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A Behavioral Characterization of Discrete Time Dynamical Systems over Directed Graphs

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Abstract

Using a directed graph, a Markov chain can be treated as a dynamical system over a compact space of bi-infinite sequences, with a flow given by the left shift of a sequence. In this paper, we show that the Morse sets of the finest Morse decomposition on this space can be related to communicating classes of the directed graph by considering lifting the communicating classes to the shift space. Finally, we prove that the flow restricted to these Morse sets is chaotic.

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References

  1. Alongi, J., Nelson, G.: Recurrence and Topology. Amer. Math. Soc, Providence, RI (2007)

  2. Ayala, J., Corbin, P., McConville, K., Colonius, F., Kliemann, W., Peters, J.: Morse decompositions, attractors, and chain recurrence. Proyecciones 25, 79–110 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  3. Ayers, K., Garcia, X., et al.: Limit and Morse Sets for Deterministic Hybrid Systems. Qualitative Theory of Dynamical Systems (2012)

  4. Katok, A., Hasselblatt, B.: Introduction to the Modern Theory of Dynamical Systems. Cambridge University Press, Cambridge (1995)

  5. Robinson, C. : Dynamical Systems: Stability, Symbolic Dynamics, and Chaos. CRC Press, Boca Raton (1995)

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Acknowledgments

We would like to thank Professor Wolfgang Kliemann, Tracy Mckay, and Geoff Tims for their assistance with this project, and the referee for helpful comments. We would also like to thank Iowa State University for their hospitality during this project. In addition, we’d like to thank Alliance and the National Science Foundation for their support of this research.

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Authors and Affiliations

  1. Department of Mathematics, Bowdoin College, Brunswick, ME, 04011, USA

    Jonathan Ackerman & Kimberly Ayers

  2. Department of Mathematics, University of Puerto Rico, Rio Piedras, PR, 00931, USA

    Eduardo J. Beltran & Joshua Bonet

  3. Department of Mathematics, Stanford University, Stanford, CA, 94309, USA

    Devin Lu

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

    Thomas Rudelius

Authors
  1. Jonathan Ackerman
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  2. Kimberly Ayers
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  3. Eduardo J. Beltran
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  4. Joshua Bonet
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  5. Devin Lu
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  6. Thomas Rudelius
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Corresponding author

Correspondence to Thomas Rudelius.

Additional information

The research of K. Ayers, D. Lu and T. Rudelius was supported by DMS 0750986. The research of E. J. Beltran and J. Bonet research was supported by DMS 0502354.

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Ackerman, J., Ayers, K., Beltran, E.J. et al. A Behavioral Characterization of Discrete Time Dynamical Systems over Directed Graphs. Qual. Theory Dyn. Syst. 13, 161–180 (2014). https://doi.org/10.1007/s12346-014-0111-2

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  • DOI: https://doi.org/10.1007/s12346-014-0111-2

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