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Analytical Routes to Chaos in Nonlinear Engineering

Analytical Routes to Chaos in Nonlinear Engineering

Albert C. J. Luo

ISBN: 978-1-118-88392-1

May 2014

280 pages



Nonlinear problems are of interest to engineers, physicists and mathematicians and many other scientists because most systems are inherently nonlinear in nature. As nonlinear equations are difficult to solve, nonlinear systems are commonly approximated by linear equations. This works well up to some accuracy and some range for the input values, but some interesting phenomena such as chaos and singularities are hidden by linearization and perturbation analysis. It follows that some aspects of the behavior of a nonlinear system appear commonly to be chaotic, unpredictable or counterintuitive. Although such a chaotic behavior may resemble a random behavior, it is absolutely deterministic.

Analytical Routes to Chaos in Nonlinear Engineering discusses analytical solutions of periodic motions to chaos or quasi-periodic motions in nonlinear dynamical systems in engineering and considers engineering applications, design, and control. It systematically discusses complex nonlinear phenomena in engineering nonlinear systems, including the periodically forced Duffing oscillator, nonlinear self-excited systems, nonlinear parametric systems and nonlinear rotor systems. Nonlinear models used in engineering are also presented and a brief history of the topic is provided.

Key features:

  • Considers engineering applications, design and control
  • Presents analytical techniques to show how to find the periodic motions to chaos in nonlinear dynamical systems
  • Systematically discusses complex nonlinear phenomena in engineering nonlinear systems
  • Presents extensively used nonlinear models in engineering

Analytical Routes to Chaos in Nonlinear Engineering is a practical 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 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