Skip to main content
SpringerLink
Log in

Limit and Morse Sets for Deterministic Hybrid Systems

  • Published:
Qualitative Theory of Dynamical Systems Aims and scope Submit manuscript

Abstract

The term “hybrid system” refers to a continuous time dynamical system that undergoes Markovian perturbations at discrete time intervals. In this paper, we find that under the right formulation, a hybrid system can be treated as a dynamical system on a compact space. This allows us to study its limit sets. We examine the Morse decompositions of hybrid systems, find a sufficient condition for the existence of a non-trivial Morse decomposition, and study the Morse sets of such a decomposition. Finally, we consider the case in which the Markovian perturbations are small, showing that trajectories in a hybrid system with small perturbations behave similarly to those of the unperturbed dynamical system.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3

Similar content being viewed by others

References

  1. Ackerman, J., Ayers, K., Beltran, E.J., Bonet, J., Devin, L., Rudelius, T.: A Behavioral Characterization of Discrete Time Dynamical Systems using Directed Graphs. Iowa State University, Ames (2011)

    Google Scholar 

  2. Ayala-Hoffmann, J., Corbin, P., McConville, K., Colonius, F., Kliemann, W., Peters, J.: Morse Decompositions, Attractors and Chain Recurrence. Iowa State University, Ames (2007)

  3. Colonius, F., Kliemann, W.: The Dynamics of Control. Birkhäuser, Boston (2000)

    Book  Google Scholar 

  4. Garcia, X., Kunze, J., Rudelius, T., Sanchez, A., Shao, S., Speranza, E., Vidden, C.: Invariant Measures for Hybrid Stochastic Systems. Iowa State University, Ames (2011)

    Google Scholar 

Download references

Acknowledgments

We wish to recognize Chad Vidden for his helpful discussions and Professor Wolfgang Kliemann for his instruction and guidance. We would like to thank the Department of Mathematics at Iowa State University for their hospitality during the completion of this work. In addition, we’d like to thank Iowa State University, Alliance, and the National Science Foundation for their support of this research.

Author information

Authors and Affiliations

  1. Department of Mathematics, Iowa State University, Ames, IA, 50014, USA

    Kimberly Ayers

  2. Department of Mathematics, University of Minnesota, Twin Cities, Minneapolis, MN, 55455, USA

    Xavier Garcia

  3. Department of Mathematics, Saint Mary’s College of Maryland, St Marys City, MD, 20686, USA

    Jennifer Kunze

  4. Department of Mathematics, Cornell University, Ithaca, NY, 14850, USA

    Thomas Rudelius

  5. Department of Mathematics, Arizona State University, Tempe, AZ, 85281, USA

    Anthony Sanchez

  6. Department of Mathematics, Iowa State University, Ames, IA, 50014, USA

    Sijing Shao

  7. Department of Mathematics, Carroll College, Helena, MT, 59625, USA

    Emily Speranza

Authors
  1. Kimberly Ayers
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Xavier Garcia
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Jennifer Kunze
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Thomas Rudelius
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Anthony Sanchez
    View author publications

    You can also search for this author in PubMed Google Scholar

  6. Sijing Shao
    View author publications

    You can also search for this author in PubMed Google Scholar

  7. Emily Speranza
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Thomas Rudelius.

Additional information

K. Ayers was supported by DMS 0502354. X. Garcia was supported by DMS 0750986 and DMS 0502354. J. Kunze was supported by DMS 0750986. T. Rudelius was supported by DMS 0750986. A. Sanchez was supported by DMS 0750986 and DMS 0502354. S. Shao was supported by Iowa State University. E. Speranza was supported by DMS 0750986.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Ayers, K., Garcia, X., Kunze, J. et al. Limit and Morse Sets for Deterministic Hybrid Systems. Qual. Theory Dyn. Syst. 12, 335–360 (2013). https://doi.org/10.1007/s12346-012-0092-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12346-012-0092-y

Keywords

Search

Navigation