Stability And Bifurcation In Two-Coupled Oscillator With Delayed Feedback And Excitatory-to-Inhibitory Connection

The focus of this thesis is to study issues related to stability and bifurcationof two-coupled oscillator with delayed feedback and excitatory-to-inhibitory con-nection, which has a refection symmetry characteristic and can be identified witha Lie group Z2. This thesis is mainly devoted to the stability and bifurcation ofthe given system based on center manifold reduction and normal form approach,which is organized as follows:Firstly, by some using linear transformations and suitable Liapunov function,we find some suffcient conditions on the existence of the synchronous solution forthe given system. In addition, we also discuss the existence and patterns of all thepossible equilibria.Secondly, linear stability is investigated by analyzing the associated character-istic transcendental equation. By means of space decomposition, we subtly discussthe distribution of zeros of the characteristic equation, and then we derive somesuffcient conditions ensuring that all the characteristic roots have negative realparts. Hence, the zero solution of the system is asymptotically stable.Thirdly, by computing the normal form following center manifold reduction,we obtain the existence condition of Hopf bifurcation, the stability of the bifurcatedperiodic solutions and the direction of Hopf bifurcation.Finally, numerical simulations of concrete examples are presented to verifyour main theoretical results.