Analytical Routes to Chaos in Nonlinear Engineering

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.

Preface ix

1 Introduction 1

1.1 Analytical Methods 1

1.1.1 Lagrange Standard Form 1

1.1.2 Perturbation Methods 2

1.1.3 Method of Averaging 5

1.1.4 Generalized Harmonic Balance 8

1.2 Book Layout 24

2 Bifurcation Trees in Duffing Oscillators 25

2.1 Analytical Solutions 25

2.2 Period-1 Motions to Chaos 32

2.2.1 Period-1 Motions 33

2.2.2 Period-1 to Period-4 Motions 35

2.2.3 Numerical Simulations 52

2.3 Period-3 Motions to Chaos 57

2.3.1 Independent, Symmetric Period-3 Motions 57

2.3.2 Asymmetric Period-3 Motions 64

2.3.3 Period-3 to Period-6 Motions 71

2.3.4 Numerical Illustrations 82

3 Self-Excited Nonlinear Oscillators 87

3.1 van del Pol Oscillators 87

3.1.1 Analytical Solutions 87

3.1.2 Frequency-Amplitude Characteristics 97

3.1.3 Numerical Illustrations 110

3.2 van del Pol-Duffing Oscillators 114

3.2.1 Finite Fourier Series Solutions 114

3.2.2 Analytical Predictions 130

3.2.3 Numerical Illustrations 143

4 Parametric Nonlinear Oscillators 151

4.1 Parametric, Quadratic Nonlinear Oscillators 151

4.1.1 Analytical Solutions 151

4.1.2 Analytical Routes to Chaos 156

4.1.3 Numerical Simulations 169

4.2 Parametric Duffing Oscillators 186

4.2.1 Formulations 186

4.2.2 Parametric Hardening Duffing Oscillators 194

5 Nonlinear Jeffcott Rotor Systems 209

5.1 Analytical Periodic Motions 209

5.2 Frequency-Amplitude Characteristics 225

5.2.1 Period-1 Motions 226

5.2.2 Analytical Bifurcation Trees 231

5.2.3 Independent Period-5 Motion 239

5.3 Numerical Simulations 246

References 261

Index 265