Singularity spectrum for period n -tupling in area-preserving maps

PHYSICAL REVIE%' A AUGUST 1, 1988 VOLUME 38, NUMBER 3 Singularity spectrum for period n-tupling in area-preserving maps Sang-Yoon Kim and Bambi Hu Department of Physics, Unioersity of Houston, Houston, Texas 77Z04 (Received 22 February 1988) f The singularity spectrum (a) and the generalized dimension D (q) of the critical orbit is numerically obtained to study the global scaling behavior of period n-tupling (n=2, 3,4) in area-preserving maps. It is found that (a) becomes quite different as n is changed, and the generalized dimension D(q) increases for all q as n increases. The global scaling behavior of conservative systems is different from that of dissipative systems. Moreover, for conservative systems, the global scaling behavior seems to depend on dimensionality. f I. INTRODUCTION of level Recently, Halsey et al. ' introduced the singularity spectrum (a) to describe the global scaling behavior of the probability measure lying on strange sets. This formalisrn has been applied to many systems, for example, period doubling in the logistic map, ' the devil's staircase of mode locking and the critical "golden-mean" orbit of the circle map, ' the spectrum of a quasiperiodic Schrodinger operator, and the critical KAM curve in the standard map. The purpose of this paper is to report on the application of the formalism to period n-tupling in the areapreserving De Vogelaere map. The De Vogelaere map T is of the following form: f T x„+]———y„+fz(x„), ~ yn+]=xn (1.1) fp(xn+]) — where fz(x) =px (1 — p)x . The local scaling behavior ~ about some special point has been studied for period dou—s — and higher period n tupling s » They ' found that the pattern of periodic orbits repeats itself asymptotically from one bifurcation to the next for even n, and to every other one for odd n. In other words, for even period n-tupling, the pattern of periodic orbits exhibits a "period-1" scaling behavior; whereas for odd period n tupling, it exhibits a "period-2" scaling behavior. The scaling behavior of the power spectrum' and the trajectory scaling function of the displacement vectors' were studied for period doubling. However, for higher period n-tupling (n & 3), the global scaling behavior is still unknown. It is therefore interesting to study the singularity spectrum of the critical orbit for period n-tupling and see how the singularity spectrum and the generalized dimension change as n is varied. In Sec. II we obtain the singularity spectra for period n-tupling (n =2, 3, 4) by the ratio method, and compare them. In Sec. III we summarize the results. bling4 II. SINGULARITY SPECTRUM FOR PERIOD n-TUPLING We now use the algorithm developed in Ref. 1 to calculate the singularity spectrum. We take the critical orbit 38 ' zk Here . Let m whose period N is n p tupling —(x]I ', yk™)=T",(zo '), P ' is the accumulation k =0, 1, . . . , n point for period n- ' ' p'= —1.266311276922 10 = —0. 477013 68427404 = —0. 689 82444028347 (n (n (n =2) =3) =4) . At the accumulation point p* and for each level m, there is one unstable orbit for n = 2 whose critical residue value R is 1.135 87. However, for higher n (n & 3), there exist two kinds of orbits, one stable and the other unstable. The critical residue values of the stable and unstable orbits for n = 3 and 4 are, respectively, ' " R*=0.7337, —0.0092 =0. 5178, —0.0277 (n (n =3) =4) . We then define the partition of level m as follows. n =2, let the length lk be the distance between zk its closest point zk+'&/2. lk (m) —zk+W/2 —zk(m) For ' and (2. 1) These lengths serve as natural scales for a partition of measure Mk 2/N attributed — to each scale. For n =3, however, the closest points to zk™ are zk+'+/3 and zk+'2Q/3 Let us denote the triangle whose zk+N/3 and zk+2N/3 by Tk . Then k there are N/3 triangles Tt', ' (k =0, 1, . . . , N/3 —1). For each triangle Tk ', there are three sides, and let the length lk . be the length of the jth side: level m with probability It, , zk+~jttt/3 — zk+(J—]]+/3 (j =1,2, 3) . (2.2) Then for n =3, these lengths serve as natural scales for a partition of level m with probability measure Mk ~ —1/N attributed to each scale. Similarly, for n =4, there are (m) (m) (m) three points zk+&/4 zk+2&/4, and zk+3N/4 which are '. Let us denote the quadrilateral whose closest to zk (m) (m) (m) (m) (m) vert]ces are zk, zk+xx4, zk+2NI4, and zk+3]vq4 by Qk Then there are N/4 quadrilaterals Qk 1534 1988 The American Physica1 Society SINGULARITY SPECTRUM FOR PERIOD n-TUPLING 38 (k =0, 1, . . . , X/4 —1). Let the length of the jth side of the kth quadrilateral: lk J (m) —zI,(m)+JNI4 —zk+() lk J be (2.3) Then, like the period-tripling case, these lengths serve as natural scales for a partition of level m with probability measure Mk —I /N attributed to each scale. The probability measure M, (l;) can be described by defining a scaling index a; of the form' M, (l;}=I j for some which depends on period n-tupling. As mentioned in Sec. I, even period n-tupling sequences exhibit a "period-1" scaling behavior, while odd-period n-tupling exhibit a "period-2" scaling behavior. ' sequences Therefore, for even-period n-tupling, =1; whereas for =2. After obtaining ~„we comodd-period n-tupling, pute a algebraically rather then differentiate ~, with respect to q. After some algebra, we obtain a= Typically, for sufficiently small I, , the scaling index a, takes on a range of values between a;„and „. Let rt (a)da be the number of singularities of type a' for all a' lying between a and a+da, and l a typical length of the partition. Then the number of singularities can be described by defining a scaling index (a) of the form' a, j j ' ' 1535 Z +J(q, r, )/Z (q, r, )=1 the length (J =1)2)3,4) . iwt'4 IN. . . where (ln(M} — ln(M} &1 (I) &. — &I (~) &. (F & & & & „ (2. 10) denotes the average value of E with respect to the probability distribution P, (m) of level m, f 'da . n(a)da=p(a)l (a) may be interpreted ' fof singularities of type a. 20 (2 5) (0) as the "dimension" of the subset I5- Now we want to know the possible values of a and the function (a) for period n-tupling. With the partition of level m defined above, we form the partition function f Z (qr} Z (q, r)= —1 N/2 g=0 Mgl„' (n =2) 0 -40 Jc N/3 I I I I -20 0 20 40 0 20 40 0 I I 20 40 I —1 3 y k=0 j=1 N/4 —1 4 Mk, JIk, J~ (n g Mk)lk, ' j=l k=0 (n =3) =4) . (2.6) (b) I50— Halsey et al. ' argued that for large m there is a unique critical function r, (q) such that r & r, (q) for forr(r 0 (q) (2.7) . 50- That is, the partition function Z (q, r) is of order unity only when r =r, (q). The critical function r, (q) is related to the generalized dimension D of Hentschel and Pro- I -40 I -20 caccia' by (q —1)D(q)=r, (q) . 20 (2.8) (c) For example, D (0) is the Hausdorff dimension of the support of the probability ineasure, while D (1) is the information dimension and D (2) the correlation dimension. ' The scaling indices transformation dr, a(q)=, (q) dg a and f (a) are given f(q)=qa(q) by a Legendre r, (q) . —9) (2. f Eliminating q gives us the singularity spectrum (a) defined in a range In practice, the solutions to Z (q, r, ) = 1 converge slowly with m. To improve the convergence, we use the ratio method. ' We can determine r, (q} by requiring that [a;„,a,„]. I5- I 0 -40 I -20 FIG. 1. The scaling function o(q) for period n =2, (b) n = 3, (c) n =4. n-tupling. (a) 38 SANG-YOON KIM AND BAMBI HU 1536 08— (2. 11) Let us define the typical length L (m) of level m by ln[L (m)]= (ln(l) ) (2. 12) f Then, since the probability measure M, is constant for period n-tupling, the scaling index a is given by ln(ni) ln[L (m)/L (m 0.2— (2. 13) +j)] 0.4— 0.0 0.2 0.4 0.8 0.6 Let us define the scaling function a (q) as the ratio of the typical length of level m to that of level m +j, L(m) (2. 14) L(m+j) ln(n J) From Eq. (2. 15), we see that by varying q we can visit the different regions with singularities of type a. The maximum and minimum values of the scaling function o and o;„determine the range „], and hence the range of D (q) [a;„,a, a,„=D( — oo )= a;„=D(oo )= ln( n i) ln(a, „) (2. 16) ln(n') ln(a, „) Here a;„corresponds to the region in the set where the probability measure is the most concentrated, while „ corresponds to the region where the probability measure is the most rarefied. The scaling functions o (q) for period n-tupling (n =2, 3, 4) are shown in Fig. 1. From o m, „and cr;„, we obtain the range a, [a;„,a, „]=[0.248, 0. 498] =[0.420, 0. 680] =[0.522, 0. 873] (n =2) =3) (n =4) (n . I n=4 0.8- 0.6- I -40 I I I I -20 0 20 40 g FIG. 2. The (n =2, 3, 4). spectrum f (a) for period n-tupling (2. 15) ln[o (q)] 02, =2, 3, 4). a becomes Then the scaling index D F1G. 3. The singularity (n generalized dimension D(q) for period n-tupling For n =2, the values of 0. ;„and o. „are equal to the local scaling factors a and P, ' where a is the scaling factor along the symmetry line and P the scaling factor across the symmetry line. However, for n =3 and 4, only cr, „ is equal to the local scaling factor P across the symmetry line. ' This means that in some other region which is not near a symmetric point, the probability measure is the most rarefied for n =3 and 4. After obtaining ~, and a, we can compute the general- , f ized dimension D(q) and the singularity spectrum (a) from Eqs. (2.8) and (2.9). The numerical result of D(q) and (a) are shown in Figs. 2 and 3. We see that D (q) is a decreasing function of q, ' and (a) is a convex function with a single maximum at q =0. The maximum value of (a) is just the Hausdorff dimension. For n =2, (a) is completely different from that of period doubling in the logistic map. ' Therefore, the global scaling behavior of conservative maps is different from that of dissipative maps. Furthermore, the left part of (a) which corresponds to the region in which q varies from 0 to ~ is quite different from that of four-dimensional symplectic maps, ' whereas the right part is only slightly different. Therefore, the global scaling behavior of period doubling in symplectic maps seems to depend on dimensionality. For n = 3 and 4, there are, as mentioned previously, two critical orbits, one stable and the other unstable. %e have studied the two critical orbits and found that the singularity spectra of the two orbits agree very well. Finally, we have studied how the singularity spectrum and the generalized dimension for period n-tupling change as n is varied. As shown in Figs. 2 and 3, the singularity spectra for different period n-tupling are quite different, and the generalized dimension increases for all q as n is changed. Therefore, different period n-tupling shows different global scaling behavior. In a recent study of the universal scaling ratio of the power spectrum in period n-tupling, Hu, Shi, and Kirn' observed that the scaling ratio increases with n. However, the rate of increase seems to slow down and approach a 1imiting value. It wi11 be interesting to see if similar behavior occurs for (a) and D (q). f f f f f f SINGULARITY SPECTRUM FOR PERIOD n-TUPLING 38 III. SUMMARY %'e have obtained the singularity spectrum and the generalized dimension of period n-tupling (n =2, 3, 4) in area-preserving maps to study the global scaling behavior of the critical orbit. It is found that the singularity spectrum (a) for period n-tupling becomes quite different as n is changed, and the generalized dimension increases for all q as n increases. The singularity spectrum of period doubling in area preserving maps is different from those of the logistic map and four-dimensional symplectic maps. Therefore, the global scaling behavior of conserva- f IN. . . 1537 tive systems is different from that of dissipative systems. Moreover, for conservative systems, the global scaling behavior seems to depend on dimensionality. ACKNO%'LED GMENTS One of us (S.Y.K.) acknowledges useful discussions with Bin Lin. This work was supported in part by the U. S. Department of Energy under Grant No. DE-FG0587ER40374 and the Korea Science and Engineering Foundation. and R. S. 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