Dynamical Complexity Research Of Artificial Neneural Network Systems

As a new research field developed in these recent years, the chaotic neural network is one of the most rapid developed, the most popular and the most attractive subjects in the nonlinear science. The complex dynamics of the chaotic neural network is different from normal neural networks which have the character of gradient descent. The chaotic neural network has plenty of dynamical behavior and different kinds of attractors, so a great deal of attention has been paid to this area. The nonlinear dynamics of neural network mainly make use of the dynamical system theory to analyze the evolution of the system and the character of the attractor, explore the cooperative behaviors of the nerve cells, and study the mechanism of information processing.As we know, chaos encryption and chaos communication is the most attractive field in the research about chaos. Meanwhile, nonlinear circuit is very helpful to simulate the real chaotic system for studying chaos encryption and chaos communication. We find that chaotic signal generators based on chaotic neural network have the great character of global stability, which overcome the shortcoming of the famous Chua's circuit. So it has very important academic and practical significance to find more chaotic neural network models which have different chaotic attractors.Chaos verification and chaos synchronization has become the focus of the research of the chaotic systems. However, the theory of dynamical complexity is profound to understand, which makes these two issues become very difficult problems. Especially, chaos verification needs not only the theory of dynamical system but also a great deal of numerical calculation, computer simulation and algorithm analysis. As for chaos synchronization, people have paid a great deal of attention on it for its important application in chaotic encryption and chaotic communication. Nevertheless now the research about chaos synchronization still base on the traditional control theory.This paper studies the dynamical complexity in continuous artificial neural network, by computer simulation, Lyapunov exponent calculation, computer-assisted verification and so on. Rich dynamical behaviors are found, such as chaos, limit cycle, equilibrium point. In this paper, chaos verifications of several chaotic systems are given by the combination of topological horseshoe theory and computer-assisted verification. Using Poincare section and Poincare map, this paper gives an effective verification for the existence the existence of attracting sets in chaotic system. The research work carried out includes the following: (1) Prove chaotic character on the Hopfield neural network model, cellular neural network model and a simple artificial neural network model strictly. The method of computer-assisted verification makes use of topological horseshoe theory developed by the Smale horseshoe of dynamical system combined with computer assisted. In these cases, we have shown this method works well and make chaos verification easily and reliability.(2) Take the famous Hopfield neural network for instance to study the dynamical complexity of the low-dimensional continuous neural network according to topology structure of the models. We are concerned with this interesting problem in dynamics of neural networks that what connection topology will prohibit chaotic behavior in continuous time neural network and to what extent a continuous time neural network described by continuous ordinary differential equations is simple enough while can still exhibit chaos. This paper shows that the existence of directed loop in connection topology is necessary for chaos to occur.(3) Present that chaos synchronization can take place in the coupled non-chaotic neural network systems. This is demonstrated by coupling two non-chaotic cellular neural networks. The couplings give rise to a synchronous chaotic dynamics and in the meanwhile the synchronous dynamics is globally asymptotically stable, thus chaos synchronization takes place under the suitable couplings. And it is interesting to show that the cellular neural network is topologically conjugate to an unfolded Chua's circuit by virtue of "The Global unfolding Theorem".(4) The two non-neural network system models which play the important roles in chaos development are reviewed:smooth Chua system and NSG system. We present the rigorous verification of chaos by giving the computer-assisted verification based on horseshoe theory of dynamical system, which has been numerically observed before.