Luo Toward Analytical Chaos in Nonlinear Systems

1. Auflage 2014
ISBN: 978-1-118-88717-2
Verlag: John Wiley & Sons
Format: EPUB
Kopierschutz: Adobe DRM (»Systemvoraussetzungen)

E-Book, Englisch, 272 Seiten, E-Book

Exact analytical solutions to periodic motions in nonlineardynamical systems are almost not possible. Since the 18th century,one has extensively used techniques such as perturbation methods toobtain approximate analytical solutions of periodic motions innonlinear systems. However, the perturbation methods cannot providethe enough accuracy of analytical solutions of periodic motions innonlinear dynamical systems. So the bifurcation trees of periodicmotions to chaos cannot be achieved analytically. The authorhas developed an analytical technique that is more effective toachieve periodic motions and corresponding bifurcation trees tochaos analytically.
Toward Analytical Chaos in Nonlinear Systemssystematically presents a new approach to analytically determineperiodic flows to chaos or quasi-periodic flows in nonlineardynamical systems with/without time-delay. It covers themathematical theory and includes two examples of nonlinear systemswith/without time-delay in engineering and physics. From theanalytical solutions, the routes from periodic motions to chaos aredeveloped analytically rather than the incomplete numerical routesto chaos. The analytical techniques presented will provide abetter understanding of regularity and complexity of periodicmotions and chaos in nonlinear dynamical systems.
Key features:
* Presents the mathematical theory of analytical solutions ofperiodic flows to chaos or quasieriodic flows in nonlineardynamical systems
* Covers nonlinear dynamical systems and nonlinear vibrationsystems
* Presents accurate, analytical solutions of stable and unstableperiodic 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 acomprehensive reference for researchers and practitioners acrossengineering, mathematics and physics disciplines, and is also auseful source of information for graduate and senior undergraduatestudents in these areas.

Autoren/Hrsg.

Luo, Albert C. J.

Inhaltsverzeichnis

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

Über Autor(innen)

Professor Luo is currently a Distinguished ResearchProfessor at Southern Illinois University Edwardsville. He is aninternational renowned figure in the area of nonlinear dynamics andmechanics. For about 30 years, Dr. Luo's contributions onnonlinear dynamical systems and mechanics lie in (i) the localsingularity theory for discontinuous dynamical systems, (ii)Dynamical systems synchronization, (iii) Analytical solutions ofperiodic and chaotic motions in nonlinear dynamical systems, (iv)The theory for stochastic and resonant layer in nonlinearHamiltonian systems, (v) The full nonlinear theory for a deformablebody. Such contributions have been scattered into 13 monographs andover 200 peer-reviewed journal and conference papers. His newresearch results are changing the traditional thinking in nonlinearphysics and mathematics. Dr. Luo has served as an editor for theJournal "Communications in Nonlinear Science and Numericalsimulation", book series on Nonlinear Physical Science (HEP)and Nonlinear Systems and Complexity (Springer). Dr. Luo is theeditorial member for two journals (i.e., IMeCh E Part K Journal ofMultibody Dynamics and Journal of Vibration and Control). He alsoorganized over 30 international symposiums and conferences onDynamics and Control.