Toward Analytical Chaos in Nonlinear Systems

Exact analytical solutions to periodic motions in nonlinear dynamical systems are almost not possible. Since the 18th century, one has extensively used techniques such as perturbation methods to obtain approximate analytical solutions of periodic motions in nonlinear systems. However, the perturbation methods cannot provide the enough accuracy of analytical solutions of periodic motions in nonlinear dynamical systems. So the bifurcation trees of periodic motions to chaos cannot be achieved analytically.  The author has developed an analytical technique that is more effective to achieve periodic motions and corresponding bifurcation trees to chaos analytically. Toward Analytical Chaos in Nonlinear Systems systematically presents a new approach to analytically determine periodic flows to chaos or quasi–periodic flows in nonlinear dynamical systems with/without time–delay.  It covers the mathematical theory and includes two examples of nonlinear systems with/without time–delay in engineering and physics. From the analytical solutions, the routes from periodic motions to chaos are developed analytically rather than the incomplete numerical routes to chaos.  The analytical techniques presented will provide a better understanding of regularity and complexity of periodic motions and chaos in nonlinear dynamical systems. Key features: Presents the mathematical theory of analytical solutions of periodic flows to chaos or quasieriodic flows in nonlinear dynamical systems Covers nonlinear dynamical systems and nonlinear vibration systems Presents accurate, analytical solutions of stable and unstable periodic flows for popular nonlinear systems Includes two complete sample systems Discusses time–delayed, nonlinear systems and time–delayed, nonlinear vibrational systems Includes real world examples Toward Analytical Chaos in Nonlinear Systems is a comprehensive reference for researchers and practitioners across engineering, mathematics and physics disciplines, and is also a useful source of information for graduate and senior undergraduate students in these areas.

Preface Chapter 1 Introduction 1 1.1 Brief history 1 1.2 boook layout 5 Chapter 2 Nonlinear Dynamical Systems 7 2.1 Continuous systems 7 2.2 Equilibrium and stability 10 2.3 Bifurcation and stability switching 20 2.3.1 Stability and switching 21 2.3.2 Bifurcations 32 Chapter 3 An Analytical Method for Periodic Flows 39 3.1 Nonlinear dynamical sysetms 39 3.1.1 Autonomous nonlinear systems 39 3.1.2 Non–autonomous nonlinear systems 51 3.2 Nonlinear vibration systems 55 3.2.1 Free vibration systems 56 3.2.2 Periodically excited vibration systems 70 3.3 Time–delayed nonlinear systems 75 3.3.1 Autonomous time–delayed nonlinear systems 75 3.3.2 Non–authonomous, time–delayed nonlinear systems 95 3.4 Time–delayed nonlinear vibration systems 96 3.4.1 Time–delayed, free vibration systems 96 3.4.2 Periodically excited vibration systems with time–delay 114 Chapter 4  Analytical Periodic to Quasi–periodic Flows 121 4.1 Nonlinear dynamical sysetms 121 4.2 Nonlinear vibration systems 137 4.3 Time–delayed nonlinear systems 147 4.4 Time–delayed, nonlinear vibration systems 160 Chapter 5 Quadratic Nonlinear Oscillators 175 5.1 Period–1 motions 175 5.1.1 Analytical solutions 175 5.1.2 Analytical predictions 180 5.1.3 Numerical illustrations 185 5.2 Period–m motions 191 5.2.1 Analytical solutions 196 5.2.2 Analytical bifurcation trees 200 5.2.3 Numiercal illustrations 185 5.3 Arbitrary periodic forcing 235 Chapter 6 Time–delayed Nonlinear Oscillators 237 6.1 Analytical solutions of period–m moitons 237 6.2 Analytical bifurcation trees 257 6.3 Illustrations of periodic motions 265 References 273 Subject index 277

Professor Luo is currently a Distinguished Research Professor at Southern Illinois University Edwardsville. He is an international renowned figure in the area of nonlinear dynamics and mechanics. For about 30 years, Dr. Luo’s contributions on nonlinear dynamical systems and mechanics lie in (i) the local singularity theory for discontinuous dynamical systems, (ii) Dynamical systems synchronization, (iii) Analytical solutions of periodic and chaotic motions in nonlinear dynamical systems, (iv) The theory for stochastic and resonant layer in nonlinear Hamiltonian systems, (v) The full nonlinear theory for a deformable body. Such contributions have been scattered into 13 monographs and over 200 peer–reviewed journal and conference papers. His new research results are changing the traditional thinking in nonlinear physics and mathematics. Dr. Luo has served as an editor for the Journal “Communications in Nonlinear Science and Numerical simulation”, book series on Nonlinear Physical Science (HEP) and Nonlinear Systems and Complexity (Springer). Dr. Luo is the editorial member for two journals (i.e., IMeCh E Part K Journal of Multibody Dynamics and Journal of Vibration and Control). He also organized over 30 international symposiums and conferences on Dynamics and Control.