E-Book Overview
The book presents the recent achievements on bifurcation studies of nonlinear dynamical systems. The contributing authors of the book are all distinguished researchers in this interesting subject area. The first two chapters deal with the fundamental theoretical issues of bifurcation analysis in smooth and non-smooth dynamical systems. The cell mapping methods are presented for global bifurcations in stochastic and deterministic, nonlinear dynamical systems in the third chapter. The fourth chapter studies bifurcations and chaos in time-varying, parametrically excited nonlinear dynamical systems. The fifth chapter presents bifurcation analyses of modal interactions in distributed, nonlinear, dynamical systems of circular thin von Karman plates. The theories, methods and results presented in this book are of great interest to scientists and engineers in a wide range of disciplines. This book can be adopted as references for mathematicians, scientists, engineers and graduate students conducting research in nonlinear dynamical systems. ?· New Views for Difficult Problems?· Novel Ideas and Concepts ?· Hilbert's 16th Problem?· Normal Forms in Polynomial Hamiltonian Systems ?· Grazing Flow in Non-smooth Dynamical Systems?· Stochastic and Fuzzy Nonlinear Dynamical Systems?· Fuzzy Bifurcation?· Parametrical, Nonlinear Systems?· Mode Interactions in nonlinear dynamical systems
E-Book Content
Preface Modeling complex natural and social dynamic phenomena has been an intellectual pursuit of mankind for centuries. Aided with modern computers, we can now develop global nonlinear models of dynamical systems and obtain solutions that defy traditional, linear thinking. Nonlinear sciences, chaotic dynamics and bifurcations have been active research topics for decades, and continue to stimulate interests from researchers all over the world. The research has found applications in a wide range of dynamical systems including civil, mechanical, electrical, control, biological, ecological, economic and financial systems. We are proud to present to readers this volume containing recent research results on bifurcation studies of complex nonlinear dynamical systems. The book contains five chapters describing the state of the art of bifurcation studies of nonlinear systems. The contributing authors of the book are all active researchers in this interesting subject area. The first two chapters deal with theoretical issues of bifurcation analysis in smooth and nonsmooth dynamical systems. The third chapter presents a numerical method for global bifurcations. The fourth chapter studies bifurcations and chaos in time-varying, parametrically excited nonlinear dynamical systems. The fifth chapter presents bifurcation analyses of modal interactions in distributed, nonlinear, dynamical systems of circular thin von Karman plates. A brief description of each chapter is presented in the following. Chapter 1 of the book considers bifurcations of nonlinear dynamical systems governed by ordinary differential equations, difference equations and time delayed differential equations. In-depth investigations of limit cycles and chaos, Hopf bifurcation control, chaos control and chaos synchronization are presented. In particular, a method unifying the center manifold theory and the method of normal forms is developed. Efficient methodologies and software based on symbolic programming are presented for normal form computations. An efficient method based on the concept of modes and “mode competition” is presented to identify the parameter values leading to chaos. The method provides necessary conditions for the existence of chaos, and can be used to systematically generate chaotic systems. Many interesting examples are included in the chapter, such as bifurcations of limit cycles, Hopf and double Hopf bifurcations, Hopf bifurcation control, chaos control and chaos synchro