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Journal of Vibration Testing and System Dynamics 7(3) (2023) 327--397 | DOI:10.5890/JVTSD.2023.09.006

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 dynamical systems possessing bivariate quadratic vector fields is presented. The bivariate quadratic vector field is a product of two different-variable univariate functions in two directions. The dynamical systems with two crossing-variable product bivariate vector fields are presented, and the corresponding global dynamics of such dynamical systems is presented. The hyperbolic and hyperbolic-secant flows with directrix flows are discussed. From the infinite-equilibriums, the inflection sink (or source) bifurcation is presented for the switching of hyperbolic flow and saddles with hyperbolic-secant flow and sink (or source). Parabola-saddle bifurcations are for the switching of saddle and hyperbolic-secant flow with center and hyperbolic flow, which is called the saddle-center switching bifurcation. Inflection diagonal-saddle bifurcations are presented for the switching of the network of saddle and sink (or saddle and source) with hyperbolic and hyperbolic-secant flows.

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.