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

Yu Xiaodan, Jia Hongjie, Wang Chengshan and Jiang Yilang

Journal of Applied Mathematics, 2014, vol. 2014, issue 1

Abstract: 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.

Date: 2014
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https://doi.org/10.1155/2014/823937

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Persistent link: https://EconPapers.repec.org/RePEc:wly:jnljam:v:2014:y:2014:i:1:n:823937

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