Bifurcations and Saddle-Sink-Source Networks in Variable-Independent Quadratic Systems

Albert C. J. Luo

Journal of Vibration Testing and System Dynamics

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

Xianguo Tuo (editor)

Bifurcations and Saddle-Sink-Source Networks in Variable-Independent Quadratic Systems

Journal of Vibration Testing and System Dynamics 7(1) (2023) 59--112 | DOI:10.5890/JVTSD.2023.03.008

Albert C. J. Luo

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

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Abstract

This paper presents a theory for nonlinear dynamics of dynamical systems possessing variable-independent univariate quadratic vector fields. The dynamical systems with a constant vector field and a variable-independent quadratic vector field are presented first, and the 1-dimensional flows discussed. Dynamical systems with linear and quadratic variable-independent univariate vector fields are discussed, and the corresponding bifurcation and global dynamics are discussed. Dynamical systems with two variable-independent univariate quadratic vector fields are analyzed, and the corresponding bifurcations and global dynamics are discussed through the first integral manifolds.

References

[1] Luo, A.C.J. (2022), A theory for singularity and stability in two-dimensional linear systems, Journal of Vibration Testing and System Dynamics, 6(1), 63-105.
[2] Luo, A.C.J. (2022), Singularity and 1-dimensional flows in 2-D single-variable quadratic systems, Journal of Vibration Testing and System Dynamics, 6(2), 107-194.