Skip to main content
SpringerLink
Log in

Dimension and Entropy Estimates for Global Attractors of Cocycles

  • Chapter
  • First Online:
Attractor Dimension Estimates for Dynamical Systems: Theory and Computation

Part of the book series: Emergence, Complexity and Computation ((ECC,volume 38))

  • 693 Accesses

Abstract

In this chapter we derive dimension and entropy estimates for invariant sets and global \(\mathcal{B}\)-attractors of cocycles in non-fibered and fibered spaces. A version of the Douady-Oesterlé theorem will be proven for local cocycles in an Euclidean space and for cocycles on Riemannian manifolds. As examples we consider cocycles, generated by the Rössler system with variable coefficients. We also introduce time-discrete cocycles on fibered spaces and define the topological entropy of such cocycles. Upper estimates of the topological entropy along an orbit of the base system are given which include the Lipschitz constants of the evolution system and the fractal dimension of the parameter dependent phase space.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 189.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 249.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 249.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Abramovich, S., Koryakin, Yu., Leonov, G., Reitmann, V.: Frequency-domain conditions for oscillations in discrete systems. I., Oscillations in the sense of Yakubovich in discrete systems. Wiss. Zeitschr. Techn. Univ. Dresden, 25(5/6), 1153 – 1163 (1977) (German)

    Google Scholar 

  2. Abramovich, S., Koryakin, Yu., Leonov, G., Reitmann, V.: Frequency-domain conditions for oscillations in discrete systems. II., Oscillations in discrete phase systems. Wiss. Zeitschr. Techn. Univ. Dresden, 26(1), 115–122 (1977) (German)

    Google Scholar 

  3. Anguiano, M., Caraballo, T.: Asymptotic behaviour of a non-autonomous Lorenz-84 system. Discrete Contin. Dynam. Syst. 34(10), 3901–3920 (2014)

    Article  MathSciNet  Google Scholar 

  4. Anikushin, M.M., Reitmann, V.: Development of the topological entropy conception for dynamical systems with multiple time. Electr. J. Diff. Equ. and Contr. Process. 4 (2016) (Russian); English Trans. J. Diff. Equ. 52(13), 1655 – 1670 (2016)

    Google Scholar 

  5. Cheban, D.N.: Nonautonomous Dyn. Springer Monographs in Mathematics, Berlin (2020)

    Book  Google Scholar 

  6. Crauel, H., Flandoli, F.: Hausdorff dimension of invariant sets for random dynamical systems. J. Dyn. Diff. Equ. 10(3), 449–474 (1998)

    Article  MathSciNet  Google Scholar 

  7. Debussche, A.: Hausdorff dimension of a random invariant set. Math. Pures Appl. 77, 967–988 (1998)

    Article  MathSciNet  Google Scholar 

  8. Douady, A., Oesterlé, J.: Dimension de Hausdorff des attracteurs. C. R. Acad. Sci. Paris, Ser. A 290, 1135–1138 (1980)

    Google Scholar 

  9. Egorova, V.E., Reitmann, V.: Estimation of topological entropy for cocycles with cellular automaton as a base system. Electron. J. Diff. Equ. Contr. Process. 3, 102–122 (2018) (Russian)

    Google Scholar 

  10. Ermakov, I.V., Kalinin, Yu.N., Reitmann, V.: Determining modes and almost periodic integrals for cocycles. Electron. J. Diff. Equ. Contr. Process. 4 (2011) (Russian); English Trans. J. Diff. Equ. 47(13), 1837–1852 (2011)

    Google Scholar 

  11. Ermakov, I.V., Reitmann, V., Skopinov, S.: Determining functionals for cocycles and application to the microwave heating problem. Abstracts, Equadiff, Loughborough, UK, 135 (2011)

    Google Scholar 

  12. Flandoli, F., Schmalfuss, B.: Random attractors for the 3D stochastic Navier-Stokes equation with multiplicative white noise. Stoch. Stoch. Rep. 59(1–2), 21–45 (1996)

    Article  Google Scholar 

  13. Giesl, P., Rasmussen, M.: Borg’s criterion for almost periodic differential equations. Nonlinear Anal. 69(11), 3722–3733 (2008)

    Article  MathSciNet  Google Scholar 

  14. Kloeden, P.E., Rasmussen, M.: Nonautonomous Dynamical Systems. Mathematical Surveys and Monographs, Amer. Math. Soc. 176 (2011)

    Google Scholar 

  15. Kloeden, P.E., Schmalfuß, B.: Nonautonomous systems, cocycle attractors and variable time-step discretization. Numer. Algorithms 14, 141–152 (1997)

    Article  MathSciNet  Google Scholar 

  16. Kolyada, S., Snoha, L.: Topological entropy of nonautonomous dynamical systems. Random Comput. Dyn. 4(2/3), 205–233 (1996)

    MathSciNet  MATH  Google Scholar 

  17. Langa, J.A., Schmalfuss, B.: Finite dimensionality of attractors for non-autonomous dynamical systems given by partial differential equations. Stoch. Dyn. 4(3), 385–404 (2004)

    Article  MathSciNet  Google Scholar 

  18. Ledrappier, F., Young, L.S.: Dimension formula for random transformations. Commun. Math. Phys. 117(4), 529–548 (1988)

    Article  MathSciNet  Google Scholar 

  19. Leonov, G.A., Reitmann, V., Slepuchin, A.S.: Upper estimates for the Hausdorff dimension of negatively invariant sets of local cocycles. Dokl. Akad. Nauk, T. 439, 6 (2011) (Russian); English Trans. Dokl. Mathematics, 84(1), 551–554 (2011)

    Google Scholar 

  20. Leonov, G., Tschschigowa, T., Reitmann, V.: A frequency-domain variant of the Belykh-Nekorkin comparison method in phase synchronisation theory. Wiss. Zeitschr. Techn. Univ. Dresden, 32(1), 51–59 (1983) (German)

    Google Scholar 

  21. Maltseva, A.A., Reitmann, V.: Global stability and bifurcations of invariant measures for the discrete cocycles of the cardiac conduction system’s equations. J. Diff. Equ. 50(13), 1718–1732 (2014)

    Article  MathSciNet  Google Scholar 

  22. Maltseva, A.A., Reitmann, V.: Existence and dimension properties of a global \({\cal{B}}\)-pullback attractor for a cocycle generated by a discrete control system. J. Diff. Equ. 53(13), 1703–1714 (2017)

    Article  MathSciNet  Google Scholar 

  23. Maricheva, A.V.: Hausdorff dimension bounds for cocycle attractors on a finite dimensional Riemannian manifold. Diploma Thesis, St. Petersburg State University (2015) (Russian)

    Google Scholar 

  24. Noack, A.: Hausdorff dimension estimates for time-discrete feedback control systems. ZAMM 77(12), 891–899 (1997)

    Article  MathSciNet  Google Scholar 

  25. Pogromsky, A.Y., Matveev, A.S.: Estimation of topological entropy via the direct Lyapunov method. Nonlinearity 24(7), 1937 (2011)

    Article  MathSciNet  Google Scholar 

  26. Reitmann, V.: About bounded and periodic trajectories in nonlinear impulse systems. Wiss. Zeitschr. Techn. Univ. Dresden, 27(2), 355–357 (1978) (German)

    Google Scholar 

  27. Reitmann, V., Anikushin, M.M., Romanov, A.O.: Dimension-like properties and almost periodicity for cocycles generated by variational inequalities with delay. Abstracts, Equadiff, Leiden, The Netherlands, 90 (2019)

    Google Scholar 

  28. Reitmann, V., Slepuchin, A.V.: On upper estimates for the Hausdorff dimension of negatively invariant sets of local cocycles. Vestn. St. Petersburg Univ. Math. 44(4), 292–300 (2011)

    Article  MathSciNet  Google Scholar 

  29. Rössler, O. E.: Different types of chaos in two simple differential equations. Z. Naturforsch. 31 a, 1664–1670 (1976)

    Google Scholar 

  30. Wakeman, D.R.: An application of topological dynamics to obtain a new invariance property for nonautonomous ordinary differential equations. J. Diff. Equ. 17(2), 259–295 (1975)

    Article  MathSciNet  Google Scholar 

  31. Wang, Y., Zhong, C., Zhou, S.: Pullback attractors of nonautonomous dynamical systems. Discrete and Cont. Dynam. Syst. 16(3), 587–614 (2006)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Applied Cybernetics, Faculty of Mathematics and Mechanics, St. Petersburg State University, St. Petersburg, Russia

    Nikolay Kuznetsov & Volker Reitmann

  2. Faculty of Information Technology, University of Jyväskylä, Jyväskylä, Finland

    Nikolay Kuznetsov

Authors
  1. Nikolay Kuznetsov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Volker Reitmann
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Nikolay Kuznetsov .

Rights and permissions

Reprints and permissions

Copyright information

© 2021 The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Kuznetsov, N., Reitmann, V. (2021). Dimension and Entropy Estimates for Global Attractors of Cocycles. In: Attractor Dimension Estimates for Dynamical Systems: Theory and Computation. Emergence, Complexity and Computation, vol 38. Springer, Cham. https://doi.org/10.1007/978-3-030-50987-3_9

Download citation

  • DOI: https://doi.org/10.1007/978-3-030-50987-3_9

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-50986-6

  • Online ISBN: 978-3-030-50987-3

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics

Search

Navigation