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Discrete Dynamics in Nature and Society
Volume 2015, Article ID 984323, 8 pages
Research Article

The Applications of Algebraic Methods on Stable Analysis for General Differential Dynamical Systems with Multidelays

1Department of Mathematics, Northeast Forestry University, Harbin 150040, China
2Department of Mathematics, Harbin Institute of Technology, Harbin 150001, China

Received 13 January 2015; Revised 31 March 2015; Accepted 3 April 2015

Academic Editor: Peng Shi

Copyright © 2015 Jian Ma and Baodong Zheng. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


The distribution of purely imaginary eigenvalues and stabilities of generally singular or neutral differential dynamical systems with multidelays are discussed. Choosing delays as parameters, firstly with commensurate case, we find new algebraic criteria to determine the distribution of purely imaginary eigenvalues by using matrix pencil, linear operator, matrix polynomial eigenvalues problem, and the Kronecker product. Additionally, we get practical checkable conditions to verdict the asymptotic stability and Hopf bifurcation of differential dynamical systems. At last, with more general case, the incommensurate, we mainly study critical delays when the system appears purely imaginary eigenvalue.