VOLUME 55, NUMBER 4 PHYSICAL REVIEW LETTERS 22 JULY 1985 Renormalixation, Unstable Manifolds, and the Fractal Structure of Mode Locking Predrag Cvitanovic~'~ Laboratory of Atomic and Solid State Physics, Corneil University, Ithaca, New York 14853 and Mogens H. Jensen and Leo P. Kadanoff The James Franck and Enrico Fermi Institutes, University of Chicago, Chicago, Illinois 60637 and Itamar Procaccia Department of Chemistry and The Jam'es Franck Institute, University of Chicago, Chicago, Illinois 60637 (Received 4 March 1985) The apparent universality of the fractal dimension of the set of quasiperiodic windings at the on- set of chaos in a wide class of circle maps is described by construction of a universal one-parameter family of maps which lies along the unstable manifold of the renormalization group. The manifold generates a universal "devil's staircase" whose dimension agrees with direct numerical calculations. Applications to experiments are discussed. PACS numbers: 05.45. +b, 03.20. +i, 47. 20. +m, 74.50. +r In the context of the transition to chaos via quasi- periodicity, most attention has been paid to the local scaling behavior at a particular irrational winding number. ' Although universal behavior has been theoretically predicted, ' ' its experimental verification has not followed, simply because minute changes in winding numbers lead to large changes in scaling behavior. It appears that of greater interest and ex- perimental accessability are those universal properties that are globa/ in the sense of pertaining to a range of winding numbers. Indeed, such a property has been found and reported by Jensen, Bak, and Bohr, and has to do with the set complementary to the "tongues" on which the dynamical system is mode locked. This set of unlocked or irrational windings has at the onset of chaos Lebesgue measure zero, and ap- parent universal fractal dimension D. Recent experi- ments on Josephson junction simulators and charge density waves7 have indicated the existence of this phenomenon and revealed results in agreement with the findings in Ref. 3. For the simple circle map 0„+& — — 0„+0 — (K/2n. ) sin(2n. H„) this transition occurs at K =1; see Fig. 1. On the plotted intervals the winding number 8'is locked on a rational value as shown. The gaps between the locked states are "full" of locked states that add up to Lebesgue measure l. The set of irrational winding numbers is the complement of the locked intervals. We calculated the dimension D of this set in a way slightly different from Ref. 3. We be- lieve that D is the same for all regions of gaps. Thus we can start with any pair of locked intervals P/Q and P'/Q'. The length of the gap between them is denoted by s. Next the locked interval (P+P')/(Q+Q') is found, and the gaps of length s& and s2 between the 1. 0 0. 8— 0. 6— P Q 0. 4— 0. 2— 0. 0 0,0 0 0.2 3 3 2 7 1 8— 3 0. 4 5 8 , —, L 7- I 0. 6 5 0.24— 0.22— 1 5 0.25 I 4f. . 2 9 3 l4 0.26 0. 8 3 I3. I 0.27 f 1. 0 FIG. 1. The mode-locking structure at K = 1 for the map (1). The "devil's staircase" is complete, and the comple- ment of the mode-locked windings is of Lebesgue measure zero and universal fractal dimension D (Ref. 3). new interval and the preceding one are found. This "Farey tree" construction is continued until a large number of gap sizes s; are found. The fractal dimen- sion D is then estimated from the formula9 g;R; = 1, where R; = s;/s. Denoting the result from the n th Farey level as D„, and the quantity min;(R;") as R", we fitted a power law D„= D' + a (R")". An excellent fit with eleven Farey levels (n = 1, . . . , 11) starting with P/Q = — „and P'/Q'= 2, was obtained. The number D, which is our direct numerical estimate of the dimension of the set, was found to be 0.868 + 0. 002, in agreement with Ref. 3. Surprisingly, the value of D&, an estimate based on only two gaps, was always very close to D" (the deviation less than 1%). The result was invariant to the choice of P/Q and P'/Q' and can be applied to any interval of the staircase on Fig. 1. Moreover, the result is invariant to the choice of dynamical system 0„+t f (iI„) as long— — 1985 The American Physical Society 343