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Toward Analytical Chaos in Nonlinear Systems

Toward Analytical Chaos in Nonlinear Systems

Albert C. J. Luo

ISBN: 978-1-118-88717-2

May 2014

272 pages



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 ix

1 Introduction 1

1.1 Brief History 1

1.2 Book Layout 4

2 Nonlinear Dynamical Systems 7

2.1 Continuous Systems 7

2.2 Equilibriums and Stability 9

2.3 Bifurcation and Stability Switching 17

2.3.1 Stability and Switching 17

2.3.2 Bifurcations 26

3 An Analytical Method for Periodic Flows 33

3.1 Nonlinear Dynamical Systems 33

3.1.1 Autonomous Nonlinear Systems 33

3.1.2 Non-Autonomous Nonlinear Systems 44

3.2 Nonlinear Vibration Systems 48

3.2.1 Free Vibration Systems 48

3.2.2 Periodically Excited Vibration Systems 61

3.3 Time-Delayed Nonlinear Systems 66

3.3.1 Autonomous Time-Delayed Nonlinear Systems 66

3.3.2 Non-Autonomous Time-Delayed Nonlinear Systems 80

3.4 Time-Delayed, Nonlinear Vibration Systems 85

3.4.1 Time-Delayed, Free Vibration Systems 85

3.4.2 Periodically Excited Vibration Systems with Time-Delay 102

4 Analytical Periodic to Quasi-Periodic Flows 109

4.1 Nonlinear Dynamical Systems 109

4.2 Nonlinear Vibration Systems 124

4.3 Time-Delayed Nonlinear Systems 134

4.4 Time-Delayed, Nonlinear Vibration Systems 147

5 Quadratic Nonlinear Oscillators 161

5.1 Period-1 Motions 161

5.1.1 Analytical Solutions 161

5.1.2 Frequency-Amplitude Characteristics 165

5.1.3 Numerical Illustrations 173

5.2 Period-m Motions 180

5.2.1 Analytical Solutions 180

5.2.2 Analytical Bifurcation Trees 184

5.2.3 Numerical Illustrations 206

5.3 Arbitrary Periodical Forcing 217

6 Time-Delayed Nonlinear Oscillators 219

6.1 Analytical Solutions 219

6.2 Analytical Bifurcation Trees 238

6.3 Illustrations of Periodic Motions 242

References 253

Index 257