An Almost Periodic Study Of Discontinuous Complex-valued Neural Networks

As we know, the connected neural networks has been widely investigated due to the successful applications in our daily life, such as signal processing, pattern recognition, associative memories, complicated optimization, and so on. These ap-plications are mainly based on dynamical behaviors of neural networks. However, in most cases, complex-valued neural networks need to satisfy certain condition that ensure the network stable. In this paper, we mainly introduce the dynamic behavior of the delayed complex-valued neural networks with discontinuous activation func-tions, which extended by real-valued neural networks in the complex domain. We define almost periodic function, and construct an equivalent discontinuous right-hand equation by decomposing real and imaginary parts of complex-valued neural networks. Applying differential inclusions theory, diagonal dominant principle and non-smooth analysis theory of generalized Lyapunov function method, we achieve a sufficient condition that ensure the existence, uniqueness and global stability of almost periodic solution for the equivalent delayed differential network. The main content of this paper is arranged as follows:The first chapter is divided into two parts. First of all, the research background and the present research situations of neural networks are introduced. Furthermore,to derive the desired result successfully, some fundamental definitions and lemmas are introduced in the preliminary knowledge part.In the second chapter, we study almost periodic dynamical behaviors for complex-valued recurrent neural networks with discontinuous activation functions and time-varying delays by applying real-valued neural networks method. We construct an equivalent discontinuous right-hand equation by decomposing real and imaginary parts of complex-valued neural networks. Based on differential inclusions theory,diagonal dominant principle and non-smooth analysis theory of generalized Lya-punov function method, we achieve the existence, uniqueness and global stability of almost periodic solution for the equivalent delayed differential network. Especially,we derive a series of results on the equivalent neural networks with discontinuous ac-tivation functions, constant coefficients as well as periodic coefficients, respectively.Finally, we give a numerical example to demonstrate the effectiveness and feasibility of the derived theoretical results. Our research method reference continuous real valued neural network.In the third chapter, we investigate almost periodic dynamics of a class of delayed complex-valued neural networks with discontinuous activation functions,which have continuous real part and discontinuous imaginary parts. Firstly, we u-tilize a method similar to the second chapter. For the convenience of research, we construct an equivalent discontinuous right-hand equation by decomposing real and imaginary parts of complex-valued neural networks. Secondly, we supposed that discontinuous function is not monotone in this chapter, which lead to weaker than previously. Depended on differential inclusions theory, diagonal dominant principle and non-smooth analysis theory of generalized Lyapunov function method, we con-clude the existence, uniqueness and global stability of almost periodic solution for the equivalent delayed differential network. Especially, we derive a series of results on the equivalent neural networks with discontinuous activation functions, constant coefficients as well as periodic coefficients, respectively. Finally, we give a numerical exainple to demonstrate the effectiveness and feasibility of the derived theoretical results.Finally, the fourth chapter summarizes the research work of this dissertation and provides some trends of future research.