Bifurcation Analysis In Two Kinds Of Delay Differential Equations

In this paper,we mainly analyze the bifurcations of two differential equations with delay,one is a neural network model with delay,the other is a symmetrically conservative two –mass system with delay.For the first classical mathematical model,firstly,by considering time delay as bifurcation parameter,we discuss the stability of the model and the existence of Hopf bifurcation.Then,after complicated calculations with Lyapunov-Schmidt reduction,we obtain the approximate analytic expressions of the bifurcation equations and periodic solutions near the Hopf bifurcation points.For the second well-known nonlinear free-vibration model,we first analyze the conditions when Hopf-pitchfork bifurcation occurs,that is,we should discuss the conditions that the parameters need to meet to make the characteristic equation of the linearized part of the system has a single zero root and a pair of purely imaginary roots,and no other roots have a zero real part.Then,we use the central manifold theory and the Faria and Magalhaes normative methods to derive the normal form with original parameters for Hopf-pitchfork bifurcation,and obtain complete bifurcation diagrams and phase portraits.Finally,through the numerical simulations,some theoretical analysis results are verified.