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Journal of Vibration Testing and System Dynamics 8(1) (2024) 77--154 | DOI:10.5890/JVTSD.2024.03.006

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, nonlinear dynamics of the product-quadratic systems with self-quadratic and crossing-quadratic vector fields is presented, which is the study continuation of product-quadratic systems with two product-quadratic vector fields. With a self-quadratic field, the stability and bifurcations of product quadratic systems are discussed. The saddle-sink bifurcation for saddle and sink is discussed, and the saddle-source bifurcation for saddle and source is presented. The saddle-saddle bifurcations \textit{of the first kind} are presented. The appearing bifurcation conditions for sink-source and saddle-saddle are discussed. The inflection sink (or source) bifurcations are presented for the switching bifurcations for hyperbolic flow and saddles with hyperbolic-secant flow and sink (or source). With a crossing-quadratic vector field, the dynamics and bifurcations for such a quadratic system are discussed. The saddle-center appearing bifurcations are presented through the parabola-saddle bifurcations. The center-center bifurcation and the saddle-saddle bifurcation of \textit{the second kind} are discussed through the hyperbolic sink-source and circular sink-source bifurcations. The up-parabola-saddle bifurcations are for the switching of the center and hyperbolic flow with saddle and hyperbolic-secant flows. The down-parabola-saddle bifurcations are for the switching of the center and hyperbolic-secant flow with saddle and hyperbolic flows. The parabola-saddle on the infinite-equilibrium is called the switching bifurcations of saddle and center.

References

1. [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. [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.

3. [3]	Luo, A.C.J. (2023), Bifurcations and saddle-sink-source networks in variable-independent quadratic systems, Journal of Vibration Testing and System Dynamics, 7(1), 59-112.

4. [4]	Luo, A.C.J. (2023), Bifurcations and saddle-limit cycle networks in crossing-variable quadratic systems, Journal of Vibration Testing and System Dynamics, 7(2), 187-252.

5. [5]	Luo, A.C.J. (2023), Dynamics and bifurcations in a quadratic nonlinear system with univariate product vector fields, Journal of Vibration Testing and System Dynamics, 7(3), 327-397.