Dynamics and Bifurcations in a Quadratic Nonlinear System with Univariate Product Vector Fields, Part II

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 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 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.