Albert C. 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.Yu Guo, Southern Illinois University, Edwardsville, USA.

"It provides a unified treatment on the topic with a sound theoretical base that is applicable to both continuous and discrete systems." (Zentralblatt MATH, 2016)

• Preface ix1 Introduction 11.1 Discrete and Discontinuous Systems 11.1.1 Discrete Dynamical Systems 11.1.2 Discontinuous Dynamical Systems 41.2 Fermi Oscillators and Impact Problems 71.3 Book Layout 92 Nonlinear Discrete Systems 112.1 Definitions 112.2 Fixed Points and Stability 132.3 Stability Switching Theory 222.4 Bifurcation Theory 383 Complete Dynamics and Fractality 473.1 Complete Dynamics of Discrete Systems 473.2 Routes to Chaos 543.2.1 One-Dimensional Maps 543.2.2 Two-Dimensional Systems 583.3 Complete Dynamics of the Henon Map 593.4 Similarity and Multifractals 643.4.1 Similar Structures in Period Doubling 653.4.2 Fractality of Chaos via PD Bifurcation 683.4.3 An Example 703.5 Complete Dynamics of Logistic Map 724 Discontinuous Dynamical Systems 854.1 Basic Concepts 854.2 G-Functions 884.3 Passable Flows 914.4 Non-Passable Flows 954.5 Grazing Flows 1074.6 Flow Switching Bifurcations 1195 Nonlinear Dynamics of Bouncing Balls 1315.1 Analytic Dynamics of Bouncing Balls 1315.1.1 Periodic Motions 1335.1.2 Stability and Bifurcation 1365.1.3 Numerical Illustrations 1405.2 Period-m Motions 1435.3 Complex Dynamics 1525.4 Complex Periodic Motions 1566 Complex Dynamics of Impact Pairs 1656.1 Impact Pairs 1656.2 Analytical, Simplest Periodic Motions 1686.2.1 Asymmetric Period-1 Motion 1706.2.2 Stability and Bifurcation 1726.2.3 Numerical Illustrations 1756.3 Possible Impact Motion Sequences 1796.4 Grazing Dynamics and Stick Motions 1826.5 Mapping Structures and Periodic Motions 1896.6 Stability and Bifurcation 1927 Nonlinear Dynamics of Fermi Oscillators 2037.1 Mapping Dynamics 2037.2 A Fermi Oscillator 2097.2.1 Absolute Description 2107.2.2 Relative Description 2127.3 Analytical Conditions 2147.4 Mapping Structures and Motions 2187.4.1 Switching Sets and Generic Mappings 2187.4.2 Motions with Mapping Structures 2217.4.3 Periodic Motions and Local Stability 2237.5 Predictions and Simulations 2267.5.1 Bifurcation Scenario 2267.5.2 Analytical Prediction 2297.5.3 Numerical Simulations 236Appendix 7.A 248References 253Index 259

Albert C. 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.Yu Guo, Southern Illinois University, Edwardsville, USA.

"It provides a unified treatment on the topic with a sound theoretical base that is applicable to both continuous and discrete systems." (Zentralblatt MATH, 2016)

• Preface ix1 Introduction 11.1 Discrete and Discontinuous Systems 11.1.1 Discrete Dynamical Systems 11.1.2 Discontinuous Dynamical Systems 41.2 Fermi Oscillators and Impact Problems 71.3 Book Layout 92 Nonlinear Discrete Systems 112.1 Definitions 112.2 Fixed Points and Stability 132.3 Stability Switching Theory 222.4 Bifurcation Theory 383 Complete Dynamics and Fractality 473.1 Complete Dynamics of Discrete Systems 473.2 Routes to Chaos 543.2.1 One-Dimensional Maps 543.2.2 Two-Dimensional Systems 583.3 Complete Dynamics of the Henon Map 593.4 Similarity and Multifractals 643.4.1 Similar Structures in Period Doubling 653.4.2 Fractality of Chaos via PD Bifurcation 683.4.3 An Example 703.5 Complete Dynamics of Logistic Map 724 Discontinuous Dynamical Systems 854.1 Basic Concepts 854.2 G-Functions 884.3 Passable Flows 914.4 Non-Passable Flows 954.5 Grazing Flows 1074.6 Flow Switching Bifurcations 1195 Nonlinear Dynamics of Bouncing Balls 1315.1 Analytic Dynamics of Bouncing Balls 1315.1.1 Periodic Motions 1335.1.2 Stability and Bifurcation 1365.1.3 Numerical Illustrations 1405.2 Period-m Motions 1435.3 Complex Dynamics 1525.4 Complex Periodic Motions 1566 Complex Dynamics of Impact Pairs 1656.1 Impact Pairs 1656.2 Analytical, Simplest Periodic Motions 1686.2.1 Asymmetric Period-1 Motion 1706.2.2 Stability and Bifurcation 1726.2.3 Numerical Illustrations 1756.3 Possible Impact Motion Sequences 1796.4 Grazing Dynamics and Stick Motions 1826.5 Mapping Structures and Periodic Motions 1896.6 Stability and Bifurcation 1927 Nonlinear Dynamics of Fermi Oscillators 2037.1 Mapping Dynamics 2037.2 A Fermi Oscillator 2097.2.1 Absolute Description 2107.2.2 Relative Description 2127.3 Analytical Conditions 2147.4 Mapping Structures and Motions 2187.4.1 Switching Sets and Generic Mappings 2187.4.2 Motions with Mapping Structures 2217.4.3 Periodic Motions and Local Stability 2237.5 Predictions and Simulations 2267.5.1 Bifurcation Scenario 2267.5.2 Analytical Prediction 2297.5.3 Numerical Simulations 236Appendix 7.A 248References 253Index 259

Albert C. 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.Yu Guo, Southern Illinois University, Edwardsville, USA.

"It provides a unified treatment on the topic with a sound theoretical base that is applicable to both continuous and discrete systems." (Zentralblatt MATH, 2016)

• Preface ix1 Introduction 11.1 Discrete and Discontinuous Systems 11.1.1 Discrete Dynamical Systems 11.1.2 Discontinuous Dynamical Systems 41.2 Fermi Oscillators and Impact Problems 71.3 Book Layout 92 Nonlinear Discrete Systems 112.1 Definitions 112.2 Fixed Points and Stability 132.3 Stability Switching Theory 222.4 Bifurcation Theory 383 Complete Dynamics and Fractality 473.1 Complete Dynamics of Discrete Systems 473.2 Routes to Chaos 543.2.1 One-Dimensional Maps 543.2.2 Two-Dimensional Systems 583.3 Complete Dynamics of the Henon Map 593.4 Similarity and Multifractals 643.4.1 Similar Structures in Period Doubling 653.4.2 Fractality of Chaos via PD Bifurcation 683.4.3 An Example 703.5 Complete Dynamics of Logistic Map 724 Discontinuous Dynamical Systems 854.1 Basic Concepts 854.2 G-Functions 884.3 Passable Flows 914.4 Non-Passable Flows 954.5 Grazing Flows 1074.6 Flow Switching Bifurcations 1195 Nonlinear Dynamics of Bouncing Balls 1315.1 Analytic Dynamics of Bouncing Balls 1315.1.1 Periodic Motions 1335.1.2 Stability and Bifurcation 1365.1.3 Numerical Illustrations 1405.2 Period-m Motions 1435.3 Complex Dynamics 1525.4 Complex Periodic Motions 1566 Complex Dynamics of Impact Pairs 1656.1 Impact Pairs 1656.2 Analytical, Simplest Periodic Motions 1686.2.1 Asymmetric Period-1 Motion 1706.2.2 Stability and Bifurcation 1726.2.3 Numerical Illustrations 1756.3 Possible Impact Motion Sequences 1796.4 Grazing Dynamics and Stick Motions 1826.5 Mapping Structures and Periodic Motions 1896.6 Stability and Bifurcation 1927 Nonlinear Dynamics of Fermi Oscillators 2037.1 Mapping Dynamics 2037.2 A Fermi Oscillator 2097.2.1 Absolute Description 2107.2.2 Relative Description 2127.3 Analytical Conditions 2147.4 Mapping Structures and Motions 2187.4.1 Switching Sets and Generic Mappings 2187.4.2 Motions with Mapping Structures 2217.4.3 Periodic Motions and Local Stability 2237.5 Predictions and Simulations 2267.5.1 Bifurcation Scenario 2267.5.2 Analytical Prediction 2297.5.3 Numerical Simulations 236Appendix 7.A 248References 253Index 259