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

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.