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 Hilberts 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.
Chapter 1 Two-dimensional Linear Dynamical Systems.
Chapter 2 Single-variable Quadratic Systems with a Self-univariate Quadratic Vector Field.
Chapter 3 Single-variable Quadratic Systems with a Non-self-univariate Quadratic Vector Field.
Chapter 4 Variable-independent quadratic systems.
Chapter 5 Variable-crossing univariate quadratic systems.
Chapter 6 Two-univariate product quadratic systems.
Chapter 7 Product-bivariate Quadratic Systems with Self-univariate Vector Fields.
Chapter 8 Product-bivariate Quadratic Systems with Non-self-univariate Vector Fields.
Prof. Albert C. J. Luo is a Distinguished Research Professor at the Department of Mechanical Engineering at Southern Illinois University Edwardsville, USA. He received his Ph.D. degree from the University of Manitoba, Canada, in 1995. His research focuses on nonlinear dynamics, nonlinear mechanics and nonlinear differential equations , and he has published over 40 books and more than 350 journal articles and conference papers in these fields. He received the Paul Simon Outstanding Scholar Award in 2008 and an ASME fellowship in 2007. He was an editor for Communications in Nonlinear Science and Numerical Simulation, and an associate editor for ASME Journal of Computational and Nonlinear Dynamics. He now serves as Co-editor of the Journal of Applied Nonlinear Dynamics and Editor of various book series, including Nonlinear Systems and Complexity and Nonlinear Physical Science.
His major contributions on nonlinear dynamical systems are:
A theory for stochastic and resonant layers in nonlinear Hamiltonian systems
A local theory and singularity for discontinuous dynamical systems
Flow barriers theory for discontinuous dynamical systems
Synchronization of continuous dynamical systems under specific constraints
Synchronization and companion of discrete dynamical systems
Analytical solutions of periodic motions in nonlinear systems
Discretization and implicit mapping dynamics in nonlinear dynamical systems
Periodic flows in time-delay systems
Memorized nonlinear dynamical systems
In addition, Luo developed accurate theories for nonlinear deformable-body dynamics, machine tool dynamics and others:
An approximate plate theory
A theory for soft structures
A nonlinear theory for beams and rods
Fluid-induced nonlinear structural vibration
A large damage theory for anisotropic materials
A generalized fractal theory
He has published over 350 peer-reviewed journal and conference papers. Luo has been an editor for the Journal Communications in Nonlinear Science and Numerical simulation, and the book series on Nonlinear Systems and Complexity (Springer), and Nonlinear Physical Science (Higher Education Press and Springer).
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