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

Quasi-synchronization of delayed coupled networks with non-identical discontinuous nodes

This paper is concerned with the quasi-synchronization issue of linearly coupled networks with discontinuous nonlinear functions in each isolated node. Under the framework of Filippov systems, the existence and boundedness of solutions for such complex networks can be guaranteed by the matrix measure approach. A design method is presented for the synchronization controllers of coupled networks with non-identical discontinuous systems. Numerical simulations on the coupled chaotic systems are given to demonstrate the effectiveness of the theoretical results

Generalized non-autonomous Cohen-Grossberg neural network model

In the present paper, we investigate both the global exponential stability and the existence of a periodic solution of a general differential equation with unbounded distributed delays. The main stability criterion depends on the dominance of the non-delay terms over the delay terms. The criterion for the existence of a periodic solution is obtained with the application of the coincide degree theorem. We use the main results to get criteria for the existence and global exponential stability of periodic solutions of a generalized higher-order periodic Cohen-Grossberg neural network model with discrete-time varying delays and infinite distributed delays. Additionally, we provide a comparison with the results in the literature and a numerical simulation to illustrate the effectiveness of some of our results.Comment: 30 page

Synchronization of Discontinuous Neural Networks with Delays via Adaptive Control

The drive-response synchronization of delayed neural networks with discontinuous activation functions is investigated via adaptive control. The synchronization of this paper means that the synchronization error approaches to zero for almost all time as time goes to infinity. The discontinuous activation functions are assumed to be monotone increasing which can be unbounded. Due to the mild condition on the discontinuous activations, adaptive control technique is utilized to control the response system. Under the framework of Filippov solution, by using Lyapunov function and chain rule of differential inclusion, rigorous proofs are given to show that adaptive control can realize complete synchronization of the considered model. The results of this paper are also applicable to continuous neural networks, since continuous function is a special case of discontinuous function. Numerical simulations verify the effectiveness of the theoretical results. Moreover, when there are parameter mismatches between drive and response neural networks with discontinuous activations, numerical example is also presented to demonstrate the complete synchronization by using discontinuous adaptive control

Almost Periodic Dynamics for Memristor-Based Shunting Inhibitory Cellular Neural Networks with Leakage Delays

We investigate a class of memristor-based shunting inhibitory cellular neural networks with leakage delays. By applying a new Lyapunov function method, we prove that the neural network which has a unique almost periodic solution is globally exponentially stable. Moreover, the theoretical findings of this paper on the almost periodic solution are applied to prove the existence and stability of periodic solution for memristor-based shunting inhibitory cellular neural networks with leakage delays and periodic coefficients. An example is given to illustrate the effectiveness of the theoretical results. The results obtained in this paper are completely new and complement the previously known studies of Wu (2011) and Chen and Cao (2002)

Further analysis of stability of uncertain neural networks with multiple time delays

This paper studies the robust stability of uncertain neural networks with multiple time delays with respect to the class of nondecreasing activation functions. By using the Lyapunov functional and homeomorphism mapping theorems, we derive a new delay-independent sufficient condition the existence, uniqueness, and global asymptotic stability of the equilibrium point for delayed neural networks with uncertain network parameters. The condition obtained for the robust stability establishes a matrix-norm relationship between the network parameters of the neural system, and therefore it can easily be verified. We also present some constructive numerical examples to compare the proposed result with results in the previously published corresponding literature. These comparative examples show that our new condition can be considered as an alternative result to the previous corresponding literature results as it defines a new set of network parameters ensuring the robust stability of delayed neural networks.Publisher's Versio

Existence and Global Uniform Asymptotic Stability of Pseudo Almost Periodic Solutions for Cohen-Grossberg Neural Networks with Discrete and Distributed Delays

This paper studies the existence and uniform asymptotic stability of pseudo almost periodic solutions to Cohen-Grossberg neural networks (CGNNs) with discrete and distributed delays by applying Schauder fixed point theorem and constructing a suitable Lyapunov functional. An example is given to show the effectiveness of the main results

Synchronization problems for unidirectional feedback coupled nonlinear systems

In this paper we consider three different synchronization problems consisting in designing a nonlinear feedback unidirectional coupling term for two (possibly chaotic) dynamical systems in order to drive the trajectories of one of them, the slave system, to a reference trajectory or to a prescribed neighborhood of the reference trajectory of the second dynamical system: the master system. If the slave system is chaotic then synchronization can be viewed as the control of chaos; namely the coupling term allows to suppress the chaotic motion by driving the chaotic system to a prescribed reference trajectory. Assuming that the entire vector field representing the velocity of the state can be modified, three different methods to define the nonlinear feedback synchronizing controller are proposed: one for each of the treated problems. These methods are based on results from the small parameter perturbation theory of autonomous systems having a limit cycle, from nonsmooth analysis and from the singular perturbation theory respectively. Simulations to illustrate the effectiveness of the obtained results are also presented.Comment: To appear in Dyn. Contin. Discrete Impuls. Syst., Ser. A, Math. Ana