Abstract
Every orientation preserving circle mapg with inflection points, including the maps proposed to describe the transition to chaos in phase-locking systems, gives occasion for a canonical fractal dimensionD, namely that of the associated set of μ for whichf μ=μ+g has irrational rotation number. We discuss how this dimension depends on the orderr of the inflection points. In particular, in the smooth case we find numerically thatD(r)=D(r −1)=r −1/8.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Huxley, A.F.: Ion movement during nerve activity. Ann. N.Y. Acad. Sci.81, 221–246 (1959)
Fitzhugh, R.: Impulses and physiological states in theoretical models of nerve membrane. Biophys. J.1, 445–446 (1961)
Best, E.N.: Null space in the Hodgkin-Huxley equations. A critical test. Biophys. J.27, 87–104 (1979)
Guttmann, R., Feldman, L., Jakobsson, E.: Frequency entrainment of squid axon membrane. J. Memb. Biol.56, 9–18 (1980)
Guevara, M.R., Glass, L.: Phase locking, period doubling bifurcations and chaos in a mathematical model of a periodically driven oscillator: A theory for the entrainment of biological oscillators and the generation of cardiac dysrhythmias. J. Math. Biol.14, 1–23 (1982)
Glass, L., Perez, R.: Fine structure of phase locking. Phys. Rev. Lett.48, 1772–1775 (1982)
Perez, R., Glass, L.: Bistability, period doubling bifurcations and chaos in a periodically forced oscillator. Phys. Lett.90 A, 441–443 (1982)
Ben-Jacob, E., Braiman, Y., Shainsky, R., Imry, Y.: Microwave-induced “devil's staircase” structure and “chaotic” behaviour in current-fed Josephson junctions. Appl. Phys. Lett.38, 822–824 (1981)
Keutz, R.L.: Chaotic states ofrf-biased Josephson junctions. J. Appl. Phys.52, 6241–6246 (1981)
Ben-Jacob, E., Goldhirsch, I., Imry, Y., Fishman, S.: Intermittent chaos in Josephson junctions. Phys. Rev. Lett.49, 1599–1602 (1982)
Levinsen, M.T.: Even and odd subharmonic frequencies and chaos in Josephson junctions: Impact on parametric amplifiers? J. Appl. Phys.53, 4294–4299 (1982)
Mawhin, J.: Periodic oscillations of forced pendulumlike equations. Lecture Notes in Mathematics, Vol. 964, pp. 458–476. Berlin, Heidelberg, New York: Springer 1982
Glass, L., Guevara, M.R., Shrier, A.: Bifurcation and chaos in a periodically stimulated cardiac oscillator. Physica 7D, 89–101 (1983)
Glass, L., Guevara, M.R.: One-dimensional Poincaré maps for periodically stimulated biological oscillators. Théorie de l'iteration et ses applications, Colloques Internationaux du CNRS332, 205–210 (1982)
Bak, P., Bohr, T., Jensen, M.H., Christiansen, P.V.: Josephson junctions and circle maps. Solid St. Commun.51, 231–234 (1984)
Shenker, S.J.: Scaling behaviour in a map of a circle onto itself: Empirical results. Physica5 D, 405–411 (1982)
Jensen, M.H., Bak, P., Bohr, T.: Complete devil's staircase, fractal dimension, and universality of mode-locking structure in the circle map. Phys. Rev. Lett.50, 1637–1639 (1983)
Feigenbaum, M.J., Kadanoff, L.P., Shenker, S.J.: Quasiperiodicity in dissipative systems: A renormalization group analysis. Physica5D, 370–386 (1982)
Feigenbaum, M.J., Hasslacher, B.: Irrational decimations and path integrals for external noise. Phys. Rev. Lett.49, 605–609 (1982)
Rand, D., Ostlund, S., Sethna, J., Siggia, E.: Universal transition from quasiperiodicity to chaos in dissipative systems. Phys. Rev. Lett.49, 132–135 (1982)
Ostlund, S., Rand, D., Sethna, J., Siggia, E.: Universal properties of the transition from quasiperiodicity to chaos in dissipative systems. Physica8D, 303–342 (1983)
Bak, P., Bohr, T., Jensen, M.H.: Transition to chaos by interaction of resonances in dissipative systems, I. Circle maps, II. Josephson junctions, Charge-density-waves, and standard maps. Phys. Rev. A30, 1960–1981 (1984)
Mandelbrot, B.: The fractal geometry of nature. San Francisco: Freeman 1982
Belykh, V.N., Pedersen, N.F., Soerensen, O.H.: Shunted-Josephson-junction model. II. The nonautonomous case. Phys. Rev. B16, 4860–4871 (1977)
Alstrøm, P., Levinsen, M.T.: The Josephson junction at the onset of chaos: A complete devil's staircase. Phys. Rev. B31, 2753 (1985)
Arnold, V.I.: Small denominators. I. Mappings of the circumference onto itself. Am. Math. Soc. Transl. Ser. 246, 213–284 (1965)
Poincaré, H.: Mémoire sur les courbes définie par une equation différentielle, I, II, III, IV. J. Math. Pures Appl. (3)7, 375–422 (1881); (3)8, 251–286 (1882); (4)1, 167–244 (1885); (4)2, 151–217 (1886)
Iooss, G.: Bifurcation of maps and applications. Amsterdam, New York, Oxford: North-Holland 1979
Herman, M.R.: Sur la conjugaison différentiable des difféomorphismes du cercle à des rotations. Publ. Math. I.H.E.S.49, 5–234 (1939)
Herman, M.R.: Mesure de Lebesgue et nombre de rotation. Lecture Notes in Mathematics, Vol.597, pp. 271–293. Berlin, Heidelberg, New York: Springer 1977
Söderberg, B.: Not published
Hua Loo Keng: Introduction to number theory. Berlin, Heidelberg, New York: Springer 1982
Henley, C.L.: Not published
Author information
Authors and Affiliations
Additional information
Communicated by O. E. Lanford
Rights and permissions
About this article
Cite this article
Alstrøm, P. Map dependence of the fractal dimension deduced from iterations of circle maps. Commun.Math. Phys. 104, 581–589 (1986). https://doi.org/10.1007/BF01211066
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01211066