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1 Introduction 1

1.1 Discontinuous systems 1

1.2 Book layout 5

References 6

2 Flow Switchability 9

2.1 Discontinuous dynamic systems 9

2.2 G-functions 11

2.3 Passable flows 15

2.4 Non-passable flows 19

2.5 Tangential flows 28

2.6 Switching bifurcations 39

References 52

3 Transversality and Sliding Phenomena 55

3.1 A controlled system 55

3.2 Transversality conditions 57

3.3 Mappings and predictions 60

3.4 Periodic and chaotic motions 67

References 75

4 A Frictional Oscillator on Time-varying Belt 77

4.1 Mechanical model 77

4.2 Analytical conditions 80

4.2.1 Equations of motion 80

4.2.2 Passable flows to boundary 83

4.2.3 Sliding flows on boundary 85

4.24 Grazing flows to boundary 89

4.3 Generic mappings and force product criteria 91

4.3.1 Generic mappings 91

4.3.2 Sliding flows and fragmentation 93

4.3.3 Grazing flows 96

4.4 Periodic motions 98

4.4.1 Mapping structures 98

4.4.2 Illustrations 100

4.5 Numerical simulations 112

References 113

5 Two Oscillators with Impacts and Stick 115

5.1 Physical problem 115

5.1.1 Introduction to problem 115

5.1.2 Equations of motion 117

5.2 Domains and vector fields 119

5.2.1 Absolute motion description 119

5.2.2 Relative motion description 125

5.3 Mechanism of stick and grazing 127

5.3.1 Analytical conditions 128

5.3.2 Physical interpretation 133

5.4 Mapping structures and motions 134

5.4.1 Switching sets and basic mappings 134

5.4.2 Mapping equations 137

5.4.3 Mapping structures 141

5.4.4 Bifurcation scenario 144

5.5 Periodic motion prediction 144

5.5.1 Approach 145

5.5.2 Impacting chatter 147

5.5.3 Impacting chatter with stick 150

5.5.4 Parameter maps 152

5.6 Numerical illustrations 153

5.6.1 Impacting chatter 153

5.6.2 Impacting chatter with stick 155

5.6.3 Further illustrations 158

References 160

6 Dynamical Systems with Frictions 161

6.1 Problem statement 161

6.2 Switching and stick motions 164

6.2.1 Equations of motion 164

6.2.2 Analytical conditions 166

6.3 Periodic motions 172

6.3.1 Switching planes and mappings 172

6.3.2 Mapping structures and motions 175

6.3.3 Bifurcation scenario 178

6.4 Numerical illustrations 181

6.4.1 Periodic motion without stick 181

6.4.2 Periodic motion with stick 184

6.4.3 Periodic motion with stick only 189

References 189

7 Principles for System Interactions 191

7.1 Two dynamical systems 191

7.1.1 Dynamical systems with interactions 191

7.1.2 Discontinuous description 195

7.1.3 Resultant dynamical systems 196

7.2 Fundamental interactions 199

7.3 Interactions with singularity 206

7.4 Interactions with corner singularity 210

References 215

Appendix 217

A.1 Basic solution 217

A.2 Stability and bifurcation 219

Index 221