Neural networks (NNs) have been useful in many fields, such as pattern recognition, image processing etc. Recently, synchronization of chaotic neural networks (CNNs) has drawn increasing attention due to the high security of neural networks. In this chapter, the problem of synchronization and parameter identification for a class of chaotic neural networks with stochastic perturbation via state and output coupling, which involve both the discrete and distributed time-varying delays has been investigated. Using adaptive feedback techniques, several sufficient conditions have been derived to ensure the synchronization of stochastic chaotic neural networks. Moreover, all the connection weight matrices can be estimated while the lag synchronization and complete synchronization is achieved in mean square at the same time. The corresponding simulation results are given to show the effectiveness of the proposed method.
TopIntroduction
It is widely believed that chaos synchronization has played a more and more significant role in nonlinear science (Chen & Dong, 1998; Pecora & Carroll, 1990; Tang & Fang, 2008; Tang et al, 2008; Tang et al, 2009a; Tang et al, 2009b; Tang et al, 2009c; Ojalvo & Roy, 2001; Yang & Chua, 1997; Boccaletti et al, 2002; Shahverdiev et al, 2002). Since Pecora and Carroll (Pecora & Carroll, 1990) synchronized two identical systems with different initial conditions, chaos synchronization has drawn much attention due to its potential applications in many fields, such as secure communication, pattern recognition, biological systems, and so on. So far, a wide variety of synchronization phenomena have been discovered such as complete synchronization, lag synchronization (Shahverdiev et al, 2002). Lag synchronization (LS) occurs as a coincidence of shifted-in-time states of two systems
, where δ is a propagation delay. It is worth mentioning that, in many practical situations, a propagation delay will appear in the electronic implementation of dynamical systems. Therefore, it is very important and necessary to investigate the lag synchronization from the view of applications.
The past two decades have witnessed significant progress on the study of dynamical characteristics of neural networks because of their wide applications in many areas. There exist many works which are devoted to achieve the synchronization problems of neural networks. With respect to some recent representative works on this topic, we refer the reader to (Tang et al, 2009a; Tang et al, 2008; Tang et al, 2009b; Lu & Cao, 2007; Yu & Cao, 2007; Cao et al, 2006; He et al, 2008; Lou & Cui, 2008; Lu & Chen, 2004; Li & Chen, 2006; Li et al, 2007; Chen et al, 2004; Cheng et al, 2005) and references therein.
On the other hand, in the past few years, there has been an increasing interest in the research of neural networks with stochastic perturbations in the neural network community (Wang et al, 2007; Wang et al, 2006; Wan & Sun, 2005; Sun & Cao, 2007). It has been shown that, in real nervous systems, the transmission is a noisy process brought on by random fluctuations from the release of neurotransmitters and other probabilistic causes. Therefore, the synchronization and stability analysis problem for stochastic neural networks has been an important research issue. Recently, there have been some initial studies on the synchronization of neural networks (Yu & Cao, 2007; Sun & Cao, 2007). Very recently, in Ref. (Sun & Cao, 2007), the adaptive synchronization scheme has been developed to synchronize the stochastic neural networks with constant delay.