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Discontinuity and Local Singularity

  • Albert C. J. Luo
Chapter
Part of the Nonlinear Systems and Complexity book series (NSCH, volume 3)

Abstract

In this chapter, a general theory for the passability of a flow to a specific boundary in discontinuous dynamical systems will be discussed. The concepts of real and imaginary flows will be introduced. The G-functions for discontinuous dynamical systems will be presented to describe the passability of a flow to the boundary. Based on the G-function, the passability of a flow from a domain to an adjacent one will be discussed. With the concepts of real and imaginary flows, the full- and half-sink and source flows to the boundary will be discussed as well. A flow to the boundary in a discontinuous dynamical system can be passable or non-passable. Thus, all possible switching bifurcations between the passable and non-passable flows will be presented.

References

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    Luo ACJ (2011) Discontinuous dynamical systems. HEP-Springer, HeidelbergGoogle Scholar
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    Luo ACJ (2008) On the differential geometry of flows in nonlinear dynamic systems. ASME J Comput Nonlinear Dyn 3:021104-1–021104-10Google Scholar
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    Luo ACJ (2008) Global transversality, resonance and chaotic dynamics. World Scientific, SingaporeMATHCrossRefGoogle Scholar
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    Luo ACJ (2005) Imaginary, sink and source flows in the vicinity of the separatrix of non-smooth dynamical systems. J Sound Vib 285:443–456MATHCrossRefGoogle Scholar
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    Luo ACJ (2006) Singularity and dynamics on discontinuous vector fields. Elsevier, AmsterdamMATHGoogle Scholar
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    Luo ACJ (2005) A theory for non-smooth dynamic systems on the connectable domains. Commun Nonlinear Sci Numer Simul 10:1–55MathSciNetMATHCrossRefGoogle Scholar
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    Luo ACJ (2008) A theory for flow switchability in discontinuous dynamical systems. Nonlinear Anal Hybrid Syst 2(4):1030–1061MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2013

Authors and Affiliations

  • Albert C. J. Luo
    • 1
  1. 1.School of EngineeringSouthern Illinois University EdwardsvilleEdwardsvilleUSA

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