Skip to main content
SpringerLink
Log in

Time Rescaling of Lyapunov Exponents

  • Chapter
  • First Online:
Advances in Dynamics, Patterns, Cognition

Part of the book series: Nonlinear Systems and Complexity ((NSCH,volume 20))

  • 940 Accesses

Abstract

For systems with zero Lyapunov exponents we introduce and study the notion of scaled Lyapunov exponents which is used to characterize the sub-exponential separation of nearby trajectories. We briefly discuss the abstract theory of such exponents and discuss some examples.

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 84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
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. Barreira, L., Pesin, Y.: Introduction to smooth ergodic theory. In: GSM, vol. 148. AMS, Providence, RI (2013)

    Google Scholar 

  2. Barreira, L., Valls, C.: Growth rates and nonuniform hyperbolicity. Discret. Contin. Dynam. Syst. 22 (3), 509–528 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  3. Corabel, N., Barkai, E.: Pesin-type identity for intermittent dynamics with zero Lyapunov exponent. Phys. Rev. Lett. 102, 050601 (2009)

    Article  MathSciNet  Google Scholar 

  4. Gaggero-Sager, L.M., Pujals, E.R., Sotolongo-Costa, O.: Infinite ergodic theory and non-extensive entropies. Braz. J. Phys. 41, 297–303 (2011)

    Article  Google Scholar 

  5. Halász, G.: Remarks on the remainder in Birkhoff’s ergodic theorem. Acta Math. Acad. Sci. Hungar. 28 (3–4), 389–395 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  6. Katok, A., Hasselblatt, B.: Introduction to the modern theory of dynamical systems. In: Encyclopedia of Mathematics and Its Applications, vol. 54. Cambridge University Press, Cambridge (1995)

    Google Scholar 

  7. Pesin, Y.: Dimension theory in dynamical systems. In: Contemporary Views and Applications. University of Chicago Press, Chicago (1997)

    Book  Google Scholar 

  8. Pesin, Y., Zhao, Y.: Scaled entropy for dynamical systems. J. Stat. Phys. 158, 447–475 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  9. Rabinovich, M., Volkovskii A., Lecanda, P., Huerta, R., Abarbanel, H.: Dynamical encoding by networks of competing neuron groups: winnerless competition. Phys. Rev. Lett. 87, 068102-1-068102-4 (2001)

    Google Scholar 

  10. Rabinovich, M., Simmons, A., Varona, P.: Dynamical bridge between brain and mind. Trends Cogn. Sci. 19, 453–461 (2015)

    Article  Google Scholar 

  11. Rabinovich, M., Tristan, I., Varona, P.: Hierarchical nonlinear dynamics of human attention. Neurosci. Biobehav. Rev. 55, 18–35 (2015)

    Article  Google Scholar 

  12. Zelerowicz, A.: Time Rescaling of Birkhoff sums, Preprint, PSU (2016)

    Google Scholar 

Download references

Acknowledgements

Ya. P. and A. Z. are partially supported by NSF grant DMS-1400027. Y. Z. is partially supported by NSFC 1137127. A part of this work was done when the authors visited Brown University Institute for Computational and Experimental Research in Mathematics (ICERM). We would like to thank the institute for their hospitality. We are also grateful to V. Afraimovich for useful discussions and comments.

Author information

Authors and Affiliations

  1. Department of Mathematics, Pennsylvania State University, University Park, PA, 16802, USA

    Yakov Pesin & Agnieszka Zelerowicz

  2. Department of Mathematics, Soochow University, Suzhou, Jiangsu, 215006, People’s Republic of China

    Yun Zhao

Authors
  1. Yakov Pesin
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Agnieszka Zelerowicz
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Yun Zhao
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yakov Pesin .

Editor information

Editors and Affiliations

  1. Pennsylvania State University, University Park, Pennsylvania, USA

    Igor S. Aranson

  2. Institute for Physics and Astronomy, University of Potsdam, Potsdam, Germany

    Arkady Pikovsky

  3. BioCircuits Institute, University of California San Diego, La Jolla, California, USA

    Nikolai F. Rulkov

  4. BioCircuits Institute, University of California San Diego, La Jolla, California, USA

    Lev S. Tsimring

Rights and permissions

Reprints and permissions

Copyright information

© 2017 Springer International Publishing AG

About this chapter

Cite this chapter

Pesin, Y., Zelerowicz, A., Zhao, Y. (2017). Time Rescaling of Lyapunov Exponents. In: Aranson, I., Pikovsky, A., Rulkov, N., Tsimring, L. (eds) Advances in Dynamics, Patterns, Cognition. Nonlinear Systems and Complexity, vol 20. Springer, Cham. https://doi.org/10.1007/978-3-319-53673-6_3

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-53673-6_3

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-53672-9

  • Online ISBN: 978-3-319-53673-6

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics

Search

Navigation