169 research outputs found
Exponential stability of the Euler-Maruyama method for neutral stochastic functional differential equations with jumps
The exponential stability of trivial solution and the numerical solution for neutral stochastic functional differential equations (NSFDEs) with jumps is considered. The stability includes the almost sure exponential stability and the mean-square exponential stability. New conditions for jumps are proposed by means of the Borel measurable function to ensure stability. It is shown that if the drift coefficient satisfies the linear growth condition, the Euler-Maruyama method can reproduce the corresponding exponential stability of the trivial solution. A numerical example is constructed to illustrate our theory
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
Stability of the numerical scheme for stochastic McKean-Vlasov equations
This paper studies the infinite-time stability of the numerical scheme for
stochastic McKean-Vlasov equations (SMVEs) via stochastic particle method. The
long-time propagation of chaos in mean-square sense is obtained, with which the
almost sure propagation in infinite horizon is proved by exploiting the
Chebyshev inequality and the Borel-Cantelli lemma. Then the mean-square and
almost sure exponential stabilities of the Euler-Maruyama scheme associated
with the corresponding interacting particle system are shown through an
ingenious manipulation of empirical measure. Combining the assertions enables
the numerical solutions to reproduce the stabilities of the original SMVEs. The
examples are demonstrated to reveal the importance of this study
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