Characteristics And Chaos Control Of Neural Networks Using Globally Coupled Map

A neural network called globally coupled map (GCM) model is a special kind of chaotic neural network composed of chaotic neurons, whose dynamical behavior depend on the chaotic map. It has cluster frozen attractors (CFA) which can be taken to represent information, therefore it can be used to information processing and associative memory and so on. Motivated by creating and developing GCM model to be effectively applied to information processing, we investigate the dynamic behaviors of several different GCM models and apply the networks under certain control methods to implement dynamic associative memory.The contributions can be concluded as four main points listed as follows:1. The characteristics of several chaotic map models are investigated. First, discrete chaotic systems iterated by a newly presented cubic Logistic map, sinusoidal (cosine) map and a special nonlinear map are considered. Their chaotic dynamic behaviors are demonstrated by calculating Lyapunov exponents and showing bifurcation diagrams, and their parameter regions are also given. Second, their features are illustrated from the point view of distribution of probability density. It is observed that both ergodicity of distribution and the shape of probability density function vary according to different bifurcation parameters. At last, different characteristics of different chaotic maps are developed from the view of relativity, power spectrum and chi-square test of mean distribution and so on. All the analyses lay the foundations for the improved GCM models iterated by the chaotic maps above.2. The macro and micro characteristics of the networks are analyzed. First, it indicates that cluster frozen attractor is a common feature of different GCM and it doesn’t change according to the variance of information transfer function. Second, the macro features of different GCM with two newly presented ways of time-delay coupling are illustrated. It can be seen that the cluster frozen attractor doesn’t change with the introduction of time delay and the networks exhibit richer running behaviors. Finally, the micro features of the networks and the influence on the neurons’ behaviors by coupling are investigated from the view of iteration of neurons. It can be concluded that the chaotic dynamics of GCM networks originate from every neuron’s chaotic running behavior and the ways of coupling mainly influence clusters.3. The dynamics of GCM networks are demonstrated. First, the existence of equilibrium points is proved, which has nothing to do with the information transfer functions and is fit for all GCM models. Second, sufficient conditions on the asymptotical stability of zero equilibrium point for S-GCM and CL-GCM are given respectively. Furthermore, the stability of equilibrium points and bifurcation for one and two dimensional systems are analyzed. Numerical simulation results demonstrate the correctness and effectiveness of the analysis above.4. Two control methods and the applications of the GCM model are given. On the one hand, GCM models are controlled by a kind of feedback control method which sends deviations of the states back instead of introducing outer information to realize the control of fixed points and periods by adjusting parameters. It is indicated that the control method is fit for most of GCM models. On the other hand, a control method presented as an improvement of Ishii’s parameter modulated control method is given to control GCM models by adjusting parameter threshold. Unfortunately, it isn’t suitable for all GCM models. However, it is delighted to find that more excellent characteristics are exhibited by time-delay GCM model than by common GCM model. Finally, the systems’ associative memory is illustrated by the improved parameter modulated control method. It suggests that CL-GCM and SI-GCM can not only output fixed patterns but also output periodical patterns which contain correct pattern. So the associative memory is successful.