16,219 research outputs found
Mathematical problems for complex networks
Copyright @ 2012 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. This article is made available through the Brunel Open Access Publishing Fund.Complex networks do exist in our lives. The brain is a neural network. The global economy
is a network of national economies. Computer viruses routinely spread through the Internet. Food-webs, ecosystems, and metabolic pathways can be represented by networks. Energy is distributed through transportation networks in living organisms, man-made infrastructures, and other physical systems. Dynamic behaviors of complex networks, such as stability, periodic oscillation, bifurcation, or even chaos, are ubiquitous in the real world and often reconfigurable. Networks have been studied in the context of dynamical systems in a range of disciplines. However, until recently there has been relatively little work that treats dynamics as a function of network structure, where the states of both the nodes and the edges can change, and the topology of the network itself often evolves in time. Some major problems have not been fully investigated, such as the behavior of stability, synchronization and chaos control for complex networks, as well as their applications in, for example, communication and bioinformatics
Spiking Optical Patterns and Synchronization
We analyze the time resolved spike statistics of a solitary and two mutually
interacting chaotic semiconductor lasers whose chaos is characterized by
apparently random, short intensity spikes. Repulsion between two successive
spikes is observed, resulting in a refractory period which is largest at laser
threshold. For time intervals between spikes greater than the refractory
period, the distribution of the intervals follows a Poisson distribution. The
spiking pattern is highly periodic over time windows corresponding to the
optical length of the external cavity, with a slow change of the spiking
pattern as time increases. When zero-lag synchronization between the two lasers
is established, the statistics of the nearly perfectly matched spikes are not
altered. The similarity of these features to those found in complex interacting
neural networks, suggests the use of laser systems as simpler physical models
for neural networks
Complex and Adaptive Dynamical Systems: A Primer
An thorough introduction is given at an introductory level to the field of
quantitative complex system science, with special emphasis on emergence in
dynamical systems based on network topologies. Subjects treated include graph
theory and small-world networks, a generic introduction to the concepts of
dynamical system theory, random Boolean networks, cellular automata and
self-organized criticality, the statistical modeling of Darwinian evolution,
synchronization phenomena and an introduction to the theory of cognitive
systems.
It inludes chapter on Graph Theory and Small-World Networks, Chaos,
Bifurcations and Diffusion, Complexity and Information Theory, Random Boolean
Networks, Cellular Automata and Self-Organized Criticality, Darwinian
evolution, Hypercycles and Game Theory, Synchronization Phenomena and Elements
of Cognitive System Theory.Comment: unformatted version of the textbook; published in Springer,
Complexity Series (2008, second edition 2010
Relay synchronization in multiplex networks of discrete maps
Complex multiplex networks consist of several subnetwork layers, which
interact via pairwise inter-layer connections. Relay synchronization between
distant layers which are not directly connected, but only via a relay layer,
can be observed in multiplex networks. We study three-layer networks of
discrete logistic maps, where each individual layer is a nonlocally coupled
ring, and demonstrate scenarios of relay synchronization of complex patterns in
the outer layers which interact via an intermediate layer. We find regimes of
relay synchronization for chimera states, i.e., patterns of coexisting coherent
and incoherent domains, and a transition from phase chimeras to amplitude
chimeras for increasing inter-layer coupling. We determine analytically the
approximate critical coupling strengths for the existence of phase chimeras
Pinning Complex Networks by a Single Controller
In this paper, without assuming symmetry, irreducibility, or linearity of the
couplings, we prove that a single controller can pin a coupled complex network
to a homogenous solution. Sufficient conditions are presented to guarantee the
convergence of the pinning process locally and globally. An effective approach
to adapt the coupling strength is proposed. Several numerical simulations are
given to verify our theoretical analysis
Complex partial synchronization patterns in networks of delay-coupled neurons
We study the spatio-temporal dynamics of a multiplex network of delay-coupled FitzHugh–Nagumo oscillators with non-local and fractal connectivities. Apart from chimera states, a new regime of coexistence of slow and fast oscillations is found. An analytical explanation for the emergence of such coexisting partial synchronization patterns is given. Furthermore, we propose a control scheme for the number of fast and slow neurons in each layer.DFG, 163436311, SFB 910: Kontrolle selbstorganisierender nichtlinearer Systeme: Theoretische Methoden und Anwendungskonzept
Synchronization in an array of linearly stochastically coupled networks with time delays
This is the post print version of the article. The official published version can be obtained from the link - Copyright 2007 Elsevier LtdIn this paper, the complete synchronization problem is investigated in an array of linearly stochastically coupled identical networks with time delays. The stochastic coupling term, which can reflect a more realistic dynamical behavior of coupled systems in practice, is introduced to model a coupled system, and the influence from the stochastic noises on the array of coupled delayed neural networks is studied thoroughly. Based on a simple adaptive feedback control scheme and some stochastic analysis techniques, several sufficient conditions are developed to guarantee the synchronization in an array of linearly stochastically coupled neural networks with time delays. Finally, an illustrate example with numerical simulations is exploited to show the effectiveness of the theoretical results.This work was jointly supported by the National Natural Science Foundation of China under Grant 60574043, the Royal Society of the United Kingdom, the Natural Science Foundation of Jiangsu Province of China under Grant BK2006093, and International Joint Project funded by NSFC and the Royal Society of the United Kingdom
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