Abstract
A method is presented for constructing a stochastic return map from a stochastic differential equation containing a locally stable limit cycle and small-amplitude [O(ε)] additive Gaussian colored noise. The construction is valid provided the correlation time isO(ε) orO(1). The effective noise in the return map has nonzeroO(ε 2) mean and is state dependent. The method is applied to a model dynamical system, illustrating how the effective noise in the return map depends on both the original noise process and the local deterministic dynamics.
Similar content being viewed by others
References
L. Arnold,Stochastic Differential Equations: Theory and Applications (Wiley, New York, 1974).
Z. Schuss,Theory and Applications of Stochastic Differential Equations (Wiley, New York, 1980).
N. G. Van Kampen,Stochastic Processes in Physics and Chemistry (North-Holland, Amsterdam, 1981).
L. Arnold and R. Lefever, eds.,Stochastic Nonlinear Systems in Physics, Chemistry, and Biology (Springer-Verlag, Berlin, 1981).
W. Horsthemke and R. Lefever,Noise-Induced Transitions: Theory and Applications in Physics, Chemistry, and Biology (Springer-Verlag, Berlin, 1984).
B. S. White,SIAM J. Appl. Math. 32:666 (1977).
M. Mangel,SIAM J. Appl. Math. 38:120 (1980).
H. D. Vollmer and H. Risken,Z. Physik B 37:343 (1980).
E. Ben-Jacob, D. J. Bergman, B. J. Matkowsky, and Z. Schuss,Phys. Rev. A 26:2805 (1982).
H. D. Vollmer and H. Risken,Z. Physik B 52:259 (1983).
P. Jung and H. Risken,Z. Physik B 54:357 (1984).
R. Cristiano and P. Silvestrini,J. Appl. Phys. 59:1401 (1986).
R. Graham and T. Tél,Phys. Rev. A 33:1322 (1986).
J. Guckenheimer and P. Holmes,Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields (Springer-Verlag, Berlin, 1983).
G. Meyer-Kress and H. Haken,J. Stat. Phys. 26:149 (1981).
J. P. Crutchfield, J. D. Farmer, and B. A. Huberman,Phys. Rep. 92:45 (1982).
K. Wiesenfeld,J. Stat. Phys. 38:1071 (1985).
C. Knessl, B. J. Matkowsky, Z. Schuss, and C. Tier,J. Stat. Phys. 42:169 (1986).
J. B. Weiss,Phys. Rev. A 35:879 (1987).
P. Talkner, P. Hänggi, E. Freidkin, and D. Trautmann,J. Stat. Phys. 48:231 (1987).
E. Knobloch and J. B. Weiss, inNoise in Nonlinear Dynamical Systems, Vol. 2, F. Moss and P. V. E. McClintock, eds. (Cambridge University Press, Cambridge, 1989).
A. Carverhill, inLyapunov Exponents, L. Arnold and V. Wihstutz, eds. (Springer-Verlag Berlin, 1986), p. 292.
C. W. Gardiner,Handbook of Stochastic Methods (Springer-Verlag, Berlin, 1985).
J. B. Weiss, Stochastic and Deterministic Oscillations, Thesis, University of California, Berkeley (1988).
F. Ledrappier and L.-S. Young,Commun. Math. Phys. 117:529 (1988).
H. Kunita, inProc. Ecole d' Eté de Probabilités de Saint-Flour XII 1982, P.-L. Hennequin, ed. (Springer-Verlag, Berlin, 1984).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Weiss, J.B., Knobloch, E. A stochastic return map for stochastic differential equations. J Stat Phys 58, 863–883 (1990). https://doi.org/10.1007/BF01026555
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01026555