Stability and Periodic Solutions for a Neural Network Model with Multiple Time Delays

Abstract

A three-triangle neural network model with seven neurons and time delays has been investigated by several researchers. The stability and bifurcating periodic solution were discussed by using the central manifold theorem and the normal form theory. However, by means of the delay as a bifurcation parameter, the authors must make a specific restrictive condition such that the network model with seven delays can be changed to only one delay system. In other words, the stability and the Hopf bifurcation of the three-triangle neural network model were studied under a very specific restrictive condition to the time delays. The present paper also considers the stability and the existence of periodic oscillations for this neural network model. Two theorems are provided to guarantee the stability and the existence of periodic oscillations for this three-triangle neural network model by using of the mathematical analysis method, which is simpler than bifurcation method. Also, our method avoids dealing with a complex bifurcating equation. It does not have any restrictions on the time delays in the model. Thus, our result is an extension of the literature. The criteria for selecting the parameters in this network are provided. Computer simulation examples are presented to demonstrate the correctness of this method. Our computer simulation indicates that the criteria in this paper are only sufficient conditions.