Stability and Hopf bifurcations of an optoelectronic time-delay feedback system

Abstract

The local dynamics around the trivial solution of an optoelectronic time-delay feedback system is investigated in the paper, and the effect of the feedback strength on the stability is addressed. The linear stability analysis shows that as the feedback strength varies, the system undergoes exactly two times of stability switch from a stable status to an unstable status or vice versa, and at each of the two end points of the stable interval, a Hopf bifurcation occurs. To gain insight of the bifurcated periodic solution, the Lindstedt-Poincaré method that involves easy computation, rather than the center manifold reduction that involves a great deal of tedious computation as done in the literature, is used to calculate the bifurcated periodic solution, and to determine the direction of the bifurcation. Two case studies are made to demonstrate the efficiency of the method.

Nonlinear analysis of a maglev system with time-delayed feedback control

“…Since A.H. Nayfeh [14,15] first presented the method of multiple scales, it has been successfully used in practical design engineering [16,17,18]. The method of multiple scales not only reveals the complex dynamic of nonlinear systems, but also provides useful approximate formulas for the periodic solution.…”