The Research On Bifurcation Of A Class Of Discrete-time Neural Network Model

The neural networks(often shortened as NN) were firstly proposed as a method to describe present world by a class of differential and difference equations in the 40s of last century. Many models from neurobiology, biological populations, and evolution theory, are concrete forms of neural network(NN), among which, there are Cohen-Grossber neural network(CGNN), the bidirectional associative memory neural network and the cellular neural network. They have been intensively used in real application, such as parallel computation, signal processing, associative memory, and so on, which has become an attracting and focused topic.In this thesis, we investigate the bifurcation analyzation of a discrete-time neural network model with generalized input-output function, based on classical theoretical works developed by Saber N.Elaydi[49], Clark Robinson[55] and Yuri A.Kuznetsov[82]. To a certain extent, this model can be regarded as a discrete form of transient chaotic neural networks proposed by Aihara et al. Therefore it expands the bound of input-output functions, especially from monotone logistic function and period sinusoidal function, to more general function in this thesis. First of all, within this more general function, we discuss saddle-node bifurcation and period-doubling bifurcation under 1-D situation, as well as theoretical bifurcation results under 2-D situation. Secondly, we obtain bifurcation both in 1-D and 2-D of sinusoidal input-output function case, when changing the weight parameter in transient chaotic neural networks from constant to time-varying.This thesis is organized as follows: In Chapter 1, we introduce the solid background of the artificial neural networks and their characteristics in the theoretical researches and applications. Furthermore, several mathematical models in the field of neural networks are showed as examples. Meanwhile, we simply review the bifurcation theory, and list some bifurcation types.In Chapter2, we firstly introduce a discrete-time neural network with generalized input-output function. According to sinusoidal function, this model extent the bound of input-output functions in transient chaotic neural network. Second of all, we study the bifurcation of this model under 1-D and 2-D situations, when the bifurcation parameter changes. At last, several simulations illustrated as concrete examples, to sustain our results.In Chapter 3, we focus on a discrete-time neural network model with sinusoidal input-output function. But more specifically, a time-varying weights case is considered here. Finally, we obtain theoretical results in 1-D and 2-D situations by using classical theories in bifurcation analysis.At the end of this dissertation, we make a fully review of this thesis and propose some important topics and prospective work on the discrete-time neural networks and the on chaotic dynamics.There are three creative points:(1)We discuss the bifurcations with more generalized input-output function in Aihara's transient discrete-time neural networks. Specially, we obtain the sufficient condition of saddle-node bifurcation and period-doubling bifurcation under 1-D situation in this model. As there is no special requirements for the periodicity and symmetry of the input-output functions, it really expands range of them. Refer to chapter 2 for further information. (2)We research bifurcation of time-varying weights in discrete-time neural networks, with a typical input-output function, under the 1-D situation. Within the result, we asymptotically approach the original model of transient discrete-time neural networks. Refer to chapter3 for further information. (3)To adjust our model to be close with the original transient discrete-time NN, we focus on the dynamical property of bifurcation with more generalized input-output function case as well as time-varying weights case, under 2-D situation. Refer to relevant information both in chapter 2 and 3.