Stability of Solutions to the Stochastic Oskolkov Equation and Stabilization

We present two qualitative results concerning the solutions of the following equation: ; the first result covers the stochastic asymptotic stability of the zero solution for the above equation in case p ≡ 0, while the second one discusses the uniform stochastic boundedness of all solutions in case p≢0. Sufficient conditions for the stability and boundedness of solutions for the considered equation are obtained by constructing a Lyapunov functional. Two examples are also discussed to illustrate the efficiency of the obtained results.

This paper presents stochastic stability and stochastic boundedness to certain second-order nonlinear neutral stochastic differential equations. The second-order differential equation is weakened to a neutral stochastic system of first-order equations and used together with a second-order quadratic function to obtain perfect Lyapunov-Krasovskii functional. This functional is adapted and applied to obtain criteria on the nonlinear functions to ensure novel results on stochastic stability and stochastic asymptotic stability of the zero solution. Furthermore, when the forcing term is nonzero, fresh results on stochastic boundedness and uniform stochastic boundedness of solutions are obtained. The results of this paper are original, new, essentially improving, complementing, and simplifying several related ones in the literature. Two special cases of the theoretical results are supplied to demonstrate the applicability of the hypothetical results.

In this paper, we study the stability of Caputo-type fractional stochastic differential equations. Stochastic stability and stochastic asymptotical stability are shown by stopping time technique. Almost surly exponential stability and pth moment exponentially stability are derived by a new established Itô’s formula of Caputo version. Numerical examples are given to illustrate the main results.

Stochastic stability and bifurcation in a macroeconomic model

Stochastic asymptotical stability for stochastic impulsive differential equations and it is application to chaos synchronization

In this paper, we present sufficient conditions to ensure the stochastic asymptotic stability of the zero solution for a specific type of fourth-order stochastic differential equation (SDE) with constant delay. By reducing the fourth-order SDE to a system of first-order SDEs, we utilize a fourth-order quadratic function to derive an appropriate Lyapunov functional. This functional is then employed to establish standard criteria for the nonlinear functions present in the SDE. The stability result obtained in this study is novel and extends the existing findings on stability in fourth-order differential equations. Additionally, we provide an illustrative example to demonstrate the significance and accuracy of our main result.

We consider a stochastically perturbed Nowak-May model of virus dynamics within a host. We prove the global existence of unique strong solution. Using the Lyapunov method, we found sufficient conditions for the stochastic asymptotic stability of equilibrium solutions of this model.

Output feedback stabilization of stochastic planar nonlinear systems with output constraint

It is well known that stochastic coupled oscillator network (SCON) has been widely applied; however, there are few studies on SCON with bidirectional cross-dispersal (SCONBC). This paper intends to study stochastic stability for SCONBC. A new and suitable Lyapunov function for SCONBC is constructed on the basis of Kirchhoff's matrix tree theorem in graph theory. Combining stochastic analysis skills and Lyapunov method, a sufficient criterion guaranteeing stochastic stability for the trivial solution of SCONBC is provided, which is associated with topological structure and coupling strength of SCONBC. Furthermore, some numerical simulation examples are given in order to illustrate the validity and practicability of our results.

A stochastic nonlinear dynamical model is proposed to describe the vibration of rectangular thin plate under axial inplane excitation considering the influence of random environment factors. Firstly, the model is simplified by applying the stochastic averaging method of quasi-nonintegrable Hamilton system. Secondly, the methods of Lyapunov exponent and boundary classification associated with diffusion process are utilized to analyze the stochastic stability of the trivial solution of the system. Thirdly, the stochastic Hopf bifurcation of the vibration model is explored according to the qualitative changes in stationary probability density of system response, showing that the stochastic Hopf bifurcation occurs at two critical parametric values. Finally, some explanations are given in a simple way on the potential applications of stochastic stability and bifurcation analysis.

In this paper, the problem of stability in terms of two measures is considered for a class of stochastic partial differential delay equations with switching. Sufficient conditions for stability in terms of two measures are obtained based on the technique of constructing a proper approximating strong solution system and conducting a limiting type of argument to pass on stability of strong solutions to mild ones. In particular, the stochastic stability under the fixed‐index sequence monotonicity condition and under the average dwell‐time switching are considered.

Multidegree-of-freedom dynamic systems subjected to parametric excitation are analyzed for stochastic stability. The variation of excitation intensity with time is described by the sum of a harmonic function and a stationary random process. The stability boundaries are determined by the stochastic averaging method. The effect of random parametric excitation on the stability of trivial solutions of systems of differential equations for the moments of phase variables is studied. It is assumed that the frequency of harmonic component falls within the region of combination resonances. Stability conditions for the first and second moments are obtained. It turns out that additional parametric excitation may have a stabilizing or destabilizing effect, depending on the values of certain parameters of random excitation. As an example, the stability of a beam in plane bending is analyzed.

The stability of equilibrium solutions of a deterministic linear system of delay differential equationscan be investigated by studying the characteristic equation. For stochastic delay differential equations stability analysis is usually based on Lyapunov functional or Razumikhin type results, or Linear Matrix Inequality techniques. In [7] the authors proposed a technique based onthe vectorisation of matrices and the Kronecker product to transform the mean-square stability problem of a system of linear stochastic differential equations into a stability problem for a system of deterministic linear differential equations. In this paper we extend this method to the case of stochastic delay differential equations, providingsufficient and necessary conditions for the stability of the equilibrium. We apply our results to a neuron model perturbed by multiplicative noise. We study the stochastic stability properties of the equilibrium of this system andthen compare them with the same equilibrium in the deterministic case. Finally the theoretical results are illustrated by numerical simulations.

The well-known class of Nicholson's blowflies equations is considered under stochastic perturbations of the white noise type. We are concerned about the stability of the zero solution which means the extinction of the species of Nicholson's blowflies, and the positive equilibrium which means their persistence. Using appropriate Lyapunov functionals, sufficient conditions of stochastic stability, uniform stability and stochastic global exponential mean-square stability are derived. Moreover, we develop a new way of constructing a delayed-deterministic system by Lyapunov functional that leads to the extinction in the sense of the mean-square. Areas of stability with some numerical simulations are given to illustrate our results.

One stochastic nonlinear dynamical model has been proposed to describe the vibration of flexible beam under axial excitation considering the influence of the environment random factors. Firstly, the model has been simplified applying the stochastic average theory of quasi-integral Hamilton system .Secondly, we utilize the methods of Lyapunov exponent and boundary classification associated with diffusion process respectively to analyze the stochastic stability of the trivial solution of system. Thirdly, we explore the stochastic Hopf bifurcation of the vibration model according to the qualitative changes in stationary probability density of system response. It is concluded that the stochastic Hopf bifurcation occurs at two critical parametric values. Finally, some explanations are given in a simply way on the potential applications of stochastic stability and bifurcation analysis.