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Journal of Vibration Testing and System Dynamics

C. Steve Suh (editor), Pawel Olejnik (editor),

Xianguo Tuo (editor)

Pawel Olejnik (editor)

Lodz University of Technology, Poland


C. Steve Suh (editor)

Texas A&M University, USA


Xiangguo Tuo (editor)

Sichuan University of Science and Engineering, China


A Theory for Singularity and Stability in Two-dimensional Linear Systems

Journal of Vibration Testing and System Dynamics 6(1) (2022) 63--105 | DOI:10.5890/JVTSD.2022.03.005

Albert C. J. Luo

Department of Mechanical and Mechatronics Engineering, Southern Illinois University Edwardsville, Edwardsville, IL62026-1805, USA

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In this paper, two-dimensional dynamical systems with constant and linear vector fields are presented. Dynamical systems with one-vari- able vector fields are presented and the singular dynamics of two-dimensional linear systems was discussed. Based on the variable-inde- pendent and variable-crossing linear vector fields, the dynamics based on saddle, sink, source and center equilibriums are discussed. Two-dimensional linear dynamical systems with two linear vector fields are discussed, and local dynamics of the saddle and focus equilibriums are presented. The dynamics and stability of two-dimensional linear dynamical systems are presented comprehensively for a better understanding of dynamical behaviors of the two-dimensional linear systems.


  1. [1]  Luo, A.C.J. (2011), Regularity and Complexity in Dynamical Systems, Springer: New York.
  2. [2]  Luo, A.C.J. (2019), On stability and bifurcation of equilibrium in nonlinear systems, Journal of Vibration Testing and System Dynamics, 3(2), 147-232.
  3. [3]  Luo, A.C.J. (2020), On dynamics of infinite-equilibrium systems, International Journal of Dynamics and Control, 8, 21-43.
  4. [4]  Luo, A.C.J. (2019), Bifurcation and stability in Nonlinear Dynamical Systems, Springer: New York.