Universal multifractal properties of circle maps from the point of view of critical phenomena I. Phenomenology |
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Authors: | B Fourcade A -M S Tremblay |
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Institution: | (1) Département de Physique et Centre de Recherche en Physique du Solide, Université de Sherbrooke, J1K 2R1 Sherbrooke, Québec, Canada;(2) Present address: Department of Physics, Simon Fraser University, V5A 156 Burnaby British Columbia, Canada;(3) Institut von Laue. Paul Langevin, B. P. 156X, 38042 Grenoble Cedex, France |
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Abstract: | The strange attractor for maps of the circle at criticality has been shown to be characterized by a remarkable infinite set of exponents. This characterization by an infinite set of exponents has become known as the multifractal approach. The present paper reformulates the multifractal properties of the strange attractor in a way more akin to critical phenomena. This new approach allows one to study the universal properties of both the critical point and of its vicinity within the same framework, and it allows universal properties to be extracted from experimental data in a straightforward manner. Obtaining Feigenbaum's scaling function from the experimental data is, by contrast, much more difficult. In addition to the infinite set of exponents, universal amplitude ratios here appear naturally. To study the crossover region near criticality, a correlation time, which plays a role analogous to the correlation length in critical phenomena, is introduced. This new approach is based on the introduction of a joint probability distribution for the positive integer moments of the closest-return distances. This joint probability distribution is physically motivated by the large fluctuations of the multifractal moments with respect to the choice of origin. The joint probability distribution has scaling properties analogous to those of the free energy close to a critical point. |
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Keywords: | Onset of chaos circle map quasiperiodic route to chaos multifractals infinite set of exponents critical phenomena universality |
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