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- new interval and the preceding one are found. This periodicity, most attention has been paid to the local "Farey tree" construction is continued until a large scaling behavior at a particular irrational winding number of gap sizes s; are found. The fractal dimen- number. ' Although universal behavior has been sion D is then estimated from the formula9 g;R; = 1, theoretically predicted, ' ' its experimental verification where R; = s;/s. Denoting the result from the n th has not followed, simply because minute changes in Farey level as D„, and the quantity min;(R;") as R", winding numbers lead to large changes in scaling we fitted a power law D„=D' + a (R")". An excellent behavior. It appears that of greater interest and ex- fit with eleven Farey levels (n = 1, . , 11) starting perimental accessability are those universal properties with P/Q = —„and P'/Q'= 2, was obtained. The that are globa/ in the sense of pertaining to a range of number D, which is our direct numerical estimate of winding numbers. Indeed, such a property has been the dimension of the set, was found to be found and reported by Jensen, Bak, and Bohr, and 0.868 + 0.002, in agreement with Ref. 3. Surprisingly, has to do with the set complementary to the the value of D&, an estimate based on only two gaps, "tongues" on which the dynamical system is mode was always very close to D" (the deviation less than locked. This set of unlocked or irrational windings has 1%). The result was invariant to the choice of P/Q at the onset of chaos Lebesgue measure zero, and ap- and P'/Q' and can be applied to any interval of the parent universal fractal dimension D. Recent experi- staircase on Fig. 1. Moreover, the result is invariant to ments on Josephson junction simulators and charge the choice of dynamical system 0„+t f(iI„) as long—— density waves7 have indicated the existence of this phenomenon and revealed results in agreement with the findings in Ref. 3. 1.0 — . For the simple circle map 0„+& 0„+0 5 4f. 0.8— —(K/2n. sin(2n. this transition occurs at K =1; 5 ) H„) 8 see Fig. 1. On the plotted intervals the winding , —, L 6— 7- number 8'is locked on a rational value as shown. The 0. P 3 0.24— 2 3 gaps between the locked states are "full" of locked Q 3 7 I3. 0.4— 1 8— 2 3 9 states that add up to Lebesgue measure l. The set of 0.22— 3 l4 irrational winding numbers is the complement of the 2— 0. 1 5 I locked intervals. We calculated the dimension D of 0 0.25 0.26 0.27 this set in a way slightly different from Ref. 3. We be- 0.0 I I f 0,0 0.2 0.4 0.6 0.8 1.0 lieve that D is the same for all regions of gaps. Thus we can start with any pair of locked intervals P/Q and FIG. 1. The mode-locking structure at K = 1 for the map P'/Q'. The length of the gap between them is denoted (1). The "devil's staircase" is complete, and the comple- by s. Next the locked interval (P+P')/(Q+Q') is ment of the mode-locked windings is of Lebesgue measure found, and the gaps of length s& and s2 between the zero and universal fractal dimension D (Ref. 3). 1985 The American Physical Society 343 VOLUME 55, NUMBER 4 PHYSICAL REVIEW LETTERS 22 JUL+ 1985 as f'(8) has a cubic inflection point. found empirically that the set of interest was invariant To understand the apparent universality we turned to the choice of initial P/0 and P'/0' of the Farey-tree to a renormalization-group formulation. The analysis construction, we can as well pick values according to given below will provide convincing evidence that all two Fibonacci-number ratios. In this way we shall the D's constructed in a small neighborhood of any make full use of the work that has been done on the "golden" winding number —one whose continued local scaling properties near the golden mean. Define fraction ends in an infinite string of ones —will have now the very same value of D. Since these numbers are f(n) n+1( dense in the interval [0, 1], one has the first step in an f ) F argument that D is truly universal. Denoting by A„ the value of 0 for which fn(") (x) has Previously, ' ' the renormalization-group formula- a superstable cycle with winding number F„,/F„, we tion has been used in this context to study the local define scaling properties of golden winding numbers like ' N"= (JS—1)/2. A series of rational approximants g()'"' (x) = c„,f("',(x/c„,). w„= was constructed F„/F„+) by using Fibonacci = numbers F„+1=F„+F„1,Fo=0, F1 = 1. Defining By construction x 0 is a superstable fixed point of go(") (x). We want now the parameter range that spans f (x) =f "+'(x) —F f" =n"f (o. "x) the distance between the superstable cycle F„&/F„ and the next one to be rescaled to one finds F„/F„+, the interval [0, 1]. We do so by turning g(")(x) into a one-param- — — f(ll +1) f(N)( f(n ))( 2x)) eter family by defining In the limit n ~ one obtains the fixed-point equa- =uf'(nf'(o. The solution to this tion f'(x) 'x)). — equation and its linearized version yields the relevant where ~„ is picked such that 0„+1—0„+6„. Ac- scaling parameters n and 5, which are the exponents in cordingly, for p = 0 g~" has a superstable fixed point. x space and in parameter space, respectively. ' ' Un- The value p = 1 corresponds to the next Fibonacci lev- fortunately in this formulation the dependence on the el (F„+2) superstable cycle of the original map. No- parameter 0 is lost, and the universal mode-locking tice that in Eq. (1) c„+) is an arbitrary scale factor. () structure cannot be investigated. What is needed is a We fix it by picking the normalization g, (0) =1. formulation that maintains the dependence on a Writing now the composition parameter. Such a formulation is achieved by parametrization of the unstable manifold. " %'e construct the unstable manifold by starting with any given one-parameter family of functions fn (x) which have a cubic inflection point at x = 0. Since we we use Eq. (1) to obtain the exact result l' (n) (n —1) ( yg —2) g& (x) = 0!zgt+~/g (~~ —)g$ ~)/5„&+p/8„&5„(x/on on —1) )~ (3) where g„=5„/5„+) and n„=c„+&/c„. After infinitely many o. rescalings of x space around the inflection tio w„. However, around p = 1 there is another locked = and infinitely 5 shifts and rescalings point x 0, many state which corresponds to the next locked region in we reach the universal one- in parameter space, the sequence and the width of this region is scaled family which lies on the un- parameter of maps g~(x) down by 5 compared to the first (we remember" that stable manifold and is invariant under rescaling and the meaning of 5 is that 0„=OG~+a/5"). Around two-cycle composition. From Eq. (3) we get the exact p= 1+1/8 there is another, scaled down by 8 com- result pared to the first, etc. Thus, by studying the stability of the fixed point of we can find an infinity of g (x) =~g)+ /s(ag, ,/, /, (x/'n')). (4) g~ mode-locked states which are universally located. The normalization conditions are go(0) = 0, g) (0) = 1. However, these are not ali the locked ranges. In Fig. 2 We use now the universal object g~(x) to investigate we plot the largest locking ranges that can be obtained the structure of mode lockings. As noted before, for in the way just described, and also indicate some of p = 0 g~(x) has a superstable fixed point at x = 0. The those that do not fall into this category, since they cor- range of p around zero for which g~ (x) still has a fixed respond to winding numbers that are not F„/F„+&. point is the range of parameters for which the original These are indeed needed to determine the fractal map is locked on some ("infinitely" high) winding ra- dimension D. How can we get them from the univer- 344 VOLUME 55, NUMBER 4 PHYSICAL REVIEW LETTERS 22 JuLY 1985 sal object g~(x) '? Consider for example the range denoted (F„+F„+2)/(F„+ t + F„+3) .