84 research outputs found
Stability of Equilibrium Points of Fractional Difference Equations with Stochastic Perturbations
Get PDFIt is supposed that the fractional difference equation xn+1=(μ+∑j=0kajxn−j)/(λ+∑j=0kbjxn−j), n=0,1,…, has an equilibrium point x^ and is exposed to additive stochastic perturbations type of Ã(xn−x^)ξn+1 that are directly proportional to the deviation of the system state xn from the equilibrium point x^. It is shown that known results in the theory of stability of stochastic difference equations that were obtained via V. Kolmanovskii and L. Shaikhet general method of Lyapunov functionals construction can be successfully used for getting of sufficient conditions for stability in probability of equilibrium points of the considered stochastic fractional difference equation. Numerous graphical illustrations of stability regions and trajectories of solutions are plotted
Mean Square Summability of Solution of Stochastic Difference Second-Kind Volterra Equation with Small Nonlinearity
Get PDF Stochastic difference second-kind Volterra equation with continuous time and small nonlinearity is considered. Via the general method of Lyapunov functionals construction, sufficient conditions for uniform mean square summability of solution of the considered equation are obtained.</p
Analysing Social Epidemics by Delayed Stochastic Models
Get PDFWe investigate the dynamics of a delayed stochastic mathematical model to understand the evolution of the alcohol consumption in Spain. Sufficient condition for stability in probability of the equilibrium point of the dynamic model with aftereffect and stochastic perturbations is obtained via Kolmanovskii and Shaikhet general method of Lyapunov functionals construction. We conclude that alcohol consumption in Spain will be constant (with stability) in time with around 36.47% of nonconsumers, 62.94% of nonrisk consumers, and 0.59% of risk consumers. This approach allows us to emphasize the possibilities of the dynamical models in order to study human behaviour
Stability of the Positive Point of Equilibrium of Nicholson's Blowflies Equation with Stochastic Perturbations: Numerical Analysis
Get PDFKnown Nicholson's blowflies equation
(which is one of the most important models in
ecology) with stochastic perturbations is considered. Stability of the positive (nontrivial)
point of equilibrium of this equation and also a capability of its discrete analogue to
preserve stability properties of the original differential equation are studied. For this purpose,
the considered equation is centered around the positive equilibrium and linearized.
Asymptotic mean square stability of the linear part of the considered equation is used to
verify stability in probability of nonlinear origin equation. From known previous results
connected with B. Kolmanovskii and L. Shaikhet, general method of Lyapunov functionals
construction, necessary and sufficient condition of stability in the mean square sense in
the continuous case and necessary and sufficient conditions for the discrete
case are deduced. Stability conditions for the discrete analogue allow to determinate an admissible step of discretization for numerical simulation of solution trajectories. The trajectories of stable and unstable solutions of considered equations are simulated numerically
in the deterministic and the stochastic cases for different values of the parameters and of the
initial data. Numerous graphical illustrations of stability regions and solution trajectories are plotted
Two unsolved problems in the stability theory of stochastic differential equations with delay
No full textAbstractTwo unsolved problems of the stability theory for stochastic differential equations with delay are offered for consideration
About Some Unsolved Problems in the Stability Theory of Stochastic Differential and Difference Equations
No full textThis paper continues a series of papers by the author devoted to unsolved problems in the theory of stability and optimal control for stochastic systems. A delay differential equation with stochastic perturbations of the white noise and Poisson’s jump types is considered. In contrast with the known stability condition, in which it is assumed that stochastic perturbations fade on the infinity quickly enough, a new situation is studied, in which stochastic perturbations can either fade on the infinity slowly or not fade at all. Some unsolved problem in this connection is brought to readers’ attention. Additionally, some unsolved problems of stabilization for one stochastic delay differential equation and one stochastic difference equation are also proposed
Stochastic Volterra integro-differential equations: stability and numerical methods
No full textWe consider the reliability of some numerical methods in preserving the stability properties of the linear stochastic functional differential equation � t ˙x(t) = αx(t) + β x(s)ds + σx(t − τ) ˙ W(t), 0 where α, β, σ, τ ≥ 0 are real constants, and W(t) is a standard Wiener process. We adopt the shorthand notation of ˙x(t) to represent the differential dx(t) etc. Our choice of test equation is a stochastic perturbation of the classical deterministic Brunner & Lambert test equation for σ = 0 and so our investi-gation may be thought of as an extension of their work. Sufficient conditions for the asymptotic mean square stability of solutions to both the differential equation and discrete analogues are derived using the general method of Lyapunov functionals construction proposed by Kol-manovskii & Shaikhet which has previously been successfully employed for deterministic and stochastic differential and difference equations with delay. The areas of the regions of asymptotic stability for each θ method, in-dicated by the sufficient conditions for the discrete system, are shown to be equal and we show that an upper bound can be put on the time-step parameter for the numerical method fo which the system is asymptotically mean-square stable. We illustrate our results by means of numerical experiments and various stability diagrams. We examine the extent to which the continuous system can tolerate stochastic perturbations before losing its stability properties and we illustrate how one may accurately choose a numerical method to preserve the stability properties of the original problem in the numerical solution. Our numerical experiments also indicate that the quality of the sufficient conditions is very high
Lyapunov functionals and stability of stochastic functional differential equations
No full textStability conditions for functional differential equations can be obtained using Lyapunov functionals. Lyapunov Functionals and Stability of Stochastic Functional Differential Equations describes the general method of construction of Lyapunov functionals to investigate the stability of differential equations with delays. This work continues and complements the author’s previous book Lyapunov Functionals and Stability of Stochastic Difference Equations, where this method is described for discrete- and continuous-time difference equations. The text begins with a description of the peculiarities of deterministic and stochastic functional differential equations. There follow basic definitions for stability theory of stochastic hereditary systems, and a formal procedure of Lyapunov functionals construction is presented. Stability investigation is conducted for stochastic linear and nonlinear differential equations with constant and distributed delays. The proposed method is used for stability investigation of different mathematical models such as: • inverted controlled pendulum; • Nicholson's blowflies equation; • predator-prey relationships; • epidemic development; and • mathematical models that describe human behaviours related to addictions and obesity. Lyapunov Functionals and Stability of Stochastic Functional Differential Equations is primarily addressed to experts in stability theory but will also be of interest to professionals and students in pure and computational mathematics, physics, engineering, medicine, and biology
Stability of Positive Equilibrium Point of Nonlinear System of Type of Predator–Prey with Aftereffect and Stochastic Perturbations
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