A Method to Determine Oscillation Emergence Bifurcation in Time-Delayed LTI System with Single Lag

Abstract EN

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.