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

IF 1.2 Q2 MATHEMATICS, APPLIED

Journal of Applied Mathematics Pub Date : 2014-07-03 DOI:10.1155/2014/823937

Yu Xiaodan, Jia Hongjie, Wang Chengshan, Jiang Yilang

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引用次数: 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.

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单滞后时滞LTI系统振荡出现分岔的一种确定方法

研究了时滞线性时不变系统中存在的一类分支——振荡涌现分支(OEB)。根据特征值变化，初步提出了OEB的定义。结果表明，当发生OEB时，一个实特征值分裂为一对共轭复特征值，这意味着系统特征值的个数增加一个，并产生新的振荡模式。其次，提出了一种基于Lambert W函数确定单滞后时滞LTI系统OEB分岔的方法。首先利用一维(1-dim)时滞系统来解释OEB分岔的机理。然后，推导了一维、二维和高维时滞LTI系统中OEB分岔的确定方法。仿真结果验证了所提方法的正确性和有效性。由于OEB分岔是随着新的振荡模态的出现而发生的，因此本文的工作对于探索时滞动态系统的复杂现象和稳定性是有帮助的。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Journal of Applied Mathematics MATHEMATICS, APPLIED-

CiteScore

2.70

自引率

0.00%

发文量

审稿时长

3.2 months

期刊介绍： Journal of Applied Mathematics is a refereed journal devoted to the publication of original research papers and review articles in all areas of applied, computational, and industrial mathematics.