Crossing and Product Polynomial Systems

Albert C. J. Luo
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Albert C. J. Luo: Southern Illinois University Edwardsville, Department of Mechanical and Mechatronics Engineering

Chapter Chapter 1 in Two-dimensional Crossing and Product Polynomial Systems, 2026, pp 1-125 from Springer

Abstract: Abstract In this chapter, the dynamics of crossing and product polynomial systems will be presented through a theorem. The singular equilibriums and 1-dimensional flows with infinite-equilibriums in product polynomial systems are presented in the theorem. The singular equilibriums are singular saddles and centers, parabola-saddles and inflections-saddles. The singular 1-dimensional flows are singular hyperbolic-flows, hyperbolic-to-hyperbolic-secant flows, inflection-source and sink flows, inflection-saddle flows. The corresponding first integral manifolds are obtained. The hybrid networks of singular equilibriums and 1-dimensional flows in the crossing and product polynomial systems are discussed in the theorem, and the corresponding infinite-equilibriums are given for switching bifurcations. The hybrid network of simple equilibriums and 1-dimensional flows with infinite-equilibriums in the crossing and product polynomial systems is stated in the theorem. The appearing and switching bifurcations of simple and singular 1-dimensional flows and equilibriums are stated in the theorem. Finally, the switching bifurcations of hybrid networks of simple and singular equilibriums and flows is also stated. The corresponding dynamical behaviors are presented through the following theorem.

Date: 2026
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DOI: 10.1007/978-981-96-5715-5_1

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