Qualitative Study Of A Bidirectional Ring Network Model

Since neural networks were proposed, their base study and applications have advanced with exceptional speed. For their great effectiveness on associative mem-ories, optimal computation, automatic control, etc., the qualitative analysis of neu-ral networks attracts a large number of experts. The focus of this thesis is to study issues related to stability and bifurcation of equilibria in a ring of neurons with one delay, which has been found in a lot of neural structures, such as neocortex, cerebellum, hippocampus, and even in chemistry and electrical design, and can know the basic mechanisms of recurrent network by studying the ring network.The paper consists of the following six parts:In the first chapter, the basic concept, properties of artificial neural networks, their current research situation and quality research are presented. Numerous works in this field are cited. Then, the main contribution of this paper is also simply introduced.In the second chapter, some relevant knowledge including bifurcation theory, especially Hopf bifurcation theory, which is needed in our study, will be given in detail.In the third chapter, we discuss and obtain some sufficient conditions ensuring the absolute synchronization of the system by using the Lyapunov function.In the fourth chapter, linear stability of the system is investigated by ana-lyzing the associated characteristic transcendental equation. By means of space decomposition, we subtly discuss the distribution of zeros of the characteristic equation, and then derive some sufficient conditions ensuring that all the charac-teristic roots have negative real parts. Therefore, the trivial solution of the system is asymptotically stable.In the fifth chapter, by regarding eigenvalues of the connection matrix of the system as bifurcation parameters, which is different from traditional method con-sidering the delays of signal transmission, we discuss Hopf bifurcation of equilibria. Meanwhile, with the help of center manifold reduction and normal form theory, we study Hopf bifurcation of equilibria, and obtain detailed information about the bifurcation direction and stability of various bifurcated periodic solutions.In the sixth chapter, some numerical simulations are given to demonstrate the obtained results using the computer simulation.