Skip to main content
SpringerLink
Log in

Statistical Description of Deterministic Systems

  • Chapter
  • First Online:
Statistical Physics of Complex Systems

  • 1302 Accesses

Abstract

This chapter presents some elementary notions on dynamical systems, concerning in particular fixed points and their stability, the more general concept of attractor, as well as the notion of bifurcation. A discussion on the comparison between deterministic and stochastic dynamics is provided, in connection with coarse-graining issues. Then, the case of globally coupled population of low-dimensional dynamical systems is investigated through the analysis of two different cases, the restabilization of unstable fixed points by the coupling, and the synchronization transition in the Kuramoto model of coupled oscillators.

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 49.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    This equation may be thought of as a Fokker-Planck equation (see Sect. 2.3) in the zero noise limit.

References

  1. M. Cencini, M. Falcioni, E. Olbrich, H. Kantz, A. Vulpiani, Chaos or noise: difficulties of a distinction. Phys. Rev. E 62, 427 (2000)

    Article  ADS  Google Scholar 

  2. S. De Monte, F. d’Ovidio, E. Mosekilde, Coherent regimes of globally coupled dynamical systems. Phys. Rev. Lett. 90, 054102 (2003)

    Article  ADS  Google Scholar 

  3. Y. Kuramoto, Chemical Oscillations, Waves and Turbulence (Springer, New York, 1984)

    Book  MATH  Google Scholar 

  4. J.A. Acebrón, L.L. Bonilla, C.J. Pérez, Vicente, F. Ritort, R. Spigler, The Kuramoto model: a simple paradigm for synchronization phenomena. Rev. Modern Phys. 77, 137 (2005)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. LIPhy, CNRS and Université Grenoble Alpes, Grenoble, France

    Eric Bertin

Authors
  1. Eric Bertin
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Eric Bertin .

Rights and permissions

Reprints and permissions

Copyright information

© 2016 Springer International Publishing Switzerland

About this chapter

Cite this chapter

Bertin, E. (2016). Statistical Description of Deterministic Systems. In: Statistical Physics of Complex Systems. Springer, Cham. https://doi.org/10.1007/978-3-319-42340-1_5

Download citation

Publish with us

Policies and ethics

Search

Navigation