Stability and Hopf bifurcation of time-delayed systems with complex coefficients, in: Jian-Qiao Sun, Qian Ding(Eds)
Zaihua Wang, Junyu Li
Abstract: The paper surveys recent advances in studying the stability and bifurcation of time-delayed dynamic systems with real or complex coefficients. It begins with some results about the crossing direction of the characteristic roots passing through the imaginary axis of the complex plan, by using the sign of the real part of the first order derivative of the characteristic function with respect to the delay or some concerned parameter at the critical points. Serval special cases are addressed, such as time-delay systems with a single delay or commensu-rate delays, with delay-independent coefficients or delay-dependent coefficients, with real coefficients or complex coefficients. In addition, the degenerate case is also considered, for which high order derivative of the characteristic function with respect to the delay or some concerned parameter at the critical points is required in determining the crossing direction. Next, two algorithms are pre-sented for checking the stability of time-delay systems by calculating the right-most characteristic root(s) numerically or by determining the number of stability switches graphically. Then, the pseudo-oscillator analysis is introduced for the Hopf bifurcation-induced periodic solution of scalar time-delay systems with real or complex coefficients respectively. By means of the pseudo-oscillator analysis, the amplitude of the bifurcated periodic solution and its stability can be deter-mined in a very simple way and with high accuracy. Several examples are given to demonstrate the effectiveness of the main results. Finally, some concluding remarks are drawn from the discussion.
