Hopf-zero Bifurcation Analysis Of Two Classes Of Delay Differential Equations

In this paper,the coupled van der Pol oscillator with time delays and Oregonator oscillator with time delays are studied from two aspects of theory and numerical simulation by using the central manifold theory and the normal form theory developed by Faria and Magalhaes.1.We study the Hopf-pitchfork bifurcation of coupled van der Pol oscillators with time delays.The influence of time delay on the system is analyzed,the condition of the Hopf-pitchfork bifurcation of the system is given,and the normal form of the system is simplified on the center manifold.From two aspects of theory and numerical simulation,the phenomena of a pair of non trivial equilibrium points,stable ordinary equilibrium points,stable periodic rails,a pair of stable non trivial equilibrium points and stable periodic rails will appear in the van der Pol oscillator system when the time and coefficient terms are disturbed near the critical point at that time.Finally,the numerical simulation is used to verify the theoretical results.2.We consider the Hopf-zero bifurcation of Oregonator oscillator with time delay.By analyzing the characteristic root distribution of the characteristic equation,the condition of the Hopf-zero bifurcation of the Oregonator oscillator is found,moreover,the normal form near the critical point of Hopf-zero bifurcation and thebifurcation diagrams and phase portraits of the system parameters are obtained.On this basis,it is found that saddle-node branches and pitchfork branches occur at M and N,respectively;the Hopf branches and heteroclinic branches occur at H and S,respectively.Finally,the theoretical results are verified by numerical simulation.