Abstract
In their 1993 paper [14], Schimansky-Geier and Herzel discovered numerically that the Kramers oscillator (which is identical with the Duffing oscillator forced by additive white noise) has a positive top Lyapunov exponent in the low damping regime.
In this paper, we study the Kramers oscillator from the point of view of random dynamical systems. In particular, we confirm the findings in the paper [14] about the Lyapunov exponent by performing more precise simulations, revealing that the Lyapunov exponent is positive up to a critical value of the damping, from which on it remains negative.
We then show that the Kramers oscillator has a global random attractor which in the stable regime (large damping) is just a random point and in the unstable regime (small damping) has very complicated geometrical structure. In the latter case the invariant measure supported by the attractor is a Sinai-Ruelle-Bowen measure with positive entropy. The Kramers oscillator hence undergoes a stochastic bifurcation at the critical value of the damping parameter.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
L. Arnold, Stochastic differential equations: theory and applications. Wiley, New York, 1974. Original German language edition 1973 by Oldenbourg Verlag, München; English edition reprinted by Krieger, Malabar (Florida), 1992.
L. Arnold, Random dynamical systems. Springer-Verlag, Berlin Heidelberg New York, 1998.
L. Arnold and P. Imkeller, Stochastic bifurcation of the noisy Duffing oscillator. Report, Institut für Dynamische Systeme, Universitat Bremen, 2000.
M. Dellnitz and A. Hohmann, Numer. Math., 75:293–317, 1997.
J. Guckenheimer and P. Holmes, Nonlinear oscillations, dynamical systems, and bifurcation of vector fields. Springer-Verlag, Berlin Heidelberg New York, 1983.
P. Imkeller and B. Schmalfufuß, Manuscript, Humboldt-Universitat Berlin, 1998.
H. Keller and G. Ochs, In H. Crauel and M. Gundlach, editors, Stochastic dynamics, pages 93–116. Springer-Verlag, New York, 1999.
W. Kliemann, Annals of Probability, 15:690–707, 1987.
H. A. Kramers, Physica, 7:284–304, 1940.
G. Ochs, Random attractors: robustness, numerics and chaotic dynamics. Report, Institut für Dynamische Systeme, Universität Bremen, 1999.
G. Ochs, Weak random attractors. Report 449, Institut für Dynamische Systeme, Universität Bremen, 1999.
K. R. Schenk-Hoppé, ZAMP, 47:740–759, 1996.
K. R. Schenk-Hoppé, Discrete and Continuous Dynamical Systems, 4:99–130, 1998.
L. Schimansky-Geier and H. Herzel. Journal of Statistical Physics, 70:141–147, 1993.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2000 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Arnold, L., Imkeller, P. (2000). The Kramers Oscillator Revisited. In: Freund, J.A., Pöschel, T. (eds) Stochastic Processes in Physics, Chemistry, and Biology. Lecture Notes in Physics, vol 557. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45396-2_26
Download citation
DOI: https://doi.org/10.1007/3-540-45396-2_26
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-41074-4
Online ISBN: 978-3-540-45396-3
eBook Packages: Springer Book Archive