In this chapter, nonlinear dynamics of dynamical systems with two variable-crossing univariate vector fields is discussed. The dynamical systems with a constant vector field and a variable-crossing univariate quadratic vector field are presented first, and the 1-dimensional flow is discussed as well. Dynamical systems with a variable-crossing univariate linear and quadratic vector fields are discussed, and the corresponding bifurcation and global dynamics are discussed. The saddle-center bifurcations are presented through parabola-saddles. Dynamical systems with two variable-crossing 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 appearing of the saddle-center network, and the homoclinic networks with centers are constructed. The saddle-center network with limit cycles are developed from the first integral manifolds..

Two-Dimensional Quadratic Nonlinear Systems : Volume I: Univariate Vector Fields

Luo, Albert C. J.

1st ed. 2023, Singapore, Singapore

S.l, 2022

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Referenced by: doi:10.5890/jvtsd.2023.06.006

This book focuses on the nonlinear dynamics based on the vector fields with univariate quadratic functions. This book is a unique monograph for two-dimensional quadratic nonlinear systems. It provides different points of view about nonlinear dynamics and bifurcations of the quadratic dynamical systems. Such a two-dimensional dynamical system is one of simplest dynamical systems in nonlinear dynamics, but the local and global structures of equilibriums and flows in such two-dimensional quadratic systems help us understand other nonlinear dynamical systems, which is also a crucial step toward solving the Hilbert's sixteenth problem. Possible singular dynamics of the two-dimensional quadratic systems are discussed in detail. The dynamics of equilibriums and one-dimensional flows in two-dimensional systems are presented. Saddle-sink and saddle-source bifurcations are discussed, and saddle-center bifurcations are presented. The infinite-equilibrium states are switching bifurcations for nonlinear systems. From the first integral manifolds, the saddle-center networks are developed, and the networks of saddles, source, and sink are also presented. This book serves as a reference book on dynamical systems and control for researchers, students, and engineering in mathematics, mechanical, and electrical engineering.

Nonlinear Physical Science
Erscheinungsdatum: 21.04.2023