chapter 1 introduction
　1.1. a brief history
　1.2. book layout
　references
chapter 2 introduction to flow passability
　2.1. domain accessibility
　2.2. discontinuous dynamical systems
　2.3. flow passability
　2.4. grazing flows
　2.5. switching bifurcations of passable flows
　2.6. switching bifurcations of non-passable flows
　2.7. an application: a frictional oscillator
　2.7.1. grazing phenomena
　2.7.2. sliding motion
　references

chapter 1 introduction
　1.1. a brief history
　1.2. book layout
　references
chapter 2 introduction to flow passability
　2.1. domain accessibility
　2.2. discontinuous dynamical systems
　2.3. flow passability
　2.4. grazing flows
　2.5. switching bifurcations of passable flows
　2.6. switching bifurcations of non-passable flows
　2.7. an application: a frictional oscillator
　2.7.1. grazing phenomena
　2.7.2. sliding motion
　references
chapter 3 singularity and flow passability
　3.1. real and imaginary flows
　3.2. g-functions and vector field decomposition
　3.3. passable flows
　3.4. non-passable flows
　3.5. grazing flows
　3.6. flow switching bifurcations
　3.7. first integral quantity increments
　3.8. an example
　3.8.1. conditions for sliding and grazing
　3.8.2. periodic motions
　3.9. concluding remarks
　references
chapter 4 flow barriers and switchability
　4.1. flow barriers for passable flows
　4.1.1. coming flow barriers
　4.1.2. leaving flow barriers
　4.1.3. passable flows with both flow barriers
　4.2. flow barriers for sink flows
　4.3. flow barriers for source flows
　4.3.1 boundary flow barriers
　4.3.2. leaving flow barriers
　4.4. sink flows with flow barriers
　4.5. an application
　4.5.1. switchability conditions
　4.5.2. illustrations
　4.6. concluding remarks
　references
chapter 5 transport laws and multi-valued vector fields
　5.1. discontinuity classification
　5.2. singular sets on boundary
　5.3. forbidden boundary and boundary channels
　5.3.1. forbidden boundary
　5.3.2. boundary channels
　5.4. domain and boundary classifications
　5.4.1. domain classification
　5.4.2. boundary classifications
　5.5. transport laws
　5.6. multi-valued vector fields and bouncing flows
　5.6.1. bouncing flows
　5.6.2. extended passable flows
　5.7. a controlled piecewise linear system
　5.7.1. passable and bouncing conditions
　5.7.2. illustrations
　references
chapter 6 switchability and attractivity of domain flows
　6.1. dynamical systems on edges
　6.2. edge classification and mirror domains
　6.3. domain flow properties to convex edges
　6.4. domain flow switchability to convex edges
　6.5. transverse grazing passability to concave edges
　6.6. domains flow attractivity
　6.6.1. attractivity to boundary
　6.6.2. attractivity to edge
　6.7. multi-valued vector fields switching at edges
　6.7.1. bouncing domain flows at edges
　6.7.2. extended passable domain flows to edges
　references
chapter 7 dynamics and singularity of boundary flows
　7.1. boundary flow properties
　7.2. boundary flow switchability
　7.3. switchability of boundary and domain flows
　7.4. boundary flow attractivity
　7.5. boundary dynamics with multi-valued vector fields
　7.5.1. bouncing boundary flows
　7.5.2. extended passable boundary flows
　references
chapter 8 edge dynamics and switching complexity
　8.1. edge flows
　8.2. edge flow switchability
　8.3. edge flow attractivity
　8.4. edge dynamics with multi-valued vector fields
　8.4.1. bouncing edge flows
　8.4.2. extended passable edge flows
　8.5. a frictional oscillator with two-degrees of freedom
　8.5.1. domains, edges and vector fields
　8.5.2. analytical conditions
　8.5.3. mapping structures and numerical illustrations
　references
chapter 9 dynamical system interactions
　9.1. introduction to system interactions
　9.1.1. system interactions
　9.1.2. discontinuous description
　9.1.3. resultant dynamical systems
　9.2. fundamental interactions
　9.3. interactions with singularity
　9.4. interactions at edges
　9.5. application to system synchronization
　9.5 1. discontinuous description
　9.5.2. flow switchability on boundaries
　9.5.3. synchronization invariant sets and mechanism
　9.5.4. illustrations for synchronization
　references
　subject index