Bifurcations and Saddle-Limit Cycle Networks in Crossing-Variable Quadratic Systems

Abstract

This paper presents a theory for nonlinear dynamics of dynamical systems with two variable-crossing univariate vector fields. Dynamical systems with a variable-crossing univariate linear and quadratic vector fields are discussed, and the corresponding bifurcation and global dynamics are presented. The saddle-center bifurcations are presented through parabola-saddle bifurcations. Dynamical systems with two crossing-variable univariate quadratic vector fields are discussed, and the switching and appearing bifurcations for saddles and centers are discussed through the first integral manifolds, and the homoclinic networks will be first presented. Double-inflection bifurcations are for appearance of the saddle-center network, and the homoclinic networks with centers are constructed. The saddle-center networks with limit cycles are presented from the first integral manifolds.