Skip to main content
SpringerLink
Log in

Dynamical Systems in Categories

  • Published:
Applied Categorical Structures Aims and scope Submit manuscript

Abstract

In this article we establish a bridge between dynamical systems, including topological and measurable dynamical systems as well as continuous skew product flows and nonautonomous dynamical systems; and coalgebras in categories having all finite products. We introduce a straightforward unifying definition of abstract dynamical system on finite product categories. Furthermore, we prove that such systems are in a unique correspondence with monadic algebras whose signature functor takes products with the time space. We discuss that the categories of topological spaces, metrisable and uniformisable spaces have exponential objects w.r.t. locally compact Hausdorff, strongly σ-compact or arbitrary time spaces as exponents, respectively. Exploiting the adjunction between taking products and exponential objects, we demonstrate a one-to-one correspondence between monadic algebras (given by dynamical systems) for the left-adjoint functor and comonadic coalgebras for the other. This, finally, provides a new, alternative perspective on dynamical systems.

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

Similar content being viewed by others

References

  1. Adámek, J., Herrlich, H., Strecker, G.E.: Abstract and concrete categories: the joy of cats. Repr. Theory Appl. Categ. 17, 1–507 (2006). (electronic). Reprint of the 1990 original [Wiley, New York; MR1051419]

    MathSciNet  MATH  Google Scholar 

  2. Akin, E., Glasner, E.: Residual properties and almost equicontinuity. J. Anal. Math. 84, 243–286 (2001). doi:10.1007/BF02788112

    Article  MathSciNet  MATH  Google Scholar 

  3. Akin, E., Hurley, M., Kennedy, J.A.: Dynamics of topologically generic homeomorphisms. Mem. Amer. Math. Soc. 164(783), viii+130 (2003). doi:10.1090/memo/0783

    MathSciNet  MATH  Google Scholar 

  4. Aliprantis, C.D., Border, K.C.: Infinite dimensional analysis A Hitchhiker’s Guide. 3rd edn. Springer, Berlin, Heidelberg (2006). doi:10.1007/3-540-29587-9

    Google Scholar 

  5. Arbib, M.A., Manes, E.G.: Machines in a category: an expository introduction. SIAM Rev. 16, 163–192 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  6. Arbib, M.A., Manes, E.G.: Machines in a category. J. Pure Appl. Algebra 19, 9–20 (1980). doi:10.1016/0022-4049(80)90090-0

    Article  MathSciNet  MATH  Google Scholar 

  7. Arnold, L.: Random Dynamical Systems. Springer Monographs in Mathematics. Springer, Berlin (1998)

    Book  MATH  Google Scholar 

  8. Awodey, S.: Category theory Oxford Logic Guides. 2nd edn., vol. 52. Oxford University Press, Oxford (2010)

    Google Scholar 

  9. Behrisch, M., Kerkhoff, S., Pöschel, R., Schneider, F.M., Siegmund, S.: Dynamical systems in categories. Preprint MATH-AL-05-2013, TU Dresden, Institut für Algebra, Technische Universität Dresden, 01062 Dresden, Germany (2013). http://nbn-resolving.de/urn:nbn:de:bsz:14-qucosa-129909

  10. Berger, A.: Chaos and chance An Introduction to Stochastic Aspects of Dynamics de Gruyter Textbook. Walter de Gruyter & Co., Berlin (2001)

    Google Scholar 

  11. Berger, A., Siegmund, S.: On the gap between random dynamical systems and continuous skew products. J. Dynam. Differential Equations 15(2–3), 237–279 (2003). doi:10.1023/B:JODY.0000009736.39445.c4

    Article  MathSciNet  MATH  Google Scholar 

  12. Böhm, G., Brzeziński, T., Wisbauer, R.: Monads and comonads on module categories. J. Algebra 322(5), 1719–1747 (2009). doi:10.1016/j.jalgebra.2009.06.003

    Article  MathSciNet  MATH  Google Scholar 

  13. Browne, M.C., Clarke, E.M., Grümberg, O.: Characterizing finite Kripke structures in propositional temporal logic. International Joint Conference on Theory and Practice of Software Development (Pisa, 1987) Theoret. Comput. Sci. 59(1–2), 115–131 (1988). doi:10.1016/0304-3975(88)90098-9

    MathSciNet  MATH  Google Scholar 

  14. Cassandras, C.G., Lafortune, S.: Introduction to Discrete Event Systems, 2nd edn. Springer, New York (2008). doi:10.1007/978-0-387-68612-7

    Book  MATH  Google Scholar 

  15. Corbae, D., Stinchcombe, M.B., Zeman, J.: An Introduction to Mathematical Analysis for Economic Theory and Econometrics. Princeton University Press, Princeton (2009)

    MATH  Google Scholar 

  16. Droste, M., Kuich, W., Vogler, H. (eds.): Handbook of weighted automata. Monographs in Theoretical Computer Science, An EATCS Series. Springer, Berlin (2009). doi:10.1007/978-3-642-01492-5

    Google Scholar 

  17. Droste, M., Quaas, K.: A Kleene-Schützenberger theorem for weighted timed automata. Theoret. Comput. Sci. 412(12–14), 1140–1153 (2011). doi:10.1016/j.tcs.2010.12.033

    Article  MathSciNet  MATH  Google Scholar 

  18. Dugundji, J.: Topology. Allyn and Bacon Inc., Boston (1966)

    MATH  Google Scholar 

  19. Eilenberg, S., Moore, J.C.: Adjoint functors and triples. Illinois J. Math. 9, 381–398 (1965)

    MathSciNet  MATH  Google Scholar 

  20. Esparza, J.: Decidability of model checking for infinite-state concurrent systems. Acta Inform. 34(2), 85–107 (1997). doi:10.1007/s002360050074

    Article  MathSciNet  MATH  Google Scholar 

  21. Glasner, E.: Enveloping semigroups in topological dynamics. Topology Appl. 154(11), 2344–2363 (2007). doi:10.1016/j.topol.2007.03.009

    Article  MathSciNet  MATH  Google Scholar 

  22. Godement, R.: Topologie algébrique et théorie des faisceaux. Actualités Sci. Ind. No. 1252. Publ. Math. Univ. Strasbourg. No. 13. Hermann, Paris (1958)

  23. Gottschalk, W.H., Hedlund, G.A.: Topological dynamics. American Mathematical Society Colloquium Publications, vol. 36. American Mathematical Society, Providence (1955)

    Book  MATH  Google Scholar 

  24. Herrlich, H.: Topologie [Topology]. II. Uniforme Räume. [Uniform spaces], Berliner Studienreihe zur Mathematik [Berlin Study Series on Mathematics], vol. 3. Heldermann Verlag, Berlin (1988)

    Google Scholar 

  25. Kleene, S.C.: Representation of events in nerve nets and finite automata. In: Automata Studies, Annals of Mathematics Studies, no. 34, pp 3–41. Princeton University Press, Princeton (1956)

    Google Scholar 

  26. Kripke, S.A.: Semantical considerations on modal logic. Acta Philos. Fenn. Fasc. 16, 83–94 (1963)

    MathSciNet  MATH  Google Scholar 

  27. Mac Lane, S.: Categories for the Working Mathematician, 2nd edn.. Graduate Texts in Mathematics, vol. 5. Springer, New York (1998)

    MATH  Google Scholar 

  28. McNaughton, R., Papert, S.: Counter-Free Automata. The M.I.T. Press, Cambridge (1971). With an appendix by William Henneman, M.I.T. Research Monograph, No. 65

    MATH  Google Scholar 

  29. Minsky, M.L.: Recursive unsolvability of Post’s problem of “tag” and other topics in theory of Turing machines. Ann. of Math. (2) 74(3), 437–455 (1961). http://www.jstor.org/stable/1970290

    Article  MathSciNet  MATH  Google Scholar 

  30. Minsky, M.L.: Computation: Finite and Infinite Machines. Prentice-Hall Inc., Englewood Cliffs (1967). Prentice-Hall Series in Automatic Computation

    MATH  Google Scholar 

  31. Nerurkar, M.G.: Ergodic continuous skew product actions of amenable groups. Pacific J. Math. 119(2), 343–363 (1985). http://projecteuclid.org/getRecord?id=euclid. pjm/1102706160

    Article  MathSciNet  MATH  Google Scholar 

  32. Nyikos, P.J.: Metrizability and the Fréchet-Urysohn property in topological groups. Proc. Amer. Math. Soc. 83(4), 793–801 (1981). doi:10.2307/2044256

    MathSciNet  MATH  Google Scholar 

  33. von Querenburg, B.: Mengentheoretische Topologie [Set theoretic topology], 2nd edn.. Hochschultext. [University Text]. Springer, Berlin (1979)

    Book  MATH  Google Scholar 

  34. Rabin, M.O.: Decidability of second-order theories and automata on infinite trees. Trans. Amer. Math. Soc. 141, 1–35 (1969)

    MathSciNet  MATH  Google Scholar 

  35. Rabin, M.O., Scott, D.S.: Finite automata and their decision problems. IBM J. Res. Develop. 3, 114–125 (1959)

    Article  MathSciNet  MATH  Google Scholar 

  36. Rutten, J.J.M.M.: Universal coalgebra: a theory of systems. Modern algebra and its applications (Nashville, TN, 1996) Theoret. Comput. Sci. 249(1), 3–80 (2000). doi:10.1016/S0304-3975(00)00056-6

    MathSciNet  MATH  Google Scholar 

  37. Schneider, F.M., Kerkhoff, S., Behrisch, M., Siegmund, S.: Chaotic actions of topological semigroups Semigroup Forum, pp. 1–9 (2013). doi:10.1007/s00233-013-9517-4

  38. Schneider, F.M., Kerkhoff, S., Behrisch, M., Siegmund, S.: Locally compact groups admitting faithful strongly chaotic actions on Hausdorff spaces. Int. J. Bifurcation Chaos 23(09), 135–158 (2013). doi:10.1142/S0218127413501587

    Article  MathSciNet  MATH  Google Scholar 

  39. Schützenberger, M.P.: On finite monoids having only trivial subgroups. Inf. Control. 8, 190–194 (1965)

    Article  MathSciNet  MATH  Google Scholar 

  40. Sell, G.R.: Topological dynamics and ordinary differential equations. Van Nostrand Reinhold Co., London (1971). Van Nostrand Reinhold Mathematical Studies, No. 33

    MATH  Google Scholar 

  41. Thomas, W.: Automata on infinite objects. In: Handbook of Theoretical Computer Science, vol. B, pp 133–191. Elsevier, Amsterdam (1990)

    Google Scholar 

  42. Thomas, W.: Automata theory and infinite transition systems. In: Preproceedings of EMS Summer School CANT 2006, Prépublication 06.002, pp. 1–25. Institut de Mathématiques, Université de Liège (2006)

  43. Thomas, W.: The reachability problem over infinite graphs. In: Frid, A., Morozov, A., Rybalchenko, A., Wagner, K.W. (eds.) Computer Science - Theory and Applications, Lecture Notes in Computer Science, vol. 5675, pp. 12–18. Springer, Berlin (2009). doi:10.1007/978-3-642-03351-3_2

    Google Scholar 

  44. Tornambè, A.: Discrete-event system theory An Introduction. World Scientific Publishing Co. Inc., River Edge (1995). doi:10.1142/2845

    Google Scholar 

  45. Wilker, P.: Adjoint product and hom functors in general topology. Pacific J. Math. 34(1), 269–283 (1970). http://projecteuclid.org/euclid.pjm/1102976655

    Article  MathSciNet  MATH  Google Scholar 

  46. Willard, S.: General Topology. Addison-Wesley Series in Mathematics. Dover, Mineola (2004). Reprint of the 1970 original [Addison-Wesley, Reading, MA; MR0264581]

    MATH  Google Scholar 

Download references

Author information

Author notes

  1. Mike Behrisch

    Present address: Institut für Computersprachen, Technische Universität Wien, 1040, Vienna, Austria

Authors and Affiliations

  1. Institut für Algebra, Technische Universität Dresden, 01062, Dresden, Germany

    Mike Behrisch, Sebastian Kerkhoff, Reinhard Pöschel & Friedrich Martin Schneider

  2. Institut für Analysis, Technische Universität Dresden, 01062, Dresden, Germany

    Stefan Siegmund

Authors
  1. Mike Behrisch
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Sebastian Kerkhoff
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Reinhard Pöschel
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Friedrich Martin Schneider
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Stefan Siegmund
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Mike Behrisch.

Additional information

F. M. Schneider has been supported by funding of the Excellence Initiative by the German Federal and State Governments (Institutional Strategy, measure “support the best”).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Behrisch, M., Kerkhoff, S., Pöschel, R. et al. Dynamical Systems in Categories. Appl Categor Struct 25, 29–57 (2017). https://doi.org/10.1007/s10485-015-9409-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10485-015-9409-8

Keywords

Mathematics Subject Classification (2010)

Search

Navigation