Dynamics for a class of nonlinear systems with time delay

Starting from the delay Liénard and van der Pol equations with time delay, this paper represents an attempt to give an introductory presentation of dynamics for a class of nonlinear systems with time delay and emphasizes the recent development on this field in china. It is clearly seen that the time delay occurred in systems can lead to the rich dynamical behaviors such as death island, Hopf bifurcation, double Hopf bifurcation, period-doubling bifurcation, phase locked (periodic) and phase shifting solutions, co-existing motions, quasi-periodic motion and even chaos. In the quantitative and qualitative treatment of the analytical method, a new method, called perturbation-incremental scheme (PIS), is introduced by two steps, namely perturbation and increment for continuation to the critical value at a simple Hopf bifurcation. This paper shows that time delay may be used as a simple but efficient "switch" to control motions of a system: either from order to complex motions or from complex to order motions for different applications. As well as the PIS can bypass and avoid the tedious calculation of the center manifold reduction (CMR) and normal form. © 2007 Elsevier Ltd. All rights reserved.