On the Scaling Function of Lyapunov Exponents for Intermittent Maps R. Stoop Institute of Mathematics, Swiss Federal Institute of Technology (ETH), CH-8092 Zurich, Switzerland J. Parisi Physical Institute, University of Bayreuth, W-8580 Bayreuth, Germany Z. Naturforsch. 48 a, 641-642 (1993); received February 16, 1993 The scaling function of Lyapunov exponents for intermittent systems is full of particularities if compared with hyperbolic cases or the usual, nonhyperbolic, parabola. One particularity arises when this function is calculated from finite-time Lyapunov exponents: Different scaling properties with respect to the length of the finite-time chains emerge. As expected from random walk models, the scaling of an ensemble with non-Gaussian fluctuations evolves for certain values of the external parameter. The evaluation of fractal dimensions and Lyapunov exponents has obtained wide-spread interest, espe- cially, since it was discovered that not only for models, but also for experimental data these concepts can be applied to characterize the statistical behavior of a system. A particularly difficult thing is the evaluation of the scaling function of Lyapunov exponents, 0 (A) [1], for intermittent systems (if compared with the hy- perbolic or the logistic cases). In this communication, we show scaling properties which depend on the length (i.e., the number of terms) of the finite-time Lyapunov exponents used for the evaluation of this function. We compare the hyperbolic case with differ- ent kinds of intermittent maps and find qualitatively different behavior. As the length of the finite-time Lyapunov exponents increases, the average Lyapunov exponent converges, a.e., towards the Lyapunov exponent of the system; </>(A), which describes the fluctuation of these values around the true exponent of the system, must, in this sense, be regarded as an entropy of the large-deviation property [2]. In Fig. 1, we show 0(A) for a hyperbolic system; the reduced map of a supercritical tent map has been considered (if not the reduced map is taken, a strange repeller is obtained). The scaling property of the obtainable width of 0 (A) with respect to the length Reprint requests to Prof. Dr. J. Parisi, Physical Institute, University of Bayreuth, P.O. Box 10 12 51, W-8580 Bay- reuth, Germany. n is indicated by the inset in the picture; as expected from theory, it scales as d(n) ~ n _1/2 . A completely different behavior is found for Man- neville's intermittent map in the version of Zumofen and Klafter [3]. Depending on the exponent z of the map, the results in Figs. 2 and 3 are obtained. In <t>(k) 1.3 1.0 1.3 1.5 X Fig. 1. Scaling function 0(A) of Lyapunov exponents of the reduced tent map, evaluated from 2000 chains of length n = 5. The scaling properties of the numerically obtained part of 0(A) is indicated in the inset. There, we display the width of this region versus the number of times the original length of the chain has been doubled. 0932-0784 / 93 / 0400-653 $ 01.30/0. - Please order a reprint rather than making your own copy.