Local Hopf Bifurcation Of Complex Nonlinear System With Time-Delay

A Hopf bifurcation occurs commonly in the time-delay systems described by delay differential equations (DDEs), even in first order autonomous DDEs. Compared with the intensive study of Hopf bifurcation for DDEs with real coefficients, which mainly by using the center manifold reduction, perturbation methods and so on, little effort has been made directly for the Hopf bifurcation of DDEs with complex coefficients. A pseudo-oscillator analysis is newly developed to study Hopf bifurcation for DDEs with real coefficients. It is shown that the pseudo-oscillator analysis involves easy computation only and it is more tractable compared to the current methods. The main idea of the pseudo-oscillator analysis is to construct a pseudo-oscillator associated with the original system so that the local dynamics near the Hopf bifurcation can be justified by the pseudo-power function of the generated oscillator. The aim of this paper is to generalize the pseudo-oscillator analysis so that it is applicable directly for the Hopf bifurcation of complex DDEs. For this purpose, the pseudo-oscillator analysis and Lambert W function are firstly introduced in brief. The stability conditions for first order scalar complex-coefficient DDEs are given on the basis of the Lambert W function; and the conditions that govern the existence of a Hopf bifurcation for complex-coefficient DDEs are also presented. Finally, the pseudo-oscillator analysis is generalized for the Hopf bifurcation of scalar complex nonlinear DDEs. As applications of the method, the Hopf bifurcation of the Lang-Kobayashi equation in laser dynamics and a second order scalar DDE with complex coefficients are investigated in detail. It is shown that the amplitudes of the periodic solutions near the Hopf bifurcation obtained by the pseudo-oscillator analysis are in a good agreement with the numerical results.