Study Of Multidirectional Associative Memory Neural Networks

The multidirectional associative memory(MAM) neural network extended the bidirectional associative memory(BAM) neural network, which was proposed by Hagiwara M., a Japanese scholar, in 1990. Although this type of network is a successful model which can realize a many-to-many association, there are few theoretical studies of the network as viewed from mathematics up to now. In this paper, the mathematic models are proposed which are described by delay differential equations or difference equations. The existence and global stability of equilibrium for MAM are discussed. At the same time, the existence and global exponential stability of periodic solution and almost periodic solution for MAM are also discussed. The paper consists of five chapters.The studying status quo on associative memory neural network are summarized in Chapter 1. The various BAM models as well as their researching methods are discussed in detail.In Chapter 2, the MAM neural network models are established which are described by differential equations with constant delays, time-varying delays or distributed delays. At the same time, by employing a semi-discretization technique, the discrete-time analogues of the continuous-time MAM neural networks are formulated which are described by difference equations with constant delays, time-varying delays or distributed delays.In Chapter 3, the existence, uniqueness, global asymptotical stability and global exponential stability of the equilibrium for MAM with time-varying delays are discussed at first. Assuming the boundedness, or Lipschitz of the signal transmission functions, the existence of the equilibrium for MAM is proved by using fix point theory and M matrix properties. In Lipschitz condition of the signal transmission functions, the uniqueness of the equilibrium for MAM is proved respectively by using M matrix properties and using homeomorphous map. By constructing suitable Lyapunov functionals, some delay dependent and delay independent sufficient criterions are obtained which ensure the global exponential stability of the equilibrium for the MAM.Some suitable assumptions are made to derive a sufficient and necessary condition which ensures the existence and global exponential stability of the equilibrium for the MAM. In the Chapter, the equilibrium for MAM with distributed delays is also discussed. Without assuming the Lipschitz of the signal transmission functions, the existence, uniqueness of the equilibrium for MAM is proved. The global exponential stability of the equilibrium for the discrete-time delayed MAM neural networks is discussed at the end of the Chapter. It is shown that the stability of the continuous-time MAM neural network is preserved by the discrete-time delayed MAM neural network. An example is given to illustrate that the criterions are feasible, and the computer simulation is carried out.In Chapter 4, the existence and global exponential stability of the periodic solution for delayed MAM networks are discussed. By using the continuation theorem of coincidence degree and some inequality analysis techniques, a sufficient condition is obtained to ensure the existence of periodic solution for MAM neural network with time-varying delays, and a delay independent sufficient criterion is also obtained to ensure the global exponential stability of periodic solution for this type of MAM neural network by constructing Lyapunov functionals. Also, by constructing Lyapunov functionals, using of fixed point theorem and some analysis techniques, the existence and global exponential stability of the periodic solution for MAM networks with distributed delays are discussed. The global exponential stability of the periodic solution for the discrete-time MAM neural networks with distributed delays is discussed at the end of the Chapter. Two examples are given to illustrate that the criterions are feasible, and the computer simulation is carried out.In Chapter 5, the existence and global exponential stability of the almost periodic solution for delayed MAM networks are discussed. A sufficient condition is obtained to ensure the existence and uniqueness of almost periodic solution for MAM neural network with time-varying delays by way of exponential dichotomy and contraction map theorem. Moreover, the global exponential stability of the almost periodic solution for this type of MAM neural network is proved by using Halanay inequality. Also, by using exponential dichotomy, contraction map theorem and matrix theory, some sufficient conditions are obtained to ensure the existence and global exponential stability of almost periodic solution for MAM neural network with distributed delays. An example is given to illustrate that the criterions are feasible, and the computer simulation is carried out.