Abstract
Renormalization group equations describing the phenomenon of intermittency in Hamiltonian systems are presented. All solutions satisfying certain physical constraints are obtained; they are the complete set of simple singularities. Further considerations lead to precise predictions for scaling behavior at the onset of intermittency.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Birkhoff, G. D.:Dynamical Systems. Am. Math. Soc. Colloq. Pub., vol. 9 (1927)
Arnold, V. I.:Mathematical Methods of Classical Physics. Berlin, Heidelberg, New York: Springer-Verlag 1978
Arnold, V. I.: Funct. Anal. Appl.6, 254 (1972)
Feigenbaum, M. J.: Quantitative universality for a class of nonlinear transformations, J. Stat. Phys.19, 25 (1978)
Feigenbaum, M. J.: The universal metric properties of nonlinear transformations, J. Stat. Phys.21, 669 (1979)
Greene, J. M., MacKay, R. S., Vivaldi, F., Feigenbaum, M. J.: Universal behaviour in families of area-preserving maps, Physica3D, 468 (1981)
Widom, M., Kadanoff, L. P.: Renormalization group analysis of bifurcations in area-preserving maps, Physica5D, 287 (1982)
Kadanoff, L. P.: In:Melting, localization and chaos. eds. Kalia, R. K. Vashishta, P. (eds.) p. 209. New York: Elsevier Science Publishing Co., 1982
Zisook, A. B.: Intermittency in area-preserving mappings, Phys. Rev.A25, 2289 (1982)
Zisook, A. B., Shenker, S. J., Renormalization group for intermittency in area-preserving mappings, Phys. Rev.A25, 2824 (1982)
Hu, B., Rudnick, J.: Exact solutions to the renormalization-group fixed-point equations for intermittency in two-dimensional maps, Phys. Rev.A26, 3035 (1982)
Manneville, P., Pomeau, Y.: Intermittency and the Lorenz model, Phys. Lett.75A, 1 (1979)
Manneville, P., Pomeau, Y.: Intermittent transition to turbulence in dissipative dynamical systems, Commun. Math. Phys.74, 189 (1980)
Dubois, M., Rubio, M. A., Bergé, Experimental evidence of intermittencies associated with a subharmonic bifurcation P.: Phys. Rev. Lett.51, 1446 (1983)
Gibson, G., Jeffries, C.: (submitted for publication)
Hirsch, J. E., Huberman, B. A., Scalapino, D. J.: Theory of intermittency, Phys. Rev.A25, 519 (1982)
Hirsch, J. E., Nauenberg, M., Scalapino, D. J.: Intermittency in the presence of noise: A renormalization group formulation, Phys. Lett.87A, 391 (1982)
Hu, B., Rudnick, J.: Exact solutions to the Feigenbaum renormalization-group equations for intermittency, Phys. Rev. Lett.48, 1645 (1982)
Arnold, V. I., Avez, A.:Ergodic problems of classical mechanics. New York, Amsterdam: Benjamin, 1968
Chirikov, B.: A universal instability of many-dimensional oscillator systems, Phys. Rep.52, 265 (1979)
Nonlinear dynamics and the beam-beam interaction. Month M., Herrera, J. C. (eds.), Vol. 57. Am. Inst. Phys. Conf. Proc. (1979)
Guilleman, V., and Pollack, A.:Differential topology. New Jersey: Prentice Hall, 1974
Mackay, R. S.: A renormalization approach to invariant cycles in area-preserving maps, Physica7D, 283 (1983)
Collet, P., Eckmann, J.-P., Koch, H.: On universality for area-preserving maps of the plane, Physica3D, 457 (1981)
Eckmann, J.-P., Koch, H., Wittwer, P.: Existence of a fixed point of the doubling transformation for area-preserving maps of the plane, Phys. Rev.A26, 720 (1982)
Walker, J. G., Berry, M. V., Upstill, C.: Measurement of twinkling exponents of light focused by randomly rippling water, Optica Acta30, 1001 (1983)
Berry, M. V.: Universal power-law tails for singularity-dominated strong fluctuations, J. Phys. A15, 2735 (1982)
Author information
Authors and Affiliations
Additional information
Communicated by O. E. Lanford
Rights and permissions
About this article
Cite this article
Zisook, A.B. The complete set of Hamiltonian intermittency scaling behaviors. Commun.Math. Phys. 96, 361–371 (1984). https://doi.org/10.1007/BF01214581
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01214581