Scaling symmetries in nonlinear dynamics. A view from parameter space

PHYSICA ELSEVIER

Physica D 81 (1995) 23-31

Scaling symmetries in nonlinear dynamics. A view from parameter space Henry Hurwitz, Jr. 1, Michael Frame 2, David Peak 3 Union College, Schenectady, N Y 12308, USA

Received 19 August 1993; revised 31 May 1994; accepted 28 June 1994 Communicated by C.K.R.T. Jones

Abstract

The orbit of the critical point of a discrete nonlinear dynamical system defines a family of polynomials in the parameter space of that system. We show here that for the important class of quadratic-like maps, these polynomials become indistinguishable under a suitable scaling transformation. The universal representation of these polynomials produced by such a scaling leads directly to accurate approximations for those parameter values where windows of order appear, the sizes of such windows, measures of window distortion, and the characterization of the internal structure of the windows in terms of generalized Feigenbaum numbers.

1. Introduction

Nonlinear dynamical s y s t e m s - both theoretical and physically realized - exhibit a rich variety of behaviors, often including the interleaving of periodicity and chaos as a control parameter is varied. In many practical situations periodic output is maintained in the midst of chaos by very fine tuning. On the other hand, recent investigations of physiological dynamics [1] and on controlling chaotic behavior [2] suggest that there may be advantages to a v o i d i n g periodic response. In a similar vein, we note that failures of such numerical procedures as Newton's method often result when iterated approximate solu1Deceased. 2Mathematics Department. 3Physics Department.

tions become trapped in nonfixed periodicity [3,4]. Obviously, identifying and characterizing periodic windows of nonlinear maps is a worthwhile, albeit frequently challenging, task. In this paper we report a powerful, new analytic tool (motivated by numerical studies in [5]) for probing the details of the canonical quadratic map, x,,+l = F c ( x , ) = x 2 + c, and cite examples of its utility. Since a broad class of maps reduce to quadratic-like form (on suitably chosen domains) [3], the results presented here are actually of fairly wide applicability. Our paper is based on the well-known fact that superstable points and Misiurewicz points of the function Fc(x ) can be located by solving fn(C) = 0 and f , , , ( c ) = f k ( C ) , respectively, where f~(c) is the nth composition of F c evaluated at x = 0, i.e., fn(C) =Fc°'(0). The new points presented here are:

0167-2789/95/$09.50 t~) 1995 Elsevier Science B.V. All rights reserved SSD1 0167-2789(94)00161-8

24

H. Hurwitz et al. / Physica D 81 (1995) 23-31

(i) The polynomials f,(c) are well approximated (for sufficiently large n) by 2 cos(v~), if c is real and sufficiently close to - 2 ; e is defined through the replacement e = 4"(c + 2)/6. (ii) The zeroes of cos(wrY) give the approximate locations of large numbers of the superstable cycles of F c accumulating at c = - 2 . (iii) There is a scaling relation between the set of polynomials f3,, Ln, and f,, evaluated near a zero of fn, and the set f3, f2, and fl, evaluated near the zero of fl, namely, c = 0. Two different measures of the size of the n-cycle superstable window (which incorporates that zero of f n - the "center" of the window) are obtained directly when fn is well approximated by 2 cos(x/~). (iv) Taken together, the latter measures provide a theoretical understanding for a well-known numerical observation of Yorke, et al. [6], concerning the absence of distortion in the periodic windows of the quadratic map. (v) One of the size measures cited in (iii), above, provides an expanded definition of the Feigenbaum ct and ~ scale factors. It generalizes a result of Eckmann and Epstein [7], and leads to another theoretical elucidation of a previously unexplained numerical observation of Delbourgo, et al. [8], concerning such generalized scalings. The central value of the approach described in this paper is that it focuses on renormalization in parameter space only, thus avoiding some of the complications associated with more conventional approaches [7,9-11]. The identification of the universal function 2 c o s ( ~ ) for the quadratic map organizes a substantial body of apparently unrelated results into a single theoretical framework. Finally, since our method can be applied to any analytic function, it can be used to establish analogous results for other dynamical systems.

ing the renormalization symmetries of the dynamic space of Fc(x ) [7,9-11]. Instead, our approach focuses on the affine symmetries of the dynamics in its associated parameter space. In particular we concentrate on the c-space polynomials,

f,+l(c) = [f,(c)] 2 + c

(1)

(with fl (c) = c). These polynomials carry dynamical information because they track the orbit of the critical point, x 0 = 0; that is, the nth iterate of x 0 = 0 is x , = F c ° " ( 0 ) = f , ( c ) - w h e r e F °" means n compositions of F. We restrict oUr attention to real c's in the interval [ - 2 , +1/4], the range over which the orbits of 0 are bounded. Note that f n ( - 2 ) = 2, for all n 1>2, and that c = - 2 is an accumulation point for roots of the family of parameter space polynomials. See Fig. 1. An examination of these polynomials for values of c near - 2 suggests that as n increases they differ only by a rescaling of their arguments. Let c = - 2 + e/p,, where e/p, is small compared to 1. Then we introduce the new family of polynomials g,(e) through the relation g.(e) = f . ( - 2 + e/p.) and hypothesize that for an appropriate choice of scale factor p. g. (e) ~

g(e),

where g is some universal function (that is, independent of n). We can fix the scale factor by

2. The universal function

A number of universal scaling properties of nonlinear maps have been identified by exploit-

Fig. 1. The polynomialsf6(c) (bold) and fT(c) are plotted for c near -2. As the order of the polynomialf~ increases, it has more and more roots near c = -2.

H. Hurwitz et al. / Physica D 81 (1995) 23-31

fixing the slope of g near E = 0 because f ' ( - 2 ) / Pn ~'g'(O). If we arbitrarily choose g ' ( 0 ) = - 1 , then the associated choice for p, is approximately --fn'(--2)- (Note that since f ~ ( - 2 ) = 2 , n ~> 2, g(0) must be 2.) Now, the defining recursion relation (1) between the f.'s results in similar recursions between the derivatives of the f,'s. In particular, we find that f'n+x(C) = 2fn(C) f'(c) + 1, which can be evaluated recursively to obtain f n ' ( - 2 ) = - 4 n / 6 - 1/3, for n ~>2. Thus, for large n, f ' ( - 2 ) is 4"/6. We take the scale factor p, to be 4"/6. Using this result, one can easily show that (i) is equivalent to a recursion relation between the gn'S:

g.+a(4e) = [g.(.)]2 _ 2 + e/p,.

(2)

Setting, in (2), g.(e) = E (% + yn,i/p.)e j, where the a ' s describe the n-independent part of g, produces k

25

2

0

E

Fig. 2. A few of the polynomials f n ( - 2 + Elpn) are plotted as a function of e. A s n increases (5, 6, 7, 8 , . . . ) the polynomials approach the universal behavior given by g = 2 cosx/~.

number of real roots of fn closest to c = - 2 that the cosine approximation identifies with an error of less than 1%. For n = 5 the approximation identifies one such root (the second, mentioned in association with Fig. 2, has a slightly higher error) which for n = 25 it yields in excess of 10,000.

j=O

3. Superstable points- locating midgets

k

%+1,k = 4-k+1 ~ ( 20~k-]'Yn,] "4-(Tn,lTn,k_i) /Pn) j=O

with % = 2 , a 1 = - 1 , %,0 = 0 , and y.,~ = - 1 / 3 . Straightforward, though tedious, algebra reveals that for e/p, sufficiently small

g,(e) ~ 2 eosV~ + 6(e/p,)(nx/~ sinv'~ - 2).

The roots of the polynomials f. correspond to values of the control parameter for which the 100000-

(3)

In other words, the polynomials f . ( - 2 + E/p,) are well approximated by the universal function 2 cosx/~ for values of e/p, sufficiently small. Fig. 2 illustrates the relation between the polynomials f5 through fs and g = 2 cosv'~. All of these polynomials (and therefore all with n > 8 as well) are well approximated by the cosine representation through at least their first two roots. The range - in ~ - of good approximation rapidly increases with increasing n (four folding each time n increases by 1). The figure also suggests that the cosine approximation often provides remarkably good location of the roots of the fn's. A study of the reliability of using the cosine approximation for root finding is shown in Fig. 3. In this figure we show, for every n, the

0 O0

I0000-

0 O0

0

2 "0 0 0 '~

0

1000-

O0 o OO O0

100o o o

.o

E -I

0

10-

0

Z

0 o o Q

5

10

1'5

2'0

2'5

30

n

Fig. 3. For 5~
H. Hurwitz et al. / Physica D 81 (1995) 23-31

26

orbits of 0 are stable n-cycles. Since, at these c-values, 0 is a fixed point of Fc°" and since [Fc°"]'(0) = 0, these special orbits are said to be superstable. By (3) the superstable values of c (near - 2 for n sufficiently large) are associated with the roots of cosx/~. We denote the real roots of f, by c,,j where j = 1 corresponds to the root closest to - 2 (i.e., with kneading sequence C L R "-z [10]), j = 2, to the next closest (with kneading sequence C L R " - 3 L ) , and so on. Approximating f , by 2 cosx/e yields c,,j ~ - 2 + 6"rr2(2j

1)2/4 n+l .

-

(4)

T h e data points designated by circles in Fig. 4 show how the error between actual values of c,,j

and the estimate ( 4 ) - measured relative to the distance 2 + Cn, j - - d e c r e a s e s as n increases for the j = 1 series, that is, for the left-most superstable points for each n. Note that (4) extends a result of Eckmann and Epstein [7], which identifies the positions of the j = 1 superstable series only. The attentive reader will recognize that our analysis bears on the structure of the Mandelbrot (M-) Set along its real axis. Centers of the cardioids of midgets (reduced copies of the whole set) correspond to values of c for which the associated n-cycle is superstable; (4), therefore, permits the approximate location of such centers along the spike of the M-Set over the range in which (3) is valid.

4. Heads, crises, and distortions O"

o

II 0

III o

0

[] o

Ill 0

0

[] o

0

[] o

-5

0 I,,, 0 0 ,--I

-10-

¢,

1'o

1'5

20

n Fig. 4. T h e circles represent the relative errors

[actual c,,j - approximate cn,jl/[2 ÷ actual c,,jl for j = 1, where the calculated c,.j are given by Eq. (4). The diamonds represent the relative errors (approximate center-to-head distance)/ (actual center-to-head distance) for j = 1. The squares represent the relative errors (approximate center-to-crisis distance)/ (actual center-to-crisis distance) for j = 1. "Actual" values were obtained by finding the appropriate polynomial root with Newton's method, using the calculated value as the seed.

Having generated a successful scheme for approximating the location of large numbers of windows of periodic behavior in parameter space, we next turn to the question of estimating the sizes of those windows. Attached to each n-cycle cardioid in the M-Set is a 2n-cycle disk sometimes referred to as a " h e a d . " In the entire bifurcation diagram, the range in p a r a m e t e r space from the center of the main cardioid to the center of largest head occupies almost one-half of the total range of the diagram, so finding the head-to-center distance in a midget provides a significant measure of its total (parameter space) size. What can be said about the locations of the superstable centers of the 2n-cycle heads? First, consider the consequences of the recursive relationship between the f~'s; f.+l(c.,i) =[f~(c.,])] 2 +c.,j = c.,] =fl(c.,i). This equality string can b e iterated to demonstrate the general result that f.+m(c.,i)=f,.(c,,,]). In particular, f2.(c.,i) = f.(c.,]) = 0. Consequently, the jth root of f . is also a root of f2., that is, c., i = C2.,k for some k > j . The center of the attached 2n-cycle disk occurs at the next root off2 . closer to c = - 2 , at C2n,k_ 1. We designate the latter by h,,j, the

27

H. Hurwitz et al. / Physica D 81 (1995) 23-31

position of the center of the head in the (n, ])midget. Suppose f. is well approximated by the cosine representation at c.,j; can.fz, be represented by a similar cosine function there? Again, reference to the defining recursion relation (1) tells us that, at the root c.,i, f'+l(c.,j)=2f.(c.,i) f'(c.,i)+ l = l = f [ ( c . , ] ) which when iterated leads to f'+m(c.,y) = f ' ( c . d ). Thus, not only are f. and fz. equal at c.,], but so too are their derivatives, f" and f 2 . . A direct consequence of this requirement is that whenever f. is well approximated by 2 cosv~ at c.,j, f2. is surely not. The argument proceeds by contradiction. Assume that for c near Cn,i f.(c) ~ 2 cosx/-~ and fz. ~ 2 cosV~;e'. Setting the derivatives of f. and f2. equal at c.,j leads to V~e'=2"#-~. Thus if x,~ is an odd integral multiple of "tr/2 (as it must be f o r f . to be zero at c.,i) then ~ is an even integral multiple of IT/2. That is, if the cosine representation of f. is close to zero at G,] it will be more nearly ---2 for f2. - n o t zero as is required. The assumption that both f. and f2. are well approximated by the cosine representation at c.,j is therefore false. So, we cannot immediately extract head centers by representing f2. by a cosine function. Instead, we use a different tact. The center of the main cardioid for the entire bifurcation diagram of the dynamics is determined by the value of the parameter for which f~ and f2 are simultaneously zero; the center of the head attached to the main cardioid is determined by the next zero of f2 closer to c = - 2 . The center of cardioid of the (n, j)-midget is determined by a simultaneous zero of fn and f2. and the center of the attached head is determined by the next zero off2 . closer to c = - 2 . In this regard, within the (n, ])-midget fn plays the same role as does f~ in the entire bifurcation diagram, while f2n plays the same role as f2. We observe that over the entire parameter range of interest (that is, over the entire bifurcation diagram), f~ is linear and fz is quadratic. We ask, are f. and f2. similarly (approximately) linear and quadratic, respectively, throughout the (n, ])-midget?

To find out, we write f. as a Taylor series expanded around c = c.,i: c~

L ( c ) = E (c k=l

(Remember, f.(Cn,j)= 0.) By assumption, f~ is well approximated by 2 cosv~ so all of the terms in the Tailor expansion are readily evaluated. We find that each derivative f~k)(C) is order of • (p.) k - t h a t is, 6((p.)).k Thus, if• c - c . , j is• --2 • • --1 ~7((p.) ) the magmtude of f. will be 6((p.) ) and f. will be approximately linear in c - c.,j. Now, we repeat the process for fz.. At c = Cn,j f 2 . = L and f ~ . = f ' . As for f., if C--Cn, j is ~((p.)-2) the magnitude of the linear term in the expansion for f2. will be 6((p.)-l. But, we cannot immediately determine the magnitudes of the remaining terms because we cannot immediately evaluate the higher derivatives of f2.. On the other hand, the master recursion relation (1) ?t requires that f l/.+,(Cn,]) = 2fn(C.j)f~(c.,j ! .~_ t 2 2[fnt(Cn,j)] = 2[f'(c...)] z = f~(c.,]) + 2[/'(c.,i)] • Iteration of this string produces the useful result

f"2. .(c . . . ~ = Y ' (' C n , ) + 2 " f . _ l ( c . d ) fn_2(C.d). ..

fl(Cn,j)[~(Cn,j)]

2 ,

(5) relating the second derivative of fz. (at c = c.,j) to derivatives of f.. ff f.(c.,j) is well approximated by the universal cosine function so too will fm(C.,i) for 2 ~
H. Hurwitz et al. / Physica D 81 (1995) 23-31

28

O((p.)-l). Repetition of this argument for higher derivatives of f2n leads to the realization that f 2(k) . (C) is 6((p.)k+l) and that when c - %,j is O((p.) -2) all terms beyond the quadratic contribution are negligibly small. But, is the root of Ln that we seek sufficiently close to G,j to warrant approximating fz. by a quadratic? To see, we set f z . ~ ( c - Cn,i)fr2n(Cn,i) + (C - % , i ) 2fz"(Cn,j)/2 . and note that in this approximation C2n,k_l -- C.,j( = hn,j - Cn,]) t 11 ,.~

_ 2f z.(C.,j)/f 2.(c.d ) .

Substitution for the derivatives in the latter expression yields

h . , / - c.,j ~ -6ax2(2j - 1)2/16" .

(6)

The validity of (6) as an approximation for the location of centers of heads is demonstrated by the diamond data points in Fig. 4. These represent the relative errors made by (6) compared with the exact center-to-head distances, for the j = 1 series. Note that although f2. is itself not well represented by the universal cosine function, its expansion near c.,j involves coefficients which come directly from that representation. Eq. (6) provides an explanation for and a generalization of a numerically observed result of Frame et al. [5]. A direct extension of the reasoning leading to (6) allows us to say more about the sizes of periodic windows. The tip of the main spike of the M-Set corresponds to a "crisis" Misiurewicz point where all the f.'s, n > 1, intersect off axis. In particular, this point, c = - 2 , occurs where f2 =f3 ~ 0. Each n-cycle window has a similar crisis point which in essence fixes the end of that cycle's domain of influence. The (n, j ) - cycle crisis point, G,j,* is the first value of c less than c., i for which f 2 . = f 3 . ~ 0 . Extending the observation of the previous section, we see that within the (n, j)-midget f3. plays the same role that f3 plays in the entire bifurcation diagram. We cannot find crisis point values directly from

the cosine approximation because neither f2. nor f3. are well approximated by 2 cosx/-~ near G,jOn the other hand, f2. is an approximate quadratic near c.,j and its coefficients can be evaluated indirectly using the cosine approximation. We are encouraged to hunt for a similar loworder polynomial representation for f~.. Arguments identical to those presented above for center of head points allow us to conclude that ' i S (~(Pn)' f " f 3n(Cn,j) 3n(Cn j) is e((pn)3), f "3n(Cn,j) is t~((p.)5), while f3n (k) (C,,,j) ' is e ( ( p . ) k+3) for k t> 4. As long as c - c.,j is (?((p.)-2) or smaller, then the Taylor expansion for f3. around c.,j will be approximately a quartic. Indeed, f3 is an exact quartic over the entire bifurcation diagram so within the (n, j)-midget f3. mirrors the role f3 plays over the larger range. Equating f2. in its quadratic form to f3. in its quartic, and noting that f'2n "~f;n and f~. = f 3 . (the latter can be derived by incrementing (5) by n more terms and noting that fn(G,i)(=O) is then a member of the product on the right hand side), at c..j permits us to write

* --Cn,j ~ -- 4f" Cn,h Y3nktc n,j)~/f"'[c J 3 n \ n,jY " The derivatives on the right hand side of the latter equation are evaluated in the same manner as the derivatives of f2. above. The end result is Cn*,j-- C.,j = - 12"rrz(2j - 1)2/16" .

(7)

The quality of this approximation is shown by the square data points in Fig. 4. More importantly, combining (6) and (7) yields the conclusion that for each (n,j)-midget throughout which fn is approximately 2 c o s ~ the ratio of the center-to-crisis and the center-tohead distances is nearly 2, the same as in the entire bifurcation diagram. Thus, all midgets sufficiently close to c = - 2 are free of distortion along their lengths in parameter space. This result springs directly from the fact that the Taylor expansions of f., f2., and fan are well approximated near Cn,j by polynomials of the same order and with coefficients in the same proportion as fl, fz, and f3, respectively. Fig. 5

H. Hurwitz et al. / Physica D 81 (1995) 23-31

29

(a) Fig. 5. Compares the polynomials fl, f2, and f3 in the whole bifurcation diagram, that is, the interval [c~, c1,1] (part (a)), with the polynomials f3, fr, and f9 in the 3-cycle window, that is, the interval [c'x, c3a] (part (b)). Within the latter interval f3 (like fl in the larger interval) is linear, f6 (like fz in the larger interval) is quadratic, and f9 (like f3 in the larger interval) is quartic. In (a) Q,1 = 0, hi ~ = - 1 , and c*~ = -2. In (b) c3,1 = -1.75488.. (denoted P), ha, ~ =-1.77289. (denoted Q), and c* = -1.79033. (denoted R). ,

,

•

shows an example, comparing fl, f2, and f3 over the entire parameter range with f3, f6, and f9 within the (3,1)-midget.

5. Dynamic space and generalized Feigenbaum numbers Finally, we explore what our scaling in parameter (c-)space tells us about behavior in dynamic (x-)space. First, we can characterize how the aspect ratio in the bifurcation diagram changes as we go from window to window. At the crisis, Cn,j,* the copy of the bifurcation diagram containing x = 0 has an extent in dynamic space given by Using the approximations developed above, this e x t e n t - w h i c h may also be called the "height" of this small portion of the bifurcation d i a g r a m - can be estimated as

IL(c:,j)--Ln(C*,j)I.

]fn(C*n,j) -- f2n(C:,j)] ~-"8"rr(2j -- 1)/4 n .

(8)

The "length" of this portion of the bifurcation diagram can be approximated by the center-to-

"

"

3,1

'

"

crisis distance, ] c * j - c , , ] [ , and, therefore, the ( n , j ) - w i n d o w aspect ratio (height/length) is 2 × 4n/[3(2j -- 1)~r] (provided, of course, that n is sufficiently large and j sufficiently small). Thus, the closer we approach the termination point c = - 2 , the more and more squashed the bifurcation diagram becomes along its length. We can also use the parameter space scaling to characterize the internal structure of windows. Each (n, j)-cycle window will contain multiple copies of windows homologous to the (n,j)-window. The length of one of these is approximately a factor 6,,j longer than the next smaller one. Similarly, the extent of that portion of the attractor in dynamic space centered on the critical point, x - - 0 , in one of the homologous windows is a factor a~,j longer than the next such smaller one. The constants 6,,i and a,,j are generalizations of the numbers found by Feigenbaum for the period doubling, (2,1) sequences [9]. Eckmann, Epstein, and Wittwer [10] (EEW) have shown that successive (n,1)- (homologous) windows within any given window shrink at a rate of 16n/6"rr2; they also demonstrate that

H. Hurwitz et al. / Physica D 81 (1995) 23-31

30

Ot., 1 -~4"/2"rr. Delbourgo, Hart, and Kenny [8]

( D H K ) report numerical values for a few sequences where j differs from 1. We show next that an extension of the E E W predictions that is consistent with the cited numerical studies follows immediately from the universal cosine approximation. The argument relies on our demonstrated result that (many) windows are essentially distortion free. The factor 6.,j can be approximated by the rate at which center-to-head distances (for example - other such distances will work equally well) shrink between successive homologous windows and that, in turn, is approximately [hi, 1 -cl,,[/]h., j - c . d 1. The numerator equals 1 and the denominator comes directly from (6); thus, 6.,j ~ 16"/[6rrz(2j -

1) 1.

(9)

Similarly, a.,j can be approximated by Ill(hi,l)~ f.(h.,j)[-'-l/[f.(h.,j)[. If we write f~(h.,j) as ( h . d - c.,j)f'(c.d), use (6) and the cosine approximation for f.(c), we derive Oln, j ~

4"/[2-rr(2j - 1)1.

(10)

Table 1 compares a few of the numerical values Table 1 Errors made by (9) and (10) in approximating generalized Feigenbaum numbers for windows of type (n, j) n

]

error in alpha

error in delta

2 3 4 5 5 6 6 6 7 7 7 7

1 1 1 1 2 1 2 3 1 2 3 4

0.0174 0.0979 0.0495 0.0181 0.1860 0.0105 0.0462 0.1330 0.0015 0.0127 0.0349 0.1747

0.0741 0.2515 0.1273 0.0458 0.5286 0.0150 0.1234 0.3319 0.0047 0.0335 0.0847 0.4539

The errors are with respect to exact ("actual") numerical calculations found in Ref. [8] and are defined as Iactual %,j - approximate a., j [/ actual %,j and so on.

of D H K with (9) and (10). Despite the fact that the cosine representation is only strictly valid in the large n limit, the table reveals remarkably close agreement with numerical studies even for small n. The observation by D H K , that 38,,j 2(%,j) 2, can be seen to follow immediately from

(9) and (10). Finally, we note that the positions in parameter space of the cusp of the main cardioid and the point of attachment of the head to the cardioid (the first period doubling bifurcation) are determined by the conditions

Fc(x ) = x 2 + c = x and F'c(x) = 2x = + 1 or

(cusp point)

F'~(x) = 2x = - 1 (attachment p o i n t ) .

A little algebra reveals that c for the cusp is + 1 / 4 and c for the attachment is - 3 / 4 . Within each (n, j)-midget the entire dynamics is recapitulated on a smaller scale with the cspace polynomial f, substituting for fl, f2n for f2, and so on. Equivalently, in x-space F~°" replaces Fc, Fc°2n replaces F~°2, and so on. Thus, the condition for locating the cusp and the headcardioid attachment points of the (n, j)-midget are

F~°"(x) : x and [Fc°"(x)] ' = +1

(cusp point)

or

(attachment p o i n t ) .

[F~°"(x)] ' = - 1

Within the portion of the (n, j)-midget containing the critical point x = 0, the extent of the attractor in dynamic space for any c is certainly no more than the extent of the attractor at the crisis point. That is, x is bounded by -+4"rr(2j1 ) / 4 " - s e e Eq. ( 8 ) - i n the expressions above. We expand F¢°"(x) around x = 0. R e p e a t e d application of the chain rule produces [F O.(x)] . =. 2. F~ . -1 (x)F co.-2 ( x ) . . . F~(x)x, from which all higher derivatives can be evaluated. All derivatives of odd order vanish at x = 0. When evaluated at x = 0, all derivatives of even order, k, are ~((0,)k/2). The important

H. Hurwitz et al. / Physica D 81 (1995) 23-31

terms in the expansion o f Fc°n(x) are the c o n s t a n t a n d quadratic c o n t r i b u t i o n s - b o t h ¢7((pn)-1); h i g h e r o r d e r terms are all ~((pn) -2) and smaller. I n s e r t i o n o f this quadratic a p p r o x i m a t i o n into t h e defining relations for the cusp (which we d e n o t e qn,]) and a t t a c h m e n t (an,/.) p o i n t s - a s well as use o f ( 7 ) - yields qn,y - c*,] ~ (27/2)wz(2j - 1)2/16 n

(11)

and qn,j -- an,j

~"

67ra(2j -- 1) 2/16~ •

(12)

T h e f o r m e r a p p r o x i m a t e s the total range in p a r a m e t e r space o f the (n, j ) - m i d g e t while the latter a p p r o x i m a t e s the range of the (n, j ) - c a r dioid. T h e ratio of these two lengths is approxim a t e l y 9 / 4 for all midgets ( t h r o u g h o u t which fn is well r e p r e s e n t e d by the cosine function). W e n o t e that the p r e c e d i n g a r g u m e n t provides an alternative explanation for the numerical observations o f Y o r k e , et al., concerning distortion f r e e d o m for cyclic w i n d o w s of increasing periodicity [6].

6. Summary T h e r e p r e s e n t a t i o n described here o f the orbit of the critical p o i n t for quadratic-like m a p s b y the universal cosine function allows a unified and relatively simple analytic derivation o f a n u m b e r o f related scaling properties. T h e results cited a b o v e h a r d l y exhaust the potential of this tool.

References [1] A.L. Goldberger and B.J. West, Ann. N.Y. Acad. Sci. 504 (1987) 195.

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