Dynamics Of Two Coupled Oscillators With Delayed Feedback And Excitatory-to-excitatory Connection

In this thesis, we study dynamics of a neural network model consisting of twocoupled oscillators with delayed feedback and excitatory-to-excitatory connection.We analyze in detail how the strength of the connections between the oscil-lators a?ects the dynamics of the neural network. We give a full classification ofall equilibria in the parameter space and obtain its linear stability by analyzingthe characteristic equation of the linearized system. We also show that systemmay undergo Hopf bifurcation. Moreover, the stability and bifurcation directionof the bifurcated periodic solutions are obtained by employing center manifold re-duction and normal form theory. Finally, some numerical simulations are given toillustrate the main results.The paper consists of the following six parts:In the first chapter, the background, the significance and the progress for thestudy on non-linear time-delay dynamical systems are presented. Then, the mainwork of this paper is also simply introduced.In the second chapter, the relevant knowledge including bifurcation theory,center manifold theorem and neural networks, which is needed in the study of thisproblem, will be given in detail.In the third chapter, we give a full analysis of all equilibria that may exist inthe system, and the su?cient condition on absolute synchronization of the system.The fourth chapter is devoted to the linear stability of the system and thecharacteristic equation of the linearized system.In Chapter five, we show that the system undergoes Hopf bifurcation whenthe connection strengthαpasses critical valuesαk orαI.Finally, in Chapter six, by making use of the computer simulation on Matlab,some numerical simulations are given to support our main results.