Singularity and 1-dimensional Flows in 2-D Single-variable Quadratic Systems

Albert C. J. Luo

Journal of Vibration Testing and System Dynamics

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

Xianguo Tuo (editor)

Singularity and 1-dimensional Flows in 2-D Single-variable Quadratic Systems

Journal of Vibration Testing and System Dynamics 6(2) (2022) 107--194 | DOI:10.5890/JVTSD.2022.06.001

Albert C. J. Luo

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

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Abstract

In this paper, two-dimensional nonlinear systems possessing univariate quadratic vector fields with a single-variable only are discussed, and the appearing and switching bifurcations for the 1-dimensional flows (e.g., sink and source flows, and up-parabola and down-parabola flows) will be discussed herein. The upper-saddle and lower-saddle flows are for the appearing bifurcations of the sink and source flows. The increasing-inflection and deceasing-inflection flows are for the appearing bifurcations of the up-parabola and down-parabola flows. The infinite-equilibriums are for switching bifurcations between the sink and source flows and the up-parabola and down-parabola flows. The bifurcations of the 1-dimensional flows are much richer than the bifurcations of equilibriums. The switching of the sink and source flows with parabola flows are very significant in applications, and it can be used to determine flow singularity.

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.