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About this book

This book systematically presents a fundamental theory for the local analysis of bifurcation and stability of equilibriums in nonlinear dynamical systems. Until now, one does not have any efficient way to investigate stability and bifurcation of dynamical systems with higher-order singularity equilibriums. For instance, infinite-equilibrium dynamical systems have higher-order singularity, which dramatically changes dynamical behaviors and possesses the similar characteristics of discontinuous dynamical systems. The stability and bifurcation of equilibriums on the specific eigenvector are presented, and the spiral stability and Hopf bifurcation of equilibriums in nonlinear systems are presented through the Fourier series transformation. The bifurcation and stability of higher-order singularity equilibriums are presented through the (2m)th and (2m+1)th -degree polynomial systems. From local analysis, dynamics of infinite-equilibrium systems is discussed. The research on infinite-equilibrium systems will bring us to the new era of dynamical systems and control.
Presents an efficient way to investigate stability and bifurcation of dynamical systems with higher-order singularity equilibriums;Discusses dynamics of infinite-equilibrium systems;Demonstrates higher-order singularity.

Table of Contents

Frontmatter

Chapter 1. Stability of Equilibriums

Abstract
In this chapter, basic concepts of nonlinear dynamical systems are introduced. A local theory of equilibrium stability for nonlinear dynamical systems is discussed. The spiral stability of equilibriums in nonlinear dynamical systems is presented through the Fourier series base. The higher order singularity and stability for nonlinear systems on the specific eigenvectors are developed. The Lyapunov function stability is briefly discussed, and the extended Lyapunov theory for equilibrium stability is also presented.
Albert C. J. Luo

Chapter 2. Bifurcations of Equilibrium

Abstract
In this chapter, the hyperbolic bifurcations of equilibriums on the eigenvectors in nonlinear dynamical systems are discussed, and the Hopf bifurcation of an equilibrium on a specific eigenvector plane is presented. Based on the Fourier series base, the transformation for the spiral stability is introduced for the Hopf bifurcation of equilibriums. The Hopf bifurcation of equilibriums in the second-order nonlinear dynamical systems is discussed from the Fourier series transformation.
Albert C. J. Luo

Chapter 3. Low-Dimensional Dynamical Systems

Abstract
In this chapter, low-dimensional nonlinear dynamical systems are discussed. The stability and bifurcations of the 1-dimensional systems are presented. The higher order singularity and stability for 1-dimensional nonlinear systems are developed.
Albert C. J. Luo

Chapter 4. Equilibrium Stability in 1-Dimensional Systems

Abstract
In this chapter, a global analysis of equilibrium stability in 1-dimensional nonlinear dynamical systems is presented. The classification of dynamical systems is given first, and infinite-equilibrium systems are defined. The 1-dimensional dynamical systems with single equilibrium are discussed first. The 1-dimensional dynamical systems with two and three equilibriums are discussed. Simple equilibriums and higher order equilibriums in 1-dimensional dynamical systems are analyzed, and herein a higher order equilibrium is an equilibrium with higher order singularity. The separatrix flow of equilibriums in 1-dimensional systems in phase space is illustrated for a better understanding of the global stability of equilibriums in 1-dimensional dynamical systems.
Albert C. J. Luo

Chapter 5. Low-Degree Polynomial Systems

Abstract
In this chapter, the global stability and bifurcation of equilibriums in low-degree polynomial systems are discussed. Appearing and switching bifurcations of simple and higher order equilibriums are discussed, and such bifurcations of equilibriums are not only for simple equilibriums but also for higher order equilibriums. The third-order sink and source bifurcations for simple equilibriums are presented. The third-order sink and source switching bifurcations for saddle and nodes are discovered, and the fourth-order upper saddle and lower saddle switching and appearing bifurcations are obtained for two second-order upper saddles and two second-order lower saddles, respectively. Graphical illustrations of global stability and bifurcations of equilibriums are presented.
Albert C. J. Luo

Chapter 6. (2m)th-Degree Polynomial Systems

Abstract
In this chapter, the global stability and bifurcations of equilibriums in the (2m)th-degree polynomial system are discussed for a better understanding of the complexity of bifurcations and stability of equilibriums. The appearing and switching bifurcations for simple equilibriums are presented, and the appearing and switching bifurcations for higher order equilibriums are discussed as well. The parallel appearing bifurcations, spraying-appearing bifurcations, and sprinkler-spraying-appearing bifurcations for simple and higher order equilibriums are presented. The antenna-switching bifurcations for simple and higher order equilibriums are discussed and the parallel straw-bundle-switching bifurcations and flower-bundle-switching bifurcations for simple and higher order equilibriums are presented as well.
Albert C. J. Luo

Chapter 7. (2m+1)th-Degree Polynomial Systems

Abstract
In this chapter, the global stability and bifurcations of equilibriums in the (2m+1)th-degree polynomial systems are discussed for a better understanding of the complexity of bifurcations and stability of equilibriums in such a (2m+1)th-degree polynomial system. The appearing and switching bifurcations are presented for simple equilibriums and higher-order equilibriums. The broom-appearing bifurcations, broom-spraying-appearing and broom-sprinkler-spraying-appearing bifurcations for simple and higher-order equilibriums are presented. The antenna-switching bifurcations for simple and higher-order equilibriums are discussed and the parallel straw-bundle-switching and flower-bundle-switching bifurcations for simple and higher order equilibriums are also presented.
Albert C. J. Luo

Chapter 8. Infinite-Equilibrium Systems

Abstract
In this chapter, dynamical systems with infinite equilibriums are discussed through the local analysis. A method for equilibriums in nonlinear dynamical systems is developed. The generalized normal forms of nonlinear dynamical systems at equilibriums are presented for a better understanding of singularity in nonlinear dynamical systems. The dynamics of infinite-equilibrium dynamical systems is discussed for the complexity and singularity of nonlinear dynamical systems. A few examples are presented for complexity and singularity in infinite-equilibrium systems.
Albert C. J. Luo

Backmatter

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