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.

Chapter 1 Introduction 1

1.1. Discrete and discontinuous systems 1

1.1.1 Discrete dynamical systems 2

1.1.2 Discontinuous dynamical systems 4

1.2 Fermi oscillator and impact problems 8

1.3 book layout 10

References 12

Chapter 2 Nonlinear Discrete Systems 19

2.1 Defintions 19

2.2 Fixed points and stability 21

2.3 Stability switching theory 34

2.4. Bifurcation theory 50

References 59

Chapter 3 Complete Dynamics and Fractality 61

3.1 Complete dynamics of discrete systems 61

3.2 Routes to chaos 69

3.2.1 One-dimensional maps 69

3.2.2 Two-dimensional maps 73

3.3 Complete Dynamics of Henon map 75

3.4 Simliarity and Multifractals 81

3.4.1 Similar Structures in period doubling 81

3.4.2 Fractality of chaos via PD bifurcation 86

3.4.3 An example 86

3.5 Complete dynamics of Logistic map 93

References 107

Chapter 4 Discontinuous Dynamical Systems 109

4.1 Basic concepts 109

4.2 G-functions 112

4.3 Passable flows 116

4.4 Non-passable flows 121

4.5 Grazing flows 135

4.6 Flow switching bifucations 149

References 162

Chapter 5 Nonlinear Dynamics of Bouncing Balls 163

5.1 Analytical dynamics of bouncing balls 163

5.1.1 Periodic motions 165

5.1.1 Stability and bifurcations 168

5.1.3 Numerical illustrations 175

5.2 Period-m motions 180

5.3 Complex dynamics 187

5.4 Complex periodic motions 192

References 200

Chapter 6 Complex Dynamics of Impact Pairs 201

6.1 Impact pairs 201

6.2 Analytical, simplest periodic motions 205

6.3 Possible impact notion sequences 216

6.4 Grazing dynamics and stick motions 220

6.5 Mapping structures and periodic motions 228

6.6 Stabilityand bifurcation 232

References 242

Chapter 7 Nonlinear Dynamics of Fermi Oscillators 243

7.1 Mapping dynamics 243

7.2 A Fermi oscillator 249

7.2.1 Absolute description 251

7.2.2 Relative description 257

7.3 Analytical conditions 258

7.4 Mapping structures and motions 260

7.4.1 Switching sets and generic mappings 260

7.4.2 Motions with mapping structures 263

7.4.3 Periodic motion and local stability 265

7.5 Predictions and similations 268

7.5.1 Bifurcation scenarios 268

7.5.2 Analytical predictions 271

7.5.3 Numberical illustractions 278

7.6 Appendix 291

References 295

Subject index 297

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