Discontinuity and Local Singularity

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


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.


  1. 1.
    Luo ACJ (2011) Discontinuous dynamical systems. HEP-Springer, HeidelbergGoogle Scholar
  2. 2.
    Luo ACJ (2008) On the differential geometry of flows in nonlinear dynamic systems. ASME J Comput Nonlinear Dyn 3:021104-1–021104-10Google Scholar
  3. 3.
    Luo ACJ (2008) Global transversality, resonance and chaotic dynamics. World Scientific, SingaporeMATHCrossRefGoogle Scholar
  4. 4.
    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
  5. 5.
    Luo ACJ (2006) Singularity and dynamics on discontinuous vector fields. Elsevier, AmsterdamMATHGoogle Scholar
  6. 6.
    Luo ACJ (2005) A theory for non-smooth dynamic systems on the connectable domains. Commun Nonlinear Sci Numer Simul 10:1–55MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    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

Personalised recommendations