Analytical Routes to Chaos in Nonlinear Engineering

Albert C. 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.

• Preface ix1 Introduction 11.1 Analytical Methods 11.1.1 Lagrange Standard Form 11.1.2 Perturbation Methods 21.1.3 Method of Averaging 51.1.4 Generalized Harmonic Balance 81.2 Book Layout 242 Bifurcation Trees in Duffing Oscillators 252.1 Analytical Solutions 252.2 Period-1 Motions to Chaos 322.2.1 Period-1 Motions 332.2.2 Period-1 to Period-4 Motions 352.2.3 Numerical Simulations 522.3 Period-3 Motions to Chaos 572.3.1 Independent, Symmetric Period-3 Motions 572.3.2 Asymmetric Period-3 Motions 642.3.3 Period-3 to Period-6 Motions 712.3.4 Numerical Illustrations 823 Self-Excited Nonlinear Oscillators 873.1 van del Pol Oscillators 873.1.1 Analytical Solutions 873.1.2 Frequency-Amplitude Characteristics 973.1.3 Numerical Illustrations 1103.2 van del Pol-Duffing Oscillators 1143.2.1 Finite Fourier Series Solutions 1143.2.2 Analytical Predictions 1303.2.3 Numerical Illustrations 1434 Parametric Nonlinear Oscillators 1514.1 Parametric, Quadratic Nonlinear Oscillators 1514.1.1 Analytical Solutions 1514.1.2 Analytical Routes to Chaos 1564.1.3 Numerical Simulations 1694.2 Parametric Duffing Oscillators 1864.2.1 Formulations 1864.2.2 Parametric Hardening Duffing Oscillators 1945 Nonlinear Jeffcott Rotor Systems 2095.1 Analytical Periodic Motions 2095.2 Frequency-Amplitude Characteristics 2255.2.1 Period-1 Motions 2265.2.2 Analytical Bifurcation Trees 2315.2.3 Independent Period-5 Motion 2395.3 Numerical Simulations 246References 261Index 265