46 research outputs found
Application of the general methods of Lyapunov functionals construction for Volterra difference equations
No full textAbstract: Lyapunov functionals are used usually for stability investigation of systems with aftereffect. The general method of Lyapunov functionals construction which was proposed and developed by Kolmanovskii and Shaikhet is used here for stochastic second type Volterra difference equations. It is shown
that using this method, there is a possibility to construct for a given equation, a sequence of extending stability regions.
Keywords: Difference equations, Method of Lyapunov functionals construction, Asymptotic stability.
1. STATEMEN
Stability of Equilibrium Points of Fractional Difference Equations with Stochastic Perturbations
No full textIt 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
About stability of nonlinear stochastic difference equations
No full textUsing the method of Lyapunov functionals construction, it is shown that investigation of stability in probability of nonlinear stochastic difference equation with order of nonlinearity more than one can be reduced to the investigation of asymptotic mean square stability of the linear part of this equation
About integrability of solutions of stochastic difference Volterra equations
No full text16th IMACS World Congress 2000 On Scientific Computation, Applied Mathematics a
Mean Square Summability of Solution of Stochastic Difference Second-Kind Volterra Equation with Small Nonlinearity
No full text 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
Stability of the Positive Point of Equilibrium of Nicholson's Blowflies Equation with Stochastic Perturbations: Numerical Analysis
No full textKnown 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
A nonlinear dynamic age-structured model of e-commerce in Spain: Stability analysis of the equilibrium by delay and stochastic perturbations
No full text[EN] First, we propose a deterministic age-structured epidemiological model to study the diffusion of e-commerce in Spain. Afterwards, we determine the parameters (death, birth and growth rates) of the underlying demographic model as well as the parameters (transmission of the use of e-commerce rates) of the proposed epidemiological model that best fit real data retrieved from the Spanish National Statistical Institute. Motivated by the two following facts: first the dynamics of acquiring the use of a new technology as e-commerce is mainly driven by the feedback after interacting with our peers (family, friends, mates, mass media, etc.), hence having a certain delay, and second the inherent uncertainty of sampled real data and the social complexity of the phenomena under analysis, we introduce aftereffect and stochastic perturbations in the initial deterministic model. This leads to a delayed stochastic model for e-commerce. We then investigate sufficient conditions in order to guarantee the stability in probability of the equilibrium point of the dynamic e-commerce delayed stochastic model. Our theoretical findings are numerically illustrated using real data. (C) 2018 Elsevier B.V. All rights reserved.This work has been partially supported by the Ministerio de Economia y Competitividad grant MTM2017-89664-P.Burgos-Simon, C.; Cortés, J.; Shaikhet, L.; Villanueva Micó, RJ. (2018). A nonlinear dynamic age-structured model of e-commerce in Spain: Stability analysis of the equilibrium by delay and stochastic perturbations. Communications in Nonlinear Science and Numerical Simulation. 64:149-158. https://doi.org/10.1016/j.cnsns.2018.04.022S1491586
Stability of a stochastically perturbed model of intracellular single-stranded RNA virus replication
No full textCompared to the replication of double-stranded RNA and DNA viruses, the replication of single-stranded viruses requires the production of a number of intermediate strands that serve as templates for the synthesis of genomic-sense strands. Two theoretical extreme mechanisms for replication for such single-stranded viruses have been proposed; one extreme being represented by the so-called linear stamping machine and the opposite extreme by the exponential growth. Of course, real systems are more complex and examples have been described in which a combination of such extreme mechanisms can also occur: a fraction of the produced progeny resulting from a stamping-machine type of replication that uses the parental genome as template, whereas other fraction of the progeny results from the replication of other progeny genomes. Martínez et al., Sardanyés et al. and Fornés et al. suggested and analyzed a deterministic model of single-stranded RNA (ssRNA) virus intracellular replication that incorporated variability in the replication mechanisms. To explore how stochasticity can affect this mixed-model principal properties, in this paper, we consider the stability of a stochastically perturbed model of ssRNA virus replication within a cell. Using the direct Lyapunov method, we found sufficient conditions for the stability in probability of equilibrium states for this model. This result confirms that this heterogeneous model of single-stranded RNA virus replication is stable with respect to stochastic perturbations of the environment.Leonid Shaikhet is supported by the Israel Science Foundation via grant No. 1128/14 and the Israeli Ministry of Absorption. Santiago F. Elena is supported by Spain's Ministerio de Economía,
Industria y Competitividad grant BFU2015-65037-P and by Generalitat Valenciana grant PROMETEOII/2014/021. Andrei Korobeinikov is supported by the Spain's Ministerio de Economía,
Industria y Competitividad grant MTM2015-71509-C2-1-R.Peer reviewe
