Linear Stability and Bifurcation Analysis in a Delayed Two-Coupled Oscillator

The focus of this thesis is to study issues related to the stability of the trivial equilibrium and the local bifurcation at the trivial equilibrium (including codimension one bifurcations and codimension two bifurcations) of equilibria in a delayed two-coupled oscillator with excitatory-to-excitatory as well as inhibitory-to-inhibitory connection. This thesis is organized as follows:Firstly, the background and the motivation for the study of artificial neural networks are presented. Then, occurrence of bifurcation and some traditional methods used to study bifurcation are introduced simply.Secondly, the characteristic equation of the two-coupled oscillator is a tran-scendental equation due to the time delay. So linear stability is investigated by analyzing the associated characteristic transcendental equation. By means of space decomposition, we subtly discuss the distribution of zeros of the character-istic equation, and then we derive some sufficient conditions ensuring that all the characteristic roots have negative real parts, i.e., such that the trivial solution is asymptotically stable. Furthermore, we give the distribution map of the stable region of the trivial equilibrium.Thirdly, by regarding eigenvalues of the connection matrix and the time delay as bifurcation parameters, we discuss codimension one Hopf bifurcation. Based on the normal form theory and center manifold reduction, we obtain detailed information about the Hopf bifurcation, such as the condition of the existence of periodic solutions, bifurcation direction, stability, and so on. Then, considering the occurrence of Fold bifurcation, and obtaining some dynamics behavior about this bifurcation.Fourthly, codimension two bifurcation (including Fold-Hopf bifurcations and Hopf-Hopf bifurcations) are discussed. By considering the connection between these systems and the normalized form of these systems with respect to the exis-tence and stability of equilibria and periodic solutions, we obtain some methods of analyzing the dynamics of bifurcation solutions of these systems.Finally, we choose tanh(x) as the amplification function and give some nu-merical simulations to support the obtained results.