Analysis Of Dynamic Properties Of A Class Of Delay Differential Equations

Bifurcation is one of the best important subjects in dynamical system and nonlinear differential equations.It mainly studies the phenomenon that the topological structure of a system passing through a critical value suddenly changes,when the topological structure of the mathematical model of dynamic system is relatively unstable.The Zero-Hopf bifurcation which we considered means that system equilibrium stability will change when the bifurcation parameters changed,and the periodic solutions and fixed point bifurcation will appear in the neighbourhood of the equilibrium point at the same time.This paper main study object is a class of second order scalar delay differential equations,consider the dynamic property change problem when the equation occurs Zero-Hopf bifurcation.First of all,we use center manifold reduction and Faria normal form theory to calculate the normal form of delay differential equation,then we analyse the morphology and property of the normal form.The conditions for generating Zero-Hopf bifurcation of the equation and the bifurcation phase diagrams corresponding to four different cases are summarized.Finally,we will analysis of two practical system equations,and combined with numerical simulation to show the generation of Zero-Hopf bifurcation phenomenon and the concrete solution curve state.