Advertisement

Chain Recurrence in Graph Determined Hybrid Systems

  • Kimberly AyersEmail author
Article
  • 6 Downloads

Abstract

In this paper, we consider a family of dynamical systems on the same compact metric space. We then consider the dynamics given when the given flow shifts between these different flows at regular time intervals. We further require that shifts be allowed by a given directed graph. We then define a type of set, called a chain set, that exhibits many similar properties to chain transitive sets of flows. By considering the dynamics as a skew product flow, we are able to demonstrate that chain sets can be lifted to a chain transitive set if the given graph is complete.

Keywords

Chain recurrence Skew product flows Chain controllability Hybrid systems Chain transitivity Dynamical systems on directed graphs 

Notes

References

  1. 1.
    Ackerman J., Ayers K., Beltran E.J., Bonet J., Lu D., Rudelius T. A behavioral characterization of discrete time dynamical systems over directed graphs. Qualitative Theory of Dynamical Systems 2014;13:161–180.MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Alongi J., Nelson G. Recurrence and Topology, Graduate studies in mathematics. Providence: American Mathematical Society; 2007.zbMATHGoogle Scholar
  3. 3.
    Ayers K., Garcia X., Kunze J., Rudelius T., Sanchez A., Shao S., Speranza E. Limit and morse sets for deterministic hybrid systems. Qualitative Theory of Dynamical Systems 2013;12:335–360.MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Colonius F., Kliemann W. The Dynamics of Control, Systems & Control: Foundations & Applications. Boston: Birkhäuser; 2000.zbMATHGoogle Scholar
  5. 5.
    Colonius F., Kliemann W. Dynamical Systems and Linear Algebra: Graduate Studies in Mathematics. Providence: American Mathematical Society; 2014.CrossRefzbMATHGoogle Scholar
  6. 6.
    Conley C. Isolated Invariant Sets and the Morse Index. Providence: American Mathematical Society; 1978.CrossRefzbMATHGoogle Scholar
  7. 7.
    Katok A., Hasselblatt B. Introduction to the Modern Theory of Dynamical Systems, Encyclopedia of Mathematics and its Applications. Cambridge: Cambridge University Press; 1997.zbMATHGoogle Scholar
  8. 8.
    Poincaré H., Vol. 1-3. Les Méthodes Nouvelles de la Mécanique Céleste. New York: Gautier-Villars; 1892.Google Scholar
  9. 9.
    Robinson C. Dynamical Systems: Stability, Symbolic Dynamics, and Chaos,. Boca Raton: CRC Press; 1999.zbMATHGoogle Scholar
  10. 10.
    Shorten R., Wirth F., Mason O., Wulff K., King C. Stability criteria for switched and hybrid systems. SIAM Rev. 2007;49:545–592.MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Steen L., Seebach J. Counterexamples in Topology, Dover books on mathematics. USA: Dover Publications; 2013.Google Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of MathematicsPomona CollegeClaremontUSA

Personalised recommendations