Abstract
Two different approaches are proposed to obtain explicit solutions for stochastic relaxation oscillator problems in the weak noise limit. The first method generalizes the idea of the cumulant expansion. It does not presuppose an analytical treatment of the deterministic motion. It is however restricted to the discussion of stationary situations. In the second method an adiabatic elimination of irrelevant variables allows for the computation of time dependent solutions. It can be carried through only if the deterministic limit cycle is known analytically. As special examples the stationary solutions of the stochastic van der Pol oscillator and time dependent solutions of a simple one dimensional model system have been obtained.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Broek, C. van den, Malek Mansour, M., Baras, F.: J. Stat. Phys.28, 557, 577 (1982);
Knobloch, E., Weisenfeld, K.A.: J. Stat. Phys.33, 611 (1983)
Moss, F., McClintock, P.V.E.: Noise in nonlinear dynamical systems. Vol. 1–3. Cambridge: Cambridge Univ. Press 1989
Kubo, R., Matsuo, M., Kitahara, K.: J. Stat. Phys.9 51 (1973)
Graham, R., Tel, T.: J. Stat. Phys.35, 729 (1984); Phys. Rev. A31, 1109 (1985);
Graham, R., Roeckaerts, D., Tel, T.: Phys. Rev. A31, 3364 (1985);
Jauslin, H.R.: J. Stat. Phys.42, 573 (1986)
Risken, H.: The Fokker-Planck equation: Methods of solution and applications. In: Springer Series in Synergetics. Vol. 18. Berlin, Heidelberg, New York, Tokyo: Springer 1984
Just, W., Sauermann, H.: Phys. Lett. A131, 234; (1988)
Just, W.: Physica D (to be published 1989/90)
Mishchenko, E.F., Rozov, N.Kh.: Differential equations with small parameters and relaxation oscillations. New York, London: Plenum Press 1980
Guckenheimer, J., Holmes, P.: Nonlinear oscillations, dynamical systems and bifurcations of vector fields. In: Applied Mathematical Sciences, Vol. 42 New York, Berlin, Heidelberg, Tokyo: Springer 1986
Suzuki, M.: Prog. Theor. Phys.56, 77, 477 (1976)
Sture, K., Nordholm, J., Zwanzig, R.: J. Stat. Phys.11, 143; (1974)
Grabert, H.: Projection operator techniques in nonequilibrium statistical mechanics. In: Springer Tracts in Modern Physics. Vol. 95. Berlin, Heidelberg, New York, Tokyo: Springer 1982
Zwanzig, R.: J. Chem. Phys.33, 1338 (1960)
Nayfeh, A.H.: Perturbation methods. New York: Wiley 1973
Gonzalez, D.L., Piro, O.: Phys. Rev. A36, 4402 (1987)
Author information
Authors and Affiliations
Additional information
This article is an excerpt from a dissertation presented at TH Darmstadt, Darmstädter Dissertation D17
This work was performed within a program of the Sonderforschungsbereich 185 Darmstadt-Frankfurt, FRG
Rights and permissions
About this article
Cite this article
Just, W. Stationary and dynamic solutions of stochastic relaxation oscillators. Z. Physik B - Condensed Matter 78, 513–525 (1990). https://doi.org/10.1007/BF01313336
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01313336