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Chapter 1 Introduction 1

1.1．A brief history 2

1.2．Book layout 3

References 6

Chapter 2 Introduction to Flow Passability 9

2.1．Domain accessibility 9

2.2．Discontinuous dynamical systems 11

2.3．Flow passability 12

2.4．Grazing flows 22

2.5．Switching bifurcations of passable flows 31

2.6．Switching bifurcations of non-passable flows 44

2.7．An application：A frictional oscillator 56

2.7.1．Grazing phenomena 59

2.7.2．Sliding motion 66

References 76

Chapter 3 Singularity and Flow Passability 77

3.1．Real and imaginary flows 77

3.2．G-functions and vector field decomposition 82

3.3．Passable flows 91

3.4 Non-passable flows 99

3.5．Grazing flows 114

3.6．Flow switching bifurcations 130

3.7．First integral quantity increments 150

3.8．An example 152

3.8.1．Conditions for sliding and grazing 153

3.8.2．Periodic motions 156

3.9．Concluding remarks 164

References 164

Chapter 4 Flow Barriers and Switchability 165

4.1．Flow barriers for passable flows 165

4.1.1．Coming flow barriers 167

4.1.2．Leaving flow barriers 183

4.1.3．Passable flows with both flow barriers 190

4.2．Flow barriers for sink flows 199

4.3．Flow barriers for source flows 211

4.3.1．Boundary flow barriers 212

4.3.2．Leaving flow barriers 226

4.4．Sink flows with flow barriers 238

4.5．An application 242

4.5.1．Switchability conditions 244

4.5.2．Illustrations 250

4.6．Concluding remarks 257

References 258

Chapter 5 Transport Laws and Multi-valued Vector Fields 259

5.1．Discontinuity classification 259

5.2．Singular sets on boundary 262

5.3．Forbidden boundary and boundary channels 290

5.3.1．Forbidden boundary 290

5.3.2．Boundary channels 292

5.4．Domain and boundary classifications 298

5.4.1．Domain classification 301

5.4.2．Boundary classifications 319

5.5．Transport laws 325

5.6．Multi-valued vector fields and bouncing flows 333

5.6.1．Bouncing flows 336

5.6.2．Extended passable flows 340

5.7．A controlled piecewise linear system 345

5.7.1．Passable and bouncing conditions 345

5.7.2．Illustrations 351

References 356

Chapter 6 Switchability and Attractivity of Domain Flows 357

6.1．Dynamical systems on edges 357

6.2．Edge classification and mirror domains 363

6.3．Domain flow properties to convex edges 370

6.4．Domain flow switchability to convex edges 377

6.5．Transverse grazing passability to concave edges 389

6.6．Domains flow attractivity 403

6.6.1．Attractivity to boundary 404

6.6.2．Attractivity to edge 412

6.7．Multi-valued vector fields switching at edges 432

6.7.1．Bouncing domain flows at edges 434

6.7.2．Extended passable domain flows to edges 442

References 446

Chapter 7 Dynamics and Singularity of Boundary Flows 447

7.1．Boundary flow properties 447

7.2．Boundary flow switchability 455

7.3．Switchability of boundary and domain flows 480

7.4．Boundary flow attractivity 504

7.5．Boundary dynamics with multi-valued vector fields 509

7.5.1．Bouncingboundary flows 513

7.5.2．Extended passable boundary flows 518

References 520

Chapter 8 Edge Dynamics and Switching Complexity 521

8.1．Edge flows 521

8.2．Edge flow switchability 529

8.3．Edge flow attractivity 551

8.4．Edge dynamics with multi-valued vector fields 576

8.4.1．Bouncing edge flows 580

8.4.2．Extended passable edge flows 588

8.5．A frictional oscillator with two-degrees of freedom 594

8.5.1．Domains,edges and vector fields 596

8.5.2．Analytical conditions 599

8.5.3．Mapping structures and numerical illustrations 608

References 622

Chapter 9 Dynamical System Interactions 623

9.1．Introduction to system interactions 623

9.1.1．System interactions 623

9.1.2．Discontinuous description 625

9.1.3．Resultant dynamical systems 629

9.2．Fundamental interactions 633

9.3．Interactions with singularity 641

9.4．Interactions at edges 645

9.5．Application to system synchronization 649

9.5.1．Discontinuous description 650

9.5.2．Flow switchability on boundaries 654

9.5.3．Synchronization invariant sets and mechanism 658

9.5.4．Illustrations for synchronization 661

References 683

Subject Index 685