A method to determine oscillation emergence bifurcation in time-delayed LTI system with single lag

One type of bifurcation named oscillation emergence bifurcation (oeb) found in time-delayed linear time invariant (abbr. lti) systems is fully studied. the definition of oeb is initially put forward according to the eigenvalue variation. it is revealed that a real eigenvalue splits into a pair of conjugated complex eigenvalues when an oeb occurs,which means the number of the system eigenvalues will increase by one and a new oscillation mode will emerge. next,a method to determine oeb bifurcation in the time-delayed lti system with single lag is developed based on lambert w function. a one-dimensional (1-dim) time-delayed system is firstly employed to explain the mechanism of oeb bifurcation. then,methods to determine the oeb bifurcation in 1-dim,2-dim,and high-dimension time-delayed lti systems are derived. finally,simulation results validate the correctness and effectiveness of the presented method. since oeb bifurcation occurs with a new oscillation mode emerging,work of this paper is useful to explore the complex phenomena and the stability of time-delayed dynamic systems. © 2014 yu xiaodan et al.