Synchronous Dynamics Of Two Coupled Oscillators With Inhibitory-to-Inhibitory Connection

The focus of this thesis is to study issues related to synchronous dynamics oftwo coupled oscillators with inhibitory-to-inhibitory connection, such as absolutesynchronization, existence, stability and bifurcation (including codimension onebifurcations and codimension two bifurcations) of equilibria. Synchronous dynam-ics has already been applied to each domain of the communication, the laser, theecosystem, neuron system, and can be studied to gain insight into the mechanismsunderlying the behavior of coupled oscillators system. This thesis is organized asfollows:Firstly, the background and the motivation for the study of synchronous dy-namics of coupled oscillators system are presented. Then, some known resultsof neural network models are introduced. Occurrence of bifurcation and sometraditional research methods are introduced in brief.Secondly, by using of the Lyapunov function, we obtained some suffcientconditions ensuring the absolute synchronization.Thirdly, linear stability of the model is investigated by analyzing the asso-ciated characteristic transcendental equation. By means of space decomposition,we subtly discuss the distribution of zeros of the characteristic equation, and thenwe derive some suffcient conditions ensuring that all the characteristic roots havenegative real parts. Hence, the zero solution of the model is asymptotically stable.Fourthly, by regarding eigenvalues of the connection matrix as bifurcationparameters, which is different from traditional research using the time delay asbifurcation parameters, we discuss codimension one bifurcations (including foldbifurcation and Hopf bifurcation) and codimension two bifurcation (including fold-Hopf bifurcations and Hopf-Hopf bifurcations). Based on the normal form theoryand center manifold reduction, we obtain detailed information about the bifur-cation direction and stability of various bifurcated equilibria as well as periodicsolutions with some kinds of spatio-temporal patterns.Finally, Numerical simulation is also given to support the obtained results.