Local Hopf bifurcation of Complex Nonlinear Systems with Time-Delay

A Hopf bifurcation can occur 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 was mainly done 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. In this paper, a generalization of the newly developed "pseudo-oscillator analysis" for the Hopf bifurcation of scalar real nonlinear DDEs is presented for scalar complex nonlinear DDEs. 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, in particular, the stability of the bifurcated periodic solution can be determined by the averaged power function. The pseudo-oscillator analysis involves easy computation only and it is more tractable than the current methods. As an application of the method, the Hopf bifurcation of a first order complex DDE and a second order complex DDE is investigated in detail, illustrated with numerical simulation.

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