Analysis On Takens-bogdanov Bifurcation For The Semiconductor Laser With Saturable Absorber And Delayed Optical Feedback

In1993, Yamada et al. proposed the mathematical model of semiconductor laser with saturable absorber. Based on the physical properties, introducing an optical feedback,could result in a laser system with time delay. In this paper we research the TakensBogdanov bifurcation of the semiconductor laser model with saturable absorber and delayed feedback. Taking feedback strength and the feedback time(delay) as the bifurcation parameters, using the conclusions of Takens-Bogdanov bifurcation in delay differential equations developed by Xu Y and Huang M, as well as theories from Faria T about the center manifold reduction and normal form calculation for delay differential equations,we show the critical bifurcation parameter value for the occurrence of Takens-Bogdanov bifurcation. We arrive at the explicit expression of reduced equation on the center manifold for the non-degenerate model problem: a system of ordinary differential equations with coefficient involving the bifurcation parameters. Through the investigation on the reduced ordinary differential equations, we get the bifurcation structure for the delayed semiconductor laser model in parameter plane: there exist a homoclinic orbit bifurcation curve and a Hopf point bifurcation curve emanating from the critical bifurcation parameter value.In addition, we also obtain the normal form on the center manifold for the degenerate case using the same theory of normal form calculation, further analysis on this new normal form can reveal the Takens-Bogdanov bifurcation structure of model problem if certain degenerate condition holds.