Stability Of Stochastic Dynamical Systems Research Articles

Stability of stochastic dynamic systems of a random structure with Markov switching in the presence of concentration points

<abstract><p>This article aims to investigate sufficient conditions for the stability of the trivial solution of stochastic differential equations with a random structure, particularly in contexts involving the presence of concentration points. The proof of asymptotic stability leverages the use of Lyapunov functions, supplemented by additional constraints on the magnitudes of jumps and jump times, as well as the Markov property of the system solutions. The findings are elucidated with an example, demonstrating both stable and unstable conditions of the system. The novelty of this work is in the consideration of jump concentration points, which are not considered in classical works. The assumption of the existence of concentration points leads to additional constraints on jumps, jump times and relations between them.</p></abstract>

The Stochastic Stability of Internal HIV Models with Gaussian White Noise and Gaussian Colored Noise

In this paper, the stochastic stability of internal HIV models driven by Gaussian white noise and Gaussian colored noise is analyzed. First, the stability of deterministic models is investigated. By analyzing the characteristic values of endemic equilibrium, we could obtain that internal HIV models reach a steady state under the influence of RTI and PI drugs. Then we discuss the stochastic stability of internal HIV models driven by Gaussian white noise and Gaussian colored noise, based on probability density functions. The functional methods are carried out to derive the approximate Fokker-Planck equation of stochastic internal HIV systems and further obtain the marginal probability density functions. Finally, numerical results show that the noise intensities have a great influence on uninfected cell, infected cell, and virus particles, for predicting the stability of stochastic dynamic systems subjected to Gaussian white noise and Gaussian colored noise.

Stochastic basin stability in complex networks

In this paper, we propose a novel concept, stochastic basin stability, to investigate the stability of dynamical systems under random noises. By this concept, we investigate synchronization of multistable second-order Kuramoto models under continuously acting perturbations in complex networks. By a mean-field approach and the Fokker-Planck equation, we derive formulas of the density functions of both phase and frequency. Based on them, we provide an analytical treatment of stochastic basin stability and illustrate that the theoretical results are in good agreement with numerical simulations. This proposed concept integrates the perturbations of both initial condition and random intervention, and paves a general and efficient approach for analytically and numerically investigating stability of stochastic dynamical systems.

Stability of Stochastic Systems of Random Structure with Markov Switchings and Perturbations

The authors use Lyapunov–Krasovskii functionals to investigate the stability in probability, asymptotic stochastic stability, root-mean-square stability, and general root-mean-square exponential stability of stochastic dynamic systems of random structure with Markov switchings and perturbation.

Partial-State Stabilization and Optimal Feedback Control for Stochastic Dynamical Systems

In this paper, we develop a unified framework to address the problem of optimal nonlinear analysis and feedback control for partial stability and partial-state stabilization of stochastic dynamical systems. Partial asymptotic stability in probability of the closed-loop nonlinear system is guaranteed by means of a Lyapunov function that is positive definite and decrescent with respect to part of the system state which can clearly be seen to be the solution to the steady-state form of the stochastic Hamilton–Jacobi–Bellman equation, and hence, guaranteeing both partial stability in probability and optimality. The overall framework provides the foundation for extending optimal linear-quadratic stochastic controller synthesis to nonlinear-nonquadratic optimal partial-state stochastic stabilization. Connections to optimal linear and nonlinear regulation for linear and nonlinear time-varying stochastic systems with quadratic and nonlinear-nonquadratic cost functionals are also provided. Finally, we also develop optimal feedback controllers for affine stochastic nonlinear systems using an inverse optimality framework tailored to the partial-state stochastic stabilization problem and use this result to address polynomial and multilinear forms in the performance criterion.

Chatter detection in turning using persistent homology

This paper describes a new approach for ascertaining the stability of stochastic dynamical systems in their parameter space by examining their time series using topological data analysis (TDA). We illustrate the approach using a nonlinear delayed model that describes the tool oscillations due to self-excited vibrations in turning. Each time series is generated using the Euler-Maruyama method and a corresponding point cloud is obtained using the Takens embedding. The point cloud can then be analyzed using a tool from TDA known as persistent homology. The results of this study show that the described approach can be used for analyzing datasets of delay dynamical systems generated both from numerical simulation and experimental data. The contributions of this paper include presenting for the first time a topological approach for investigating the stability of a class of nonlinear stochastic delay equations, and introducing a new application of TDA to machining processes.

Analysis of stability of stochastic dynamic systems under Poisson perturbations. II. Stability of solutions in quadratic mean of systems of linear stochastic differential equations under Poisson perturbations

Necessary and sufficient conditions of asymptotic stability in quadratic mean are obtained for trivial solutions of systems of linear stochastic differential equations under Poisson perturbations. Model problems are analyzed.

Some results on stability of stochastic dynamical systems

Abstract In this brief survey, we are concerned with the problem of stability of differential equations with randomly fluctuating parameters. Even for this specific problem, we can cover only a small portion of the available results. The increased interest on this topic has generated a vast literature. Thus, we can only touch upon a few of the significant results and techniques. We distinguish between stochastic equations with Gaussian white noise coefficients, and stochastic equations with (non-white) physical noise coefficients. We discuss a few problems for which results are available, but complete answers have not been forthcoming as yet. We treat, mainly, stochastic ordinary differential equations. However, we have also included a brief discussion of stochastic functional equations as well as discrete time stochastic equations. Functional equations are of interest in applications to materials with memory and to control of complex multicomponent structures. Discrete time equations are of interest as on-line computer models of identification algorithms.