Stability of stochastic impulsive differential equations: integrating the cyber and the physical of stochastic systems

According to Newton's second law of motion, we humans describe a dynamical system with a differential equation, which is naturally discretized into a difference equation whenever a computer is used. The differential equation is the physical model in human brains and the difference equation the cyber model in computers for the dynamical system. The physical model refers to the dynamical system itself (particularly, a human-designed system) in the physical world and the cyber model symbolises it in the cyber counterpart. This paper formulates a hybrid model with impulsive differential equations for the dynamical system, which integrates its physical model in real world/human brains and its cyber counterpart in computers. The presented results establish a theoretic foundation for the scientific study of control and communication in the animal/human and the machine (Norbert Wiener) in the era of rise of the machines as well as a systems science for cyber-physical systems (CPS)

Nonlinear analysis of dynamical complex networks

Copyright © 2013 Zidong Wang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.Complex networks are composed of a large number of highly interconnected dynamical units and therefore exhibit very complicated dynamics. Examples of such complex networks include the Internet, that is, a network of routers or domains, the World Wide Web (WWW), that is, a network of websites, the brain, that is, a network of neurons, and an organization, that is, a network of people. Since the introduction of the small-world network principle, a great deal of research has been focused on the dependence of the asymptotic behavior of interconnected oscillatory agents on the structural properties of complex networks. It has been found out that the general structure of the interaction network may play a crucial role in the emergence of synchronization phenomena in various fields such as physics, technology, and the life sciences

Existence and exponential stability for neutral stochastic integro-differential equations with impulses driven by a Rosenblatt process

The existence and uniqueness of mild solution of an impulsive stochastic system driven by a Rosenblatt process is analyzed in this work by using the Banach fixed point theorem and the theory of resolvent operator developed by R. Grimmer in R. Grimmer, Resolvent operators for integral equations in a Banach space, Trans. Amer. Math. Soc., 273 (1982), 333–349. Furthermore, the exponential stability in mean square for the mild solution to neutral stochastic integro-differential equations with Rosenblatt process is obtained by establishing an integral inequality. Finally, an example is exhibited to illustrate the abstract theory

Existence of solutions and stability for impulsive neutral stochastic functional differential equations

In this paper we prove some results on the existence of solutions and the mean square asymptotic stability for a class of impulsive neutral stochastic differential systems with variable delays by using a contraction mapping principle. Namely, a sufficient condition ensuring the asymptotic stability is proved. The assumptions do not impose any restrictions neither on boundedness nor on the differentiability of the delay functions. In particular, the results improve some previous ones in the literature. Finally, an example is exhibited to illustrate the effectiveness of the results.Ministerio de Economía y Competitividad (MINECO). EspañaEuropean Commission (EC). Fondo Europeo de Desarrollo Regional (FEDER)Junta de AndalucíaEuropean Mathematical Societ

Asymptotic properties of stochastic population dynamics

In this paper we stochastically perturb the classical Lotka{Volterra model x_ (t) = diag(x1(t); ; xn(t))[b + Ax(t)] into the stochastic dierential equation dx(t) = diag(x1(t); ; xn(t))[(b + Ax(t))dt + dw(t)]: The main aim is to study the asymptotic properties of the solution. It is known (see e.g. [3, 20]) if the noise is too large then the population may become extinct with probability one. Our main aim here is to nd out what happens if the noise is relatively small. In this paper we will establish some new asymptotic properties for the moments as well as for the sample paths of the solution. In particular, we will discuss the limit of the average in time of the sample paths

Existence and Exponential Stability of Solutions for Stochastic Cellular Neural Networks with Piecewise Constant Argument

By using the concept of differential equations with piecewise constant argument of generalized type, a model of stochastic cellular neural networks with piecewise constant argument is developed. Sufficient conditions are obtained for the existence and uniqueness of the equilibrium point for the addressed neural networks. pth moment exponential stability is investigated by means of Lyapunov functional, stochastic analysis, and inequality technique. The results in this paper improve and generalize some of the previous ones. An example with numerical simulations is given to illustrate our results