Abstract
A numerical method is presented allowing the computation of the invariant density of a time-discrete bi- or multistable map perturbed by weak noise. It permits the examination of noise-induced transitions between different stable states in the limit of weak but not amplitude-limited noise. The method is tested by comparing the results with computer experiments. For this purpose a new one-parameter family of bistable maps is introduced. It turns out that the numerics are in good agreement with the experiments. The results suggest the conjecture that in the limit of weak but transition-inducing noise the probability of being in one basin of attraction approaches one. A simple example which can be solved in closed form and which illustrates these findings is discussed.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Lorenz, E.N.: J. Atmos. Sci.20, 130 (1963)
Sparrow, C.: The Lorenz equations: bifurcations, chaos, and strange attractors. Berlin, Heidelberg, New York: Springer 1982
Guckenheimer, J., Holmes, P.: Nonlinear oscillations, dynamical systems and the bifurcation of vector fields. Berlin, Heidelberg, New York: Springer 1983
Haken, H.: Advanced synergetics. Berlin, Heidelberg, New York, Springer 1983
Singer, D.: SIAM J. Appl. Math.35, 260 (1978)
Fraser, S., Celarier, E., Kapral, R.: J. Stat. Phys.33, 341 (1983)
Celarier, E., Fraser, S., Kapral, R.: Phys. Lett.94A, 247 (1983)
Kapral, R., Celarier, E., Fraser, S.: Noise induced transitions in discrete-time systems. In: Horsthemke, W., Kondepudi, D.K. (eds.): Fluctuations and sensitivity in nonequilibrium systems. Springer Proceedings in Physics 1. Berlin, Heidelberg, New York: Springer 1984
Kifer, J.I.: Math. USSR Izvestija8, 1083 (1974)
Heldstab, J., Thomas, H., Geisel, T., Radons, G.: Z. Phys. B — Condensed Matter50, 141 (1983)
Baker, C.T.A.: The numerical treatment of integral equations. Oxford: Clarendon Press 1978
Press, W.H., Flannery, B.P., Teucholsky, S.A., Wetterling, W.T.: Numerical recipes. Cambridge, Cambridge University Press 1986
Talkner, P., Hänggi, P., Freidkin, E., Trautmann, D.: J. Stat. Phys.48, 231 (1987)
Varga, R.S.: Matrix iterative analysis. Englewood Cliffs, New Jersey: Prentice Hall 1962
Collet, P., Eckmann, J.P.: Iterated maps on the intervall as dynamical systems. Progress in Physics Series. Boston: Birkhäuser 1980
Abramowitz, M., Stegun, I.A.: Handbook of mathematical functions. New York: Dover 1964
Eckmann, J.P.: Rev. Mod. Phys.53, 643 (1981)
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Holtfort, J., Möhring, W. & Vogel, H. Computing the invariant density of bistable noisy maps. Z. Physik B - Condensed Matter 72, 115–121 (1988). https://doi.org/10.1007/BF01313118
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01313118