Advertisement

Journal of Statistical Physics

, Volume 15, Issue 4, pp 307–326 | Cite as

Successive bifurcations leading to stochastic behavior

  • John McLaughlin
Articles

Abstract

A model of interacting normal modes in a nonlinear, dissipative system is constructed in order to analyze speculations by Ruelle and Takens. The first bifurcation leads to a periodic state. The second bifurcation leads to phaselocking, if the first mode is sufficiently energetic. A third bifurcation leads to stochastic behavior. Possible relevance of these phenomena for physical systems is discussed.

Key words

Bifurcation correlation function dissipative nonlinear nonperiodic normal mode quasiperiodic stochastic turbulence 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    D. Ruelle and F. Takens,Commun. Math. Phys. 20:167 (1971).Google Scholar
  2. 2.
    V. I. Arnold and A. Avez,Ergodic Problems of Classical Mechanics, W. A. Benjamin (1968).Google Scholar
  3. 3.
    S. A. Orszag,Lectures on the Statistical Theory of Turbulence (1973).Google Scholar
  4. 4.
    E. N. Lorenz,J. Atmos. Sci. 20:130 (1963).Google Scholar
  5. 5.
    J. B. McLaughlin and P. C. Martin,Phys. Rev. A 12:186 (1975).Google Scholar
  6. 6.
    G. E. Willis and J. W. Deardorff,J. Fluid Mech. 44:661 (1970).Google Scholar
  7. 7.
    G. Ahlers,Phys. Rev. Lett. 33:1185 (1974).Google Scholar
  8. 8.
    F. H. Busse,J. Fluid Mech. 52:1, 97 (1972).Google Scholar
  9. 9.
    N. Wiener,Acta Mathematica 55:117 (1930).Google Scholar
  10. 10.
    L. Lapidus and J. H. Seinfeld,Numerical Solution of Ordinary Differential Equations, Academic Press (1971).Google Scholar
  11. 11.
    D. Ruelle, “A Measure Associated with Axiom-A Attractors,” preprint.Google Scholar

Copyright information

© Plenum Publishing Corporation 1976

Authors and Affiliations

  • John McLaughlin
    • 1
  1. 1.Physics DepartmentClarkson College of TechnologyPotsdam

Personalised recommendations