1935
Progress of Theoretical Physics, Vol. 68, No.6, December 1982
On the Scaling Behavior
in a Map of a Circle onto Itself
Hiroaki DAIDO
Research Institute for Fundamental Physics
Kyoto University, Kyoto 606
(Received July 15, 1982)
A scaling·theoretical framework for a map of a circle onto itself is proposed, and it gives
a way of understanding the appearance of unusual scaling behaviors for a particular parameter
value found by Shenker and reveals a simple relation between scaling exponents, v = 2x .
Recently Shenker!) has numerically shown the existence of remarkable
scaling behaviors in a map of a circle onto itself as follows:
T(B)=B+Q-
tr
sin 2J[B,
(1)
where Q and K are constants, and the variable B parametrizes the circle, so that
B+ 1 must be identified with B. It is no doubt important to investigate the
properties of such a map since it can be regarded as a one-dimensional version of
the standard mapping which plays an important role in the study of stochastic
instabilities in Hamiltonian dynamics. 2 )-4) Moreover, the map can be useful to
study a certain type of routes to turbulence such that the transition to a turbulent
state takes place via quasiperiodic state, i.e., torus. S )
Let us now describe Shenker's results concentrating on a part with which we
are mainly concerned in this paper. He determined a set of values of Q, Qi(K),
for which TQi( 0) - Pi = 0 holds as Q is varied with K fixed, where Q i and Pi are
the (i + 1)th and the ith Fibonacci numbers,*) respectively. Then, it was found
that in the limit i~oo, Qi(K) converge to a certain value Q(K) in the following
way:
r
Qi(K)-Qi-l(K)
iI.~Qi+l(K)-Qi(K)
(O~K <1)
(2)
(K=1)
(3)
where p=(I5+1)/2, so called the Golden mean, and y===.=2.16443.
*)
Moreover, the
Hereafter we denote the ith Fibonacci number by Fi, so that Fi+l=Fi+Fi- 1 with F 2 =F1 =1.
H. Daido
1936
distance di(K) between 8=0 and a point with period Qi closest to the former was
found to show the following behavior:
= -px=a,
where x ='70.52687.
(O~K<l)
(4 )
(K=l)
(5)
Note that
(6 )
In what follows, we denote the ratios (Qi(K)- Qi-l(K))/ (Qi+l(K)- Qi(K)) and
di-I(K)/di(K) by oi(K) and a;(K), respectively, and the gradient of T Q i(8) at
8=0 for Q=Qi(K) by Di(K) following Shenker. We call oi(K), ai(K) and
Di(K) "transient quantities" since they all contain i dependence which should
vanish in the limit i ->(X). For these quantities, he found that each of them
depends only on £Q/ for large i and small £=l-K, where v='71.053744.
In this paper, on the basis of a theoretical framework quite similar to those
for the period-doubling phenomenon,6).7l we discuss the mechanism of the appearance of the unusual scaling behaviors for K=l seen in the numerical results (2)
~ (5) and show in particular that the following relation between the scaling
exponents x and v holds:
v=2x.
In fact Shenker's results for x and v strongly suggest the validity of this relation.
Our starting point is to introduce a three-variable function/; for each i by
(8 )
where (j = 8/ di(K), Q = (Q - Q;(K))/ (Qi+I(K)- Qi(K)), and z = £[i;. The constant fii has to be chosen in such a way that the resultant function Ii depends
smoothly on z for any large i. For example, it may be determined by imposing
a 0,
2
~a
-li(O,
za8
(9 )
0, 0)=1.
A basic assumption in this paper is that the following limits exist, i.e.,
lim/;=
i---+ct::J
I,
limfi
z/ fi i-I = fJ. .
i ..... oo
(10)
Then we find that for large i,
a0,
__
a0,
__
Di(K)= atfli(O, 0, £fJ.i)~ atfl(O, 0, £fJ.;),
(11 )
ai(K)=J:-I(O, 1, £fii-I)"""I~ 1(0,1, £fii-I)"""I.
(12)
On the Scaling Behavior
1937
Later we will show f.J.=a 2, so that Eqs. (11) and (12) explain the transient scaling
laws for Di(K) and ai(K) with v=2x ((7)) since Qi~pi for large i. The
scaling law for oi(K) is derived from a recursion relation between;;'
and
;;-1
/:-2:
/:( if, Q, z)= a;(K);;-I(ai-l(K)/:-2(ai-I(K)-1
Xai(K)-lif, Q~i), f.J.i!.lf.J.i1z), Qli), f.J.i1z),
(13)
where Qli)=1 +oi(K)-IQ, Q~i)=1 +Oi-I(K)-I+Oi-I(K)-IOi(K)-IQ, and f.J.i= fli
/fli-I.*) This relation is obtained using TQi= TFi+l= TFi. TFi-l and (8). Since
;;(0,0, z )=0 by definition, we are led to
0=/:-I(ai-I(K)/:-2(0, l+o i -l(K)-\ f.J.i!.lf.J.ilz), 1, f.J.ilz),
(14)
which enables us to conclude the scaling law for oi(K) by (10) and (12). We can
obtain in principle scaling functions for Di(K), ai(K) and oi(K) as well by
solving the fixed point equation of the recursion relation (13) and using (11), (12)
and (14).
Let us then proceed to show f.J.=a 2. For this purpose, we further introduce
Ii and gi through a Taylor expansion of the l.h.s. of (8) with respect to £, i.e.,
(15)
For simplicity, we omit the Q dependence of Ii and gi.
ward to obtain
Then, it is straightfor-
li( ()) = Ii-I (fi-2( ())),
(16 )
gi( ()) = 1:-1 (fi-2( ()) )gi-2( ()) + gi-l (fi-2( ())).
(17)
Note first that li( ()) has the following form for any i:
li( ()) = ai + bi()3 + C i()5+ O( ()6),
(18)
where ai, etc. depend only on Q. This fact comes from (16) and that T(())=Q
+(27r)2()3/6-(27r)4()5/120+'" for £=0 (i.e., K=l).l) Therefore, we may put
Ji( if, Q)=/:( if, Q, 0)
(19)
and hence
J( if, Q)=/( if, Q, 0)
In the recursion relation (13), (lj{K) and Bj(K) (j=i-1, i) should be regarded as functions of
that are defined through (12) and (14) (for Bj(K) see below) although we have used aj(K) and
Bj(K) there for the sake of simplicity.
*)
Z=E:jli
H. Daido
1938
(20 )
since /;(8)=d;fi(8/di, (Q-Qi)/(Qi+l
=Qi(l). By (13), 1 satisfies
Qi), 0) by (8), where di=di(1), Qi
(21)
with ,Ql=l+o- 1 ,Q and ,Q2=I+o- 1 +o- 2,Q.
It follows from (21) that for 1(0)
== l( 0, ,Qc),
1(0)=al(al(a- 20»,
(22)
where ,Qc=lim(.Q(1)-Q;)/(Qi+l-Qi)=o/(o-l). Shenker proposed Eq. (22)
and numerically obtained the form of 1 ( 0). *)
Let us now pay attention to Si==g;'(O), which are shown to obey
S;
using (17) and (18).
= /f-l(fi-2(0»S i-2
(23)
Then, we find using (16), (18) and (23)
which leads to
6
S;
7J:- (2Jr )2
for any i ~1
(24 )
or by (18) and (19) to
6
-s;= (2Jr)2di 2 b;(Q)
(25)
due to the fact that Sz/b2=Sl/bl=6/(2Jr)2.
After all the foregoing considerations, we now deal with the transient
quantities. For the purpose of showing (7), we have only to consider Di(K) as
shown below. Using (15), (18) and (25), we obtain
= £g;' (0 ),Q=,Q, + O( £2)
=£
(2~)2 di 2b;(O)+ 0(£2)
~£
6d- b-(O) u+ O( 2)
(2Jt )2
a
£
2
for large i ,
(26)
where di=d"a- i . A comparison between (11) and (26) yields /1=a 2, which is
*)
For Eq. (22), see also Ref. 8).
.on the Scaling Behavior
1939
equivalent to (7) as mentioned previously. Note that if the condition (9) is
adopted, Iii = - D;' (1) holds. Therefore, it turns out that under the condition, the
latter half of (10) is actually satisfied with fJ- = a 2 due to (26).
Finally we give some comments. First of all it should be noted that Eq. (26)
implies the instability of the fixed point function 7 against a perturbation of 71
that has an O( (f) term in its Taylor expansion form around (f =0. On the other
hand, an argument similar to the one made to derive (26) for the case € = K - Ko
with K o< 1 shows that there occurs no such instability caused by the O( (f) term
since fi(8)Ko has an 0(8) term (ct. (18)). Therefore, our arguments leading to
(26) may be regarded as clarifying the mechanism of the appearance of the
unusual scaling behaviors for K = 1. In fact, the situation becomes much clearer
in terms of 1: since due to the assumption mentioned before, the fixed point
function equals f<{f, Q, =) except €=O (i.e., K=1). It may be worthwhile
noticing at this point that the present situation is quite analogous to that encountered in the period-doubling phenomenon in two-dimensional mappings as
the strength of dissipation is varied to or from zero. 9 ),10) A simple calculation for
the case K = 0 yields
which also satisfies (21) by (13) with a = - P and 0 = - p2. Another comment is
on the universal constant 0, in other words, the exponent y. These quantities
can be obtained by solving, say, the following eigenvalue problem derived from
(21) :
(28)
where h( {f)=a7( (f, Qc)/aQ .and il=a·o- 1 • Note that a can be obtained from
(22). Considering the rather complicated form of (28), there seems to be no
simple relation between x and y.
In this paper we have concentrated on the "local" properties of the map (1).
It is possible, however, to deal with other maps exhibiting similar behaviors in the
same way and to discuss even their "global properties" which are also reported
by Shenker/) based on our framework along the line of Kadanoff's recent work.4)
Acknowledgements
The author would like to thank Professor K. Tomita for valuable discussions
and careful reading of the manuscript and Professor Y. Kuramoto for stimulating
discussions. He would also like to thank the Yukawa Foundation for financial
support. After completion of this work, the author has received two
preprints 11 ),12) by D. Rand, S. Ostlund, ]. Sethna, E. D. Siggia and M. ]. Feigen-
1940
H. Daido
baum, L. P. Kadanoff, S. J. Shenker in both of which the same subject as that of
the present paper is discussed.
References
1) S.]. Shenker, "Scaling Behavior in a Map of a Circle onto Itself: Empirical Results", Preprint
(to appear in Physica D).
2) B. V. Chirikov, Physics Reports 52 (1979), 264.
3) ]. M. Greene, ]. Math. Phys. 20 (1979), 1173.
4) L. P. Kadanoff and S. ]. Shenker, "Critical Behavior of a KAM Surface I, II", Preprints.
5) See, e.g., ]. P. Gollub and S. V. Benson, ]. Fluid Mech. 100 (1980), 449.
6) M.]. Feigenbaum, ]. Stat. Phys. 19 (1978), 25; 21 (1979), 269.
7) H. Daido, Phys. Letters 83A (1981),246; 86A (1981),259; Prog. Theor. Phys. 67 (1982), 1698.
8) I. Tsuda, Prog. Theor. Phys. 66 (1981), 1985.
9) R. H. G. Heileman, in Fundamental Problems in Statistical Mechanics, vol. 5, ed. E. G. D.
Cohen (North-Holland, Amsterdam, 1980).
10) A. B. Zisook, Phys. Rev. A24 (1981), 1640.
11) D. Rand, S. Ostlund, ]. Sethna and E. D. Siggia, Phys. Rev. Letters 49 (1982), 132.
12) M.]. Feigenbaum, L. P. Kadanoff and S. ]. Shenker, "Quasi periodicity in Dissipative
Systems: A Renormalization Group Analysis", Preprint (Physica D, in press).