Albert C. J. Luo 
Toward Analytical Chaos in Nonlinear Systems 

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.

Table des matières

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

A propos de l’auteur

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., IMe Ch 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.