Abstract
The escape rate for one-dimensional noisy maps near a crisis is investigated. A previously introduced perturbation theory is extended to very general kinds of weak uncorrelated noise, including multiplicative white noise as a special case. For single-humped maps near the boundary crisis at fully developed chaos an asymptotically exact scaling law for the rate is derived. It predicts that transient chaos is stabilized by basically any noise of appropriate strength provided the maximum of the map is of sufficiently large order. A simple heuristic explanation of this effect is given. The escape rate is discussed in detail for noise distributions of Lévy, dichotomous, and exponential type. In the latter case, the rate is dominated by an exponentially leading Arrhenius factor in the deep precritical regime. However, the preexponential factor may still depend more strongly than any power law on the noise strength.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
H. G. Schuster,Deterministic Chaos (VHC Verlagsgesellschaft, Weinheim, 1988).
C. Grebogi, E. Ott, and Y. A. Yorke,Phys. Rev. Lett. 48:1507 (1982);Physica 7D:181 (1983).
S. Takesue and K. Kaneko,Progr. Theor. Phys. 71:35 (1984).
F. T. Arecchi, R. Badii, and A. Politi,Phys. Lett. 103A:3 (1984).
C. Grebogi, E. Ott, and Y. A. Yorke,Ergod. Theory Dynam. Syst. 5:341 (1985).
R. L. Kautz,J. Appl. Phys. 62:198 (1987).
J. Bene and P. Szépfalusy,Phys. Rev. A 37:871 (1988).
P. D. Beale,Phys. Rev. A 40:3998 (1989).
I. N. Struchkov,Sov. Phys. Tech. Phys. 37:758 (1990).
M. Franaszek,Phys. Rev. A 44:4065 (1991).
J. C. Sommerer, E. Ott, and C. Grebogi,Phys. Rev. A 43:1754 (1991).
J. C. Sommerer et al.,Phys. Rev. Lett. 66:1947 (1991); J. C. Sommerer, InProceedings of the 1st Experimental Chaos Conference, S. Vohraet al. eds. (World Scientific, Singapore, 1992).
J. C. Sommerer,Phys. Lett. 176A:85 (1993).
M. Franaszek and L. Fronzoni,Phys. Rev. E 49:3888 (1994).
A. Hamm, T. Tél, and R. Graham:Phys. Lett. 185A:313 (1994).
P. Reimann, Z. Naturforsch.49a:1248 (1994);Helv. Phys. Acta 67:235 (1994).
J. A. Blackburn, N. Gronbech-Jensen, and H. J. T. Smith,Phys. Rev. Lett. 74:908 (1995).
R. L. Kautz,J. Appl. Phys. 58:424 (1985); M. Iansiti, Qing Hu, R. M. Westervelt, and M. Tinkham,Phys. Rev. Lett. 55:746 (1985); H. Herzel and W. Ebeling,Z. Naturforsch.42a:136 (1987); V. S. Anishchenko and H. Herzel,Z. Angew. Math. Mech. 68:317 (1988).
F. T. Arecchi, R. Badii, and A. Politi,Phys. Rev. A 29:1006 (1984);32:402 (1985).
J. Güémez and M. A. Matías,Phys. Rev. E. 48:R2351 (1993);51:3059 (1995).
T. Geisel and J. Nierwetberg,Phys. Rev. Lett. 48:7 (1982).
P. Reimann,Phys. Rev. E 50:727 (1994).
E. G. Gwinn and R. M. Westervelt,Phys. Rev. Lett. 54:1613 (1985); P. Grassberger,J. Phys. A 22:3282, 1989; P. E. Phillipson and A. J. ArmstrongPhys. Lett. A 146:401 (1990); Junli Liu and S. K. Scott,J. Chem. Phys 94:4416 (1991).
P. Reimann, R. Müller, and P. Talkner,Phys. Rev. E 49:3670 (1994).
P. Reimann,J. Stat. Phys. 82:1467 (1996).
G. Györgyi and P. Szépfalusy,Z. Phys. B 55:179 (1984).
G. Györgyi and P. Szépfalusy,J. Stat. Phys. 34:451 (1984).
T. Geisel and J. Nierwetberg,Phys. Rev. Lett. 47:975 (1981); J. D. Farmer,Phys. Rev. Lett. 55:351 (1985); Z. Swiatek and M. Jakobson, Unpublished.
T. Tél, Transient chaos, inDirections in Chaos, Vol. 3, Hao Bai-Lin, ed. (World Scientific, Singapore, 1990).
B. V. Gnedenko and A. N. Kolmogorov,Limit Distributions for Sums of Independent Random Variables (Addison-Wesley, Cambridge 1954), Chapter 7.
S. Chang and J. Wright,Phys. Rev. A 23:1419 (1981).
J. M. Gutierrez, A. Iglesias, and M. A. Rodriguez,Phys. Rev. A 48:2507 (1993).
P. Reimann, Ph.D. thesis, Basel (1992), unpublished; P. Reimann and P. Talkner,Helv. Phys. Acta 66:93 (1993).
A. Hamm and R. Graham,J. Stat. Phys. 66:689 (1992).
G. Mayer-Kress and H. Haken,J. Stat. Phys. 26:149 (1981).
S. Fraser, E. Celarier, and R. Kapral,J. Stat. Phys. 33:341 (1983).
R. García-Pelayo and W. C. Schieve,J. Math. Phys. 33:570 (1992).
M. E. Fisher, InLecture Notes in Physics, Vol. 186, F. J. W. Hahne, ed. (Springer, Berlin, 1983).
A. Hamm, Ph.D. thesis, Essen (June 1993), paragraph 4.2.5 [in German].
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Reimann, P. Noisy one-dimensional maps near a crisis. II. General uncorrelated weak noise. J Stat Phys 85, 403–425 (1996). https://doi.org/10.1007/BF02174212
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/BF02174212