Global Attracting And Invariant Sets For Complex-valued Neural Networks With Time-varying Delays

The human brain is a complex system and it has the ability to organize neurons for nonlinear and parallel information processing.In the meanwhile,the human brain can process the information with a faster rate than the computer.In fact,it has been a dream for researchers in scientific and engineering applications to establish a machine or autonomous mechanism with human intelligence.Therefore,more than 50 years ago,artificial neural networks,often referred to as neural networks,have attracted much attention.Recently,neural networks once again arouse scholars' concern after AlphaGo beats the world's top Goers.The neural network is the most popular and useful one in machine learning algorithms,and it plays an important role in the development of modern science.Besides,the neural network has been widely used in deep learning,big data analysis and pattern recognition,because it adopts distributed and parallel computing.In addition,many scholars have carried on the detailed research to each aspect of the neural network.As the basis of research on the neural network theory,the dynamic characteristic analysis of the neural network is also a hot issue.Actually,complex-valued neural networks are more suitable for practical scenarios,since they introduce complex values into the state variables,connection weight matrices and activation functions.Therefore,complex-valued neural networks are able to work out problems that cannot be solved by real-valued neural networks.Moreover,we can see from the existing research that the complex-valued neural network has a faster data processing rate compared with the real-valued neural network.This is because it can deal with the complex signal directly and this feature is important for signal processing.Therefore,the dynamic behavior of the complex neural network is worthy of further exploration.At present,the dynamic behavior analysis has been discussed in many existing research results on neural networks.The common solution is to construct the appropriate Lyapunov function,and then use the inequality technique to solve this problem.However,it is very difficult to construct a Lyapunov function that matches the system.Moreover,the operation process of the linear matrix inequality(LMI)is also quite cumbersome.Therefore,the Lyapunov function and the linear matrix inequality are not used in this thesis,and the dynamic behavior of the complex-valued neural network is discussed by combining the differential inequality and the complex conjugate property.In general,the dynamics analysis and application of the differential system mainly depend on the stability of the system at the equilibrium point or the region where the equilibrium point exists,i.e.,the attracting set and invariant set.Therefore,this thesis focuses on the exponential stability of the equilibrium point of the autonomous complexvalued neural network with time-varying delay and the global attracting set and positive invariant set of the non-autonomous complex-valued neural network with time-varying delay.The main contribution of this thesis can be concluded as follows:(1)This thesis investigates an autonomous neural network with time-varying delay,and analyzes the exponential stability of the system at the equilibrium point.To be specific,we take advantage of the conjugate property of complex numbers instead of separating the activation function into real and imaginary parts,the existing differential inequality and the characteristic of M matrix to obtain the sufficient conditions for the stability of the complex-valued neural network.Numerical simulation verifies the validity of the condition.(2)This thesis studies the global attracting set and positive invariant set of nonautonomous complex-valued neural networks with time-varying delay.In this thesis,an integral-differential inequality is established on the basis of the existing differential inequalities.Combining the Hadmard product,M matrix and the property of complex numbers,the estimation of the global attracting and positive invariant set of the nonautonomous complex neural networks can be achieved.Then,we obtain the global attracting set of non-autonomous real-valued neural networks.Finally,numerical simulations are given to verify the effectiveness of the above results.