The paper deals with systems of linear differential equations with coefficients depending on the Markov process. Equations for particular density and the moment equations for given systems are derived and used in the investigation of solvability of initial problems and stability. Results are illustrated by examples. MSC: 34K50; 60H10; 60H30; 65C30
The paper deals with a system of difference equations, where coefficients depend on Markov chains. The functional equations for particular density and the moment equations for the system are derived and used in the investigation of solvability and stability. An application of the results is shown how to solve various economic problems. c 2013 Mathematical Institute, Slovak Academy of Sciences. 2010 M a t h e m a t i c s S u b j e c t C l a s s i f i c a t i o n: 34K50, 60H10, 60H30, 65C30. K e y w o r d s: stochastic systems, Markov chain, moment equations, stability.
Information technology has taken the world by storm. Its emergence has given rise to a new level of digital knowledge systems. Its application has been catalytic to the rapid changes taking place in the way people work, live and think, and is facilitating the development of our society and civilization in a new era. The current growing size and complexity of local computer networks also bring increasing demands for continuous monitoring of their proper performance, which is a prerequisite for their good efficiency, safety and reliability.The aim of this paper is to introduce a cyberattack detection model based on a system of delay nonlinear differential equations and find its equilibrium state. The proposed model is based on a system of non-linear differential equations with delay and allows us to obtain a qualitative portrait of dynamic systems using the general Poincaré-Lyapunov theory based on the knowledge of the existence of stationary points and cycles and their mutual disposition.
The paper deals with a system of nonlinear differential equations under the influence of white noise. This system can be used as a mathematical model of various real problems in finance, mathematical biology, climatology, signal theory and others. Necessary and sufficient conditions for the asymptotic mean square stability of the zero solution of this system are derived in the paper. The paper introduces a new approach to studying such problems -construction of a suitable deterministic system with the use of Lyapunov function. MSC: 34K50; 60H10; 60H30; 65C30Keywords: stochastic systems; white noise; mean square stability; Lyapunov function IntroductionWe can come across stochastic behavior while examining many important problems of a global character in various fields of research, for example, in the theory of climate change. Detailed understanding of extreme events in climate, of phenomena that are beyond our normal expectations, is a very important topic in climatology, meteorology and related fields. Common methods of studying extreme events, such as the statistical approach, the empirical-physical approach or the numerical modeling approach, have some limitations, and study of them has been largely empirical.The idea of replacing the whole deterministic system with a stochastic differential equation was introduced by Hasselmann in his work [] on stochastic climate models that appeared in . There he proposed to improve deterministic models for the 'climate' (slow variables) by incorporating the influence of the 'weather' (fast variables) in the form of random noise. The univariate linear systems that appear in the work have been successful in describing various modes of climate variability. Success of these models has inspired researchers to consider the stochastic forcing as a possible source of more complex dynamics, for example, in []. The direction of stochastic parametrizations in which the development of the climate models will be possible in the coming years is formulated, for example, in [].Hasselmann's works can be seen as the beginning of describing extreme events in climate by a stochastic system of differential equations in which random weather changes
We deal with the investigation of \(L_{2}\)-stability of the trivial solution to the system of difference equations with coefficients depending on a semi-Markov chain. In our considerations, random transformations of solutions are assumed. Necessary and sufficient conditions for \(L_{2}\)-stability of the trivial solution to such systems are obtained. A method is proposed for constructing Lyapunov functions and the conditions for its existence are justified. The dynamic system and methods discussed in the paper are very well suited for use as models for protecting information in cyberspace.
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