in combustion processes is presented to demonstrate the efficacy of the proposed framework.

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

1982 ◽  
Vol 19 (03) ◽  
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.

AN AVERAGING PRINCIPLE FOR COMBINED INTERACTION GRAPHS — CONNECTIVITY AND APPLICATIONS TO GENETIC SWITCHES

2010 ◽  
Vol 13 (03) ◽  
pp. 293-326 ◽  
Author(s):  
MARKUS KIRKILIONIS ◽  
LUCA SBANO
Keyword(s):  
Dynamical Systems ◽  
Markov Chains ◽  
Cell Biology ◽  
Stochastic Systems ◽  
Order Term ◽  
Continuous Variables ◽  
Leading Order ◽  
Loop Formation ◽  
Qualitative Properties ◽  
Model Complex

Time-continuous dynamical systems defined on graphs are often used to model complex systems with many interacting components in a non-spatial context. In the reverse sense attaching meaningful dynamics to given "interaction diagrams" is a central bottleneck problem in many application areas, especially in cell biology where various such diagrams with different conventions describing molecular regulation are presently in use. In most situations these diagrams can only be interpreted by the use of both discrete and continuous variables during the modelling process, corresponding to both deterministic and stochastic hybrid dynamics. The conventions in genetics are well known, and therefore we use this field for illustration purposes. In [25] and [26] the authors showed that with the help of a multi-scale analysis stochastic systems with both continuous variables and finite state spaces can be approximated by dynamical systems whose leading order time evolution is given by a combination of ordinary differential equations (ODEs) and Markov chains. The leading order term in these dynamical systems is called average dynamics and turns out to be an adequate concept to analyze a class of simplified hybrid systems. Once the dynamics is defined the mutual interaction of both ODEs and Markov chains can be analyzed through the (reverse) introduction of the so-called Interaction Graph, a concept originally invented for time-continuous dynamical systems, see [5]. Here we transfer this graph concept to the average dynamics, which itself is introduced as a heuristic tool to construct models of reaction or contact networks. The graphical concepts introduced form the basis for any subsequent study of the qualitative properties of hybrid models in terms of connectivity and (feedback) loop formation.

Sign in / Sign up

Export Citation Format

Share Document