Research On Dynamical Behaviors Of Several Classes Of Complex-Valued Neural Networks

Complex-valued neural networks(CVNNs)are the networks that deal with information in the complex plane,that is,their state vectors,connection weights and activation functions are complex-valued.Complex-valued neural networks are regarded as the extension of real-valued neural networks(RVNNs),but they are quite different from real-valued neural networks and have more complicated properties than RVNNs.Complex-valued neural networks have been used to solve complex-valued as well as real-valued problems in real life,also complex-valued neural networks have shown more powerful capability and advantage than realvalued neural networks.Complex-valued neural networks have the potential to solve some problems that cannot be solved with their real-valued counterparts,such as the exclusion XOR problem.So they have become increasingly popular and been paid more and more attention in the neural network community in recent years.Complex-valued neural networks have a diverse variety of applications in mathematics,physics,engineering,and other areas,a lot of problems have been solved successfully,such as function approximation,classification problems,image processing,speech recognition,signal processing,pattern recognition,secure communication,artificial neural information processing and so on.As we all know,the applications of neural networks depend on the study of its dynamical behaviors,so it is very important to analyze the dynamical behaviors of neural networks.Because of great potential and application prospect of complex-valued neural networks,recently,more and more researchers focus on their attentions and interests to CVNNs,dynamical behaviors of complex-valued neural networks are investigated deeply and have great progress,and there have been some important theories and good results.The activation function plays an important role in the dynamical behaviors of recurrent neural networks.The choice of activation function of CVNNs is quite different from real-valued neural networks.In RVNNs,their activation function is usually chosen to be a smooth,bounded and nonconstant function.However,in complex domain,according to the Liouville's theorem,every bounded analytic function will reduce to a constant.Therefore,the choice of activation function of CVNNs is an important challenge.Be similar to real-valued neural networks,complex-valued neural networks have lots of dynamical behaviors,such as stability(including asymptotic stability,exponential stability,multistability,-stability,robust stability etc.),periodicity,bifurcation,synchronization,dissipation and passivity and so on.However,the present research is far from enough,thus it is very important and worthwhile to investigate CVNNs deeply in order to explore new capabilities and higher performance both theoretically and practically.This is also the main purpose and motivation of this thesis.The main aim of this thesis is to study several dynamical behaviors of delayed complex-valued neural networks,including existence,uniqueness and global exponential stability of periodic solutions of CVNNs,synchronization,finite-time synchronization and finite-time lag synchronization of CVNNs.The main research method include Mawhin's continuation theorem of coincidence degree theory,matrix measure method,Halanay inequality,Lyapunov stability theory,finite-time stability theory and so on.The thesis is divided into five chapters.In the first chapter,we summarize the research background and significance of complex-valued neural networks,describe the current research status and development trend of complex-valued neural networks and the motivation of this thesis.And then we give a summary outlining of the main results and the innovations of the thesis.In the chapter 2,the preliminary knowledge about complex-valued neural networks are reviewed.Firstly,we introduce basic mathematical notations,which will be used throughout the thesis.Then,we introduce some basic theory of CVNNs,including several CVNNs models,the activation functions,which play an important role in CVNNs,and some basic definitions and related theorems of CVNNs,which will be used in the following chapters.In the chapter 3,a class of delayed complex-valued neural networks with impulses is investigated.By using Mawhin's continuation theorem of coincidence degree theory,a series of useful criteria on existence of periodic solution are established for the complex-valued neural networks.By constructing appropriate Lyapunov-Krasovskii functional,some new sufficient conditions are derived for the global exponential stability of periodic solutions to the complex-valued neural networks.In the chapter 4,global exponential synchronization of a class of complexvalued neural networks with time delays is investigated.Based on Halanay inequality theory,Lyapunov theory and matrix measure method,by separating complexvalued neural networks into the real part and imaginary parts,several new criteria for the global exponentially synchronization of drive-response complex-valued neural networks are presented.In the chapter 5,the problems of finite-time synchronization of two classes of complex-valued neural networks are considered.In the section 1,the problem of finite-time lag synchronization of complex-valued neural networks with time delay is investigated.By means of the Lyapunov function method and finite-time stability theory,some new sufficient conditions are derived to guarantee the finite-time lag synchronization between two delayed complex-valued neural networks by designing a feedback controller.In the section 2,the issue of finite-time synchronization of complex-valued neural networks with time delay is investigated.We don't use the method separating the real and imaginary parts,but treat the complex vector as a whole,based on the finite-time stability theory and by constructing appropriate Lyapunov function,some new sufficient conditions are derived to guarantee the finite-time synchronization between two delayed complex-valued neural networks by designing an appropriate controller.In each chapter of this thesis,some examples with numerical simulations are given to illustrate the correctness and effectiveness of our theoretical results via standard numerical software.These researches not only enrich and develop some basic theory of complex-valued neural networks,but also provide theoretical basis to solve many practical problems in science and technology.