Two-dimensional Single-Variable Cubic Nonlinear Systems, Vol VI

This book, the sixth of 15 related monographs, discusses singularity and networks of equilibriums and 1-diemsnional flows in product quadratic and cubic systems. The author explains how, in the networks, equilibriums have source, sink and saddles with counter-clockwise and clockwise centers and positive and negative saddles, and the 1-dimensional flows includes source and sink flows, parabola flows with hyperbolic and hyperbolic-secant flows. He further describes how the singular equilibriums are saddle-source (sink) and parabola-saddles for the appearing bifurcations, and the 1-dimensional singular flows are the hyperbolic-to-hyperbolic-secant flows and inflection source (sink) flows for 1-dimensional flow appearing bifurcations, and the switching bifurcations are based on the infinite-equilibriums, including inflection-source (sink), parabola-source (sink), up-down and down-up upper-saddle (lower-saddle), up-down (down-up) sink-to-source and source-to-sink, hyperbolic and hyperbolic-secant saddles. The diagonal-inflection upper-saddle and lower-saddle infinite-equilibriums are for the double switching bifurcations. The networks of hyperbolic flows with connected saddle, source and center are presented, and the networks of the hyperbolic flows with paralleled saddle and center are also illustrated. Readers will learn new concepts, theory, phenomena, and analysis techniques.

• Product-quadratic and product cubic systems


• Self-linear and crossing-quadratic product vector fields


• Self-quadratic and crossing-linear product vector fields


• Hybrid networks of equilibriums and 1-dimensional flows


• Up-down and down-up saddle infinite-equilibriums


• Up-down and down-up sink-to-source infinite-equilibriums


• Inflection-source (sink) Infinite-equilibriums 


• Diagonal inflection saddle infinite-equilibriums


• Infinite-equilibrium switching bifurcations

 

• Develops singularity and networks of equilibriums and 1-diemsnional flows in product-quadratic and cubic systems; 


• Provides dynamics of product-quadratic/ product-cubic systems through equilibrium network and first integral manifolds;


• Discovers new switching bifurcations through infinite-equilibriums of up-down upper-saddles (lower-saddles).