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Transport Laws and Multi-valued Vector Fields

In: Discontinuous Dynamical Systems

Author

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  • Albert C. J. Luo

    (Southern Illinois University Edwardsville, Department of Mechanical and Industrial Engineering)

Abstract

In this Chapter, a classification of discontinuity in discontinuous dynamical systems will be discussed first. To discuss the singularity to the boundary, the grazing and inflexional singular sets on the boundary will be presented, and the real and imaginary singular sets will be also discussed. With permanent flow barriers, the forbidden boundary and the boundary channel will be presented. The forbidden boundary will not allow any flows passing through the boundary, and the boundary channel will not allow any boundary flows getting into the corresponding domains. Further, the domain and boundary classification will be addressed. Sink and source domains will be discussed. Similarly, the sink and source boundary will be also presented. To make flows continue in discontinuous dynamical systems, transport laws are needed for C 0-discontinuity, flow barriers, isolated domains and boundary channels. Multi-valued vector fields in a single domain will be introduced. With the simplest transport law (i.e., the switching rule), bouncing flows on the boundary will be presented, and extendable flows will be discussed as well. A controlled piecewise linear system will be presented as an application, and vector fields on both sides of the boundary will be switched at the boundary. Bouncing flows will be illustrated in such a controlled piecewise linear system.

Suggested Citation

  • Albert C. J. Luo, 2012. "Transport Laws and Multi-valued Vector Fields," Springer Books, in: Discontinuous Dynamical Systems, chapter 0, pages 259-356, Springer.
    • Handle: RePEc:spr:sprchp:978-3-642-22461-4_5
    DOI: 10.1007/978-3-642-22461-4_5
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