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Symbolic dynamics of fully developed chaos III. Infinite-memory sequences and phase transitions
Physica A 186 (1992)405-440 North-Holland
~
[~1
Symbolic dynamics of fully developed chaos III. Infinite-memory sequences and phase transitions R. Kluiving and H.W. Capel lnstituut voor Theoretische Fysica, Universiteit van Amsterdam, Valckenierstraat 65, 1018 X E Amsterdam, The Netherlands
R.A. Pasmanter Koninklijk Nederlands Meteorologisch Instituut, Postbus 201, 3730 A E De Bilt, The Netherlands
Received 2 March 1992 A special type of dynamical phase transitions which arises in a particular one-dimensional fully developed chaotic iterated map is studied by means of a symbolic dynamics. Exact analytical expressions for the probabilities of words are found on both sides of the critical value, making it possible to give a statistical description of the critical behaviour at the phase transition in terms of Boltzmann entropies, correlation functions and generalized dimensions.
1. Introduction This is the third and last part of a series of articles on the application of symbolic dynamics to fully developed chaotic ( F D C ) motion in one dimension. In part 1 [1] the statistics and characteristics of general two-symbol sequences were studied. These characteristics include Boltzmann entropies and correlation functions which are functions of the probabilities of subsequences of consecutive symbols. In part 2 [2] three algorithms to extract two-symbol sequences out of a F D C iterated o n e - h u m p m a p x n+l = f ( x n ) were presented. F r o m this, three different symbolic descriptions of fully developed chaotic motion in one-dimensional o n e - h u m p maps were obtained, namely the L R - , the ( + - ) - and the AS-symbolic dynamics. All three of t h e m turned out to have their own advantages. In this third part we study the phase-transition in the bungalow-tent m a p [3] by m e a n s of the ( + - ) - s y m b o l i c dynamics. The reasons why this particular choice of symbolic description is preferred above the LR-symbolic dynamics are the following: 0378-4371/92/$05.00 (~) 1992- Elsevier Science Publishers B.V. All rights reserved
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R. Kluiving et al. / Symbolic dynamics of fully developed chaos 111
The bungalow-tent map is labeled by a control parameter a. For a = lim~+0(½ - e ) one finds a dynamical behaviour of the iterates which differs dramatically from the dynamical behaviour for a = lim~+0(½ + e), thus showing a phase transition at a = ½. To understand this phase transition in terms of the statistics of half-infinite symbol sequences, one needs to know the probability of occurrence of words of finite length k, i.e. of subsequences of k consecutive symbols in this half-infinite symbol sequence. For a =lim,,+0(½ + e) these probabilities can be determined relatively simply in the ( + - ) - s y m b o l i c description, since we will show that the half-infinite ( + - ) - s y m b o l sequence is order-1 Markovian, i.e. the memory extends over only one time step. The half-infinite LR-sequence, on the other hand, has a more complicated m e m o r y structure and therefore the evaluation of the probabilities would be much more cumbersome. But for a = lim~0( ½ - e ) things are different. The half-infinite ( + - ) - s e q u e n c e as well as the LR-sequence have an infinite memory, i.e. there does not exist a finite integer m such that the sequences are order-m Markovian. It turns out that here it is extremely difficult if not impossible to find closed analytical expressions for the probabilities of words in the LR-description, whereas in the ( + -)-symbolic description these expressions can actually be determined for a = lim~0(½ - e) (section 5). This is due to the fact that the origin of the phase transition lies in the existence of infinitely long laminar intervals of the iterates x i in the close neighbourhood of an unstable fixed point x v for a = l i m ~ ; 0 ( ½ - e ) , which are absent for a = l i m ~ 0 ( ½ + e ) . Since the ( + - ) - s y m b o l i c description uses a partition of the interval [ - 1 , 1] into regions left ( + ) and right ( - ) of the fixed point x F, it is possible to solve r e c u r s i v e l y the integrals of the probability density of the iterates over certain parts of the interval [ - 1 , 1]. This provides immediately the desired probabilities of occurrence of words in the ( + - ) - s y m b o l i c description. Thus the ( + - ) - s y m b o l i c dynamics provides exact analytical expressions for the probability of occurrence of ( + - ) - w o r d s on both sides of the critical point a = ½. In the LR-symbolic dynamics on the other hand, the fixed point x F is not specified by the partition left ( L ) and right (R) of a reference point ~ and therefore this does not lead in a natural way to a recursive scheme of the integrals. As concerned to the AS-symbolic dynamics, the following can be noted: It is not possible to derive the probabilities directly in the AS-symbolic dynamics without knowing the results in the ( + - ) - s y m b o l i c dynamics. But since it was shown in ref. [2] that there is a one-to-one correspondence between the ( + - ) - s y m b o l i c dynamics and the AS-symbolic dynamics, one may, in principle, derive expressions for the probabilities of AS-words left and *1 This point is the so-called critical point of a one-hump map.
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407
right of the critical point a = ½ from the corresponding expressions in the ( + -)-symbolic description, in this way creating an alternative exact symbolic description of the phase transition. However, since both ( ( + - ) - and ASsequences have an infinite memory at a--lim,~0( 1 - e ) , this derivation is technically complicated and therefore the ( + - ) description is more convenient than the A S description. It is worthwhile noticing that the phase transition studied in this paper is of a different nature from those usually discussed in the dynamical-systems literature [4-7]. In particular, this phase transition is caused by the variation of an external control parameter and therefore reminiscent of the more usual phase transitions in statistical mechanics. This paper is organized as follows: In section 2 the features of the bungalow-tent map are shortly reviewed. For more details about this interesting map we refer to ref. [3]. Also in section 2 we explain how to create a half-infinite ( + - ) - s e q u e n c e from the iterates of a one-hump map and how to determine the probabilities of occurrence of words in this sequence. Furthermore we enumerate the characteristics of such a sequence. Since these characteristics depend on probabilities of occurrence of words, one must know these probabilities. In section 3 these probabilities are analytically evaluated for a = limes0(½ + e), i.e. on the critical value from above. Section 4 is devoted to the ( + -)-symbolic dynamics of the bungalow-tent map for a = 1 _ e with 0 < e ~ 1, i.e. just before the critical value. In subsection 4.1 we derive an analytical expression up to order e from which the probabilities of occurrence of words can recursively be investigated. In subsection 4.2 this expression is used to give a statistical derivation for the mean length of the laminar intervals, thereby corroborating an earlier result which was based on dynamical considerations. This illustrates the power of the ( + - ) - s y m b o l i c dynamics. In subsection 4.3 we investigate the correlation function of the ( + - ) - s e q u e n c e associated with the bungalow-tent map for a = 1 _ e. It turns out that the correlations do not decay significantly for time intervals which are of the order of the mean length of the laminar intervals. Eventually the correlations decay towards a small but nonzero value, which shows that the ( + - ) - s e q u e n c e has an infinite memory. In section 5 we consider the case a = lim~0(½ - e) and compare the characteristics of the ( + - ) - s e q u e n c e with the characteristics of this sequence for a = lim,$0( ½ + e). The differences provide the statistical description of the critical behaviour at the phase transition. Section 6 presents the results of another way of characterizing the phase transition, namely by a discontinuous change in the spectrum of generalized dimensions. These generalized dimensions describe the properties of the
R. Kluiving et al. / Symbolic dynamics of fully developed chaos Ill
408
Fibonacci multifractal associated with the ( + - ) - s y m b o l i c bungalow-tent map.
d y n a m i c s of the
2. Preliminaries
2.1. The bungalow-tent map T h e b u n g a l o w - t e n t m a p [3] is defined on the interval [ - 1 , 1] as follows:
gl(x) = 1 + 2a + 2(a + 1)x
for-l~x~<-½,
g2(x) = 1 + 2(1 - a)x
for - ½ ~ x ~ < 0 ,
g3(x) = 1 + 2(a - 1)x
for 0 ~ < x ~ ½ ,
g4(x) = 1 + 2a - 2(a + 1)x
for½~x~l,
L(x) =
(1)
cf. fig. 1, w h e r e - 1 < a < 1. It is a s y m m e t r i c piece-wise-linear o n e - h u m p m a p . T h e height of the e x t r e m e left and right kink is tuned by the control p a r a m e t e r a. T h e i t e r a t e d system
Xn+1
= ]ca (Xn)
(2)
is F D C , i.e. chaotic and ergodic on the entire interval [ - 1 , 1], for - ½ < a < 1, w h e r e a s for - 1 < a < - ½ the fixed point 2 = - 1 is stable.
1
Ii 8
t
...........
O
~(x)
T
-1 -1
-0,5
8 )
B,5
X
Fig. 1. The bungalow-tent map.
1
R. Kluiving et al. / Symbolic dynamics of fully developed chaos 111
409
In this article the bungalow-tent map is studied for values of a near ½. In particular, use will be made of the expressions derived in ref. [3] for the probability density pa(x) of the iterates for a = ½ and a = ½ - e with 0 < e ~ 1: p~,2(x)= ~o(: 2 , -x)+
~O(x- ½)
(3)
and
+ e~pl(x) + C(e2), Pl/2_e(X)=
I -1 + ~o2(x)+ 6 ( e ) , ge
+ e%(x)+ 6~(f),
-l
- e,
21
½,
e
(4)
½< x < l ,
where ~ol, ~o2 and % are unknown functions. Taking the limit e~0 in eq. (4) yields lim E~O
p~/2_,(x)= 10(½-x)+ 20(x- ~)+ ~6(x- 1),
(5)
which differs from eq. (3). Thus the probability density changes discontinuously at the transition (a~'½)--~ (a = ½). This explains quantitatively the discontinuity [3] in the Lyapounov spectrum, 1
A. = f p.(x) loglf'(x)l d x ,
(6)
-1
at a = ½. In fact there is an infinite number of discontinuities in this Lyapounov spectrum, as shown in fig. 2, their locations ag being determined by the equation [2(a, + 1)]*-~(1- a , ) = ½
(k = 1,2 . . . . ).
(7)
This equation was derived using an exact scaling procedure which showed that the jump phenomena at a 2, a 3 are rescaled copies of the phenomenon at a = ½ [3]. In this article the following terminology concerning values of a close to ½will be used: . . . .
a= ½-e
withO
just before the critical value,
a = l i m (½ - e):
the critical value from below,
a= ~
the critical value from above,
e$O
cf. fig. 3.
(= lifo (½ + e)):
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410
o.~
0
-0.5
i
i
o
o,5
1
m_ O.
Fig. 2. The Lyapounov spectrum of the bungalow-tent map. The jumps occur at values a k determined by eq. (7).
1
a
Fig. 3. Visualization of the phase transition in the bungalow-tent map in terms of the jump in its Lyapounov spectrum. 1 refers to just before the critical value, 2 denotes the critical value from below and 3 refers to the critical value from above.
R. Kluiving et al. / Symbolic dynamics of fully developed chaos I11 2.2.
411
The (+ - ) - s y m b o l i c dynamics
If x = x , , , x l , x2 . . . .
(8)
is a trajectory of a F D C iterated o n e - h u m p map, then the half-infinite ( + - ) - s e q u e n c e s is defined as follows:
S = [S|,S2,. " .],
Si=
+ __
if xi > xi_ 1, ifxi
(9)
A n equivalent way of forming s is
s = [s 1, s 2 , . . . ] ,
si =
+ -
ifxi_ 1 XF ,
(10)
where x v is the extreme right fixed point of the o n e - h u m p map. The probability of finding a particular word or subsequence w ~ = [ w l , w 2. . . . .
wk],
wi=+
of k successive symbols w 1, w 2 . . . . .
or-,
k=l,2
....
,
(11)
w k in s is given by
N-k+l
P s ( W k ) = l!m= N -
1k + l
~, i=l
6w,.,i6w>,i+l...6wk.si+~ .
(12)
This probability can be shown to be equal to 1
Ps(w k) : J d x p(x) 0[sgn(w,) [f(x) - x]] 1
× 0[sgn(w2) [f(Z)(x) - f ( x ) ] ] ' ' '
0[sgn(wk)[f(k)(x)
-- f(k-l)(X)]]
,
(13)
where p(x) is the probability density of the m a p and {10 0(y) =_
if y > 0 , if y < 0 .
(14)
D u e to the nature of a o n e - h u m p m a p , a minus in the ( + - ) - s e q u e n c e s is always followed by a plus, or, equivalently, one cannot find words w k in s which possess one or m o r e pairs of neighbouring minuses. T h e symbol sequence s gives a description of the trajectory x in terms of
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412
ascending and descending behaviour and, due to the FDC-aspect of the map, the statistical properties of s are independent of the starting position x 0 (as long as x 0 does not belong to an unstable cycle), as follows from eq. (13). The level-k (Boltzmann) entropies h (k) [1] of a half-infinite sequence s are given by
h~g)(s) = kS(k)(s) - ( k -
1)5(k-')(8),
(15)
where 1
S(k)(s) =_ _ ~ ~ es(w k) logzPs(w k)
(16t
wk
is the average Shannon entropy [8, 9, 1] associated with the given probabilities of words with length k. Words which have a zero probability do not give a contribution to the summation in the r.h.s, of eq. (16), in accordance with limx+ 0 x l o g x = 0 . The level-k entropy htk)(s) is a measure for remaining uncertainty about s after being informed of all the probabilities Ps(W k) of words with length k, i.e. it is the logarithm of the number of questions per symbol that should be asked to know which particular sequence is under consideration. The limit
h(s) =
lira
k~
h~k)(s)= kl i~m S(k)(s)
(17)
is called the Entropy of the sequence s [1]. It measures the uncertainty about s after being informed of all possible statistical information about s. If s is order-m Markovian, i.e. the memory extends only over the last m symbols, then one can derive [1] that
h(s) = h
+')(s).
(18)
The four symbolic correlation functions y+,+(n), 7+, (n), 7_ +(n) and 7_ _(n) of two symbols at distance n of a sequence s are defined in the following way [1]. If n = 0 then
yw~.w2(O) = 6w~,w2ps([wt]) - Ps([Wt]) Ps([w2]),
(19)
and if n/> 1 then
y,,,.w2(n) = es([w , , n - 1, w2] ) - Ps([w,]) Ps([Wz]),
(20)
where the underbrace denotes the summation over the intermediate n - 1 symbols:
R. Kluiving et al. / Symbolic dynamics of fully developed chaos 111 P,([w,,n-l,
~
w2])-==
"'"
o,=+,-
~ P,([w,,v,,v v._,=+,-
2. . . . .
413
v._,,w2]).
(21)
In ref. [1] it was shown that (22)
= -y_+(n),
%.+(n) = %._(n) = -y+_(n)
i.e. one needs only one of the four correlation functions to know the correlations between two symbols over a distance n. Another characteristic of s is the mixing measure/z defined by /*(s) = 2Ps([+,-])
(23)
E [0, 1].
It measures the degree of mixing of the symbols + and - . If/z(s) = 0, we say that the symbols + and - are not mixed. If/x(s) = 1, then we say that the mixing is maximal.
3. The critical value from above
We will apply the symbolic dynamics presented above to the bungalow-tent map with a = ½. For this map the fixed point x F is equal to ½, and the probability density is given by eq. (3). The probability of finding a plus in the ( + -)-symbol sequence s associated with the map is then XF
P,([+])
=
f
Pl/2(X) dx =
(24)
3. 2 = 3 .
-1
More generally, omitting in f the subscript a -- ~ indicating the value of the control parameter and using the short-hand notation
~Owk(X) : q'tw,,w, ...... ,j(x) = 0[sgn(w,) ...
0[sgn(wk
If(x)
- x]] 0[sgn(w2)
[f(2)(X) --f(x)]] • " "
) [f(k)(x ) _ f(k-')(x)]],
one derives 1/2
PAl+, +
, w 3 , . . . ,wkl ) = f -1
~01+,w3...... kl(f(x))dx
(25)
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414
f(I/2)
=5
l(f
f(1/2)
f(O)
f
+
f(I)
f(1/2)
i)2[~4~÷,w~
-
J
. . ..,. ~I(Y)
dY]
f(O)
I/2
1 (
: 5
2
J
1
5~Ol+.w3...... , ( x l d x = 5 Ps([+' w3 . . . . .
(26)
w,]),
f(1)
from which one obtains the recursion relation wk]) : Q ( W l , w2)(2.J.&.',rl , . 3 ,~+.,.', - ~P.([w2 . . . . , . , . ] ) , (27)
P s ( [ W l , w2 . . . . .
relating the probabilities of level k to the probabilities of level k - 1. Here Q, being a function of two neighbouring symbols w i and wi+l, is defined by (~
ifwi=-andwi+l otherwise.
Q(wi' wi+I)=
=-,
(28)
Relating level k to level l, i.e. applying the recursion formula eq. (27) k - 1 times, yields e s ( [ w l , w2 . . . . :
, w,])
O ( w i , wi+i
2)
-."'~t!',-'-' (6+.w-6 "' ')Ps([wkl). ,.3;
(29)
Using P,([wk] ) = 3(32-)6 -,',,
(30)
cf. eq. (24), and k
k
(31)
2 8 .... = k - E & . . , , j=l
j:l
we arrive at the final result: p s ( w * ) = _5
Q ( w i , wi+l ) (2
. ,(~)
. ~
.
-.,
(32)
for the probability of finding a particular word wk in the ( + - ) - s y m b o l sequence s associated with the bungalow-tent map with a = ½.
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415
F o r the conditional probability
es(Iw, . . . .
-=
, wA--, [w~+ll)
e,([w,,..., wk+d) es([14dl,...,
(33)
14"k] )
o n e finds
Ps([Wl
. . . .
,
w,,]---~ [Wk+l] =
Q(wk, Wk+,)(-~)'~-''~+'(-{) ~+'w~+'-~-'''
= P,([wA--" [w~+,]).
(34)
T h u s the ( + - ) - s y m b o l s e q u e n c e s of the (a = 1)_bungalow_tent m a p is order-1 M a r k o v i a n , i.e. the m e m o r y extends only o v e r one time step. In this case the E n t r o p y of the s e q u e n c e reduces to h(s) = h(2)(s) = 2S(2)(s) - S ~' ~(s)
(35)
= _ 3[½ log2~ + 21og2~ ] = 0 . 5 5 0 9 7 7 . . . .
cf. eqs. ( 1 5 ) - ( 1 8 ) . Since the m a x i m u m possible value for the E n t r o p y of a half-infinite ( + - ) - s e q u e n c e is equal to the capacity d ( s ) = log2[½(1 + X/5)] = 0.6942419 . . . . cf. refs. [1, 2], the ( + - ) - s e q u e n c e s of the (a = ½)-bungalowtent m a p does not c o r r e s p o n d to m a x i m a l uncertainty.
O.3
o
-
o3
,
0
,
,
20
Fig. 4. The correlation function y+ +(n) of the (+-)-sequence s associated with the (a = ½)bungalow-tent map.
416
R. Kluiving et al. / Symbofic dynamics of fully developed chaos 111
For the correlation function 7+,+(n), cf. eqs. (19) and (20), one derives from eq. (32) 74.+(n) = ( - 1 ) " 9 ( 2 ) "41 ,
(36)
cf. fig. 4. The other three correlation functions follow using eq. (22). Notice that the correlations decay exponentially.
4. Just before the critical value 4.1. The probabilities Ps(w ~)
For a = ½ - e, with 0 < e ~ 1, the probability density of the iterates is given by eq. (4). With the fixed point 1 _ ½ _ ½e + 6(e z) xv - 2(1 + e)
(37)
the following expressions for the probabilities up to level 2 are easily derived: xF
Ps([+]) = f p,/z_~(x) dx = 7 + be + G(e2),
(38)
1
P~([-])
= 1 - P~([+])
(39)
= ~ - be + 6(d),
P s ( [ - , +1) = P ~ ( [ - ] ) ,
(40)
Ps([+, -1) = P s ( [ - , +1),
(41)
Ps([ +, +]) = P ~ ( [ + ] ) - Ps([ +, -1) = ~6+ 2 b e + O(e2),
(42)
Ps([-, -])=0.
(43)
Here b is an unknown constant. We now turn to the probabilities of level 3. Omitting in f the subscript ~ - e indicating the value of the control parameter and using again the short-hand notation (25), one has 1
XF
/*
P~([+, +, +]) = J p,~-~(~) ¢[+.+.+j(x)dx = J p,,~ ~(x)¢[+.+~(f(x))dx -I
= Jl + J2 + J3 + J4,
-I
(44)
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417
with -1/2
f dx [-~ + ~,p,(xll~,~+, +l(f(xl) + ~(d),
J1
(45)
-1 0
dx [½ +
J2 = f
E~I(X)]I~/[+" + ] ( f ( x ) ) +
~(2),
(46)
-1/2 l/2-e
dx [½ + eq~l(X)]qJ[+' +](f(x)) + ~'(e2) ,
J3
(47)
0 xF 14
dx(1
f
+q~2(x))4jl+,+l(f(x))+
~(2).
(48)
l/2-e
Introducing the new variable y = f(x) and using qJ[+.+l(y) = 0
ify>xv,
(49)
dy [1 + e~,(y)l~Oi+,+l(y) + ~(82) ,
(50)
dy ½OI+,+I(Y) + ~7(e2) ,
(51)
one finds l/2-e
1
(
Jl - 3 - 2 e
J -1
xF
1 f
J2- l+2e
1/2-e
J3=J4=O,
(52)
~l(g,(Y)) = q~l(Y),
(53)
gl(y) = 1 +2(½ - e ) + 2 ( { - e)y,
(54)
with
cf. eq. (1). With the help of the notation 1/2-e
1~+'+j=- f -1
[q~l(Y) --
~o~(Y)]q'i+,+] dy
(55)
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418
we can write xF
,f
dy Pl/2-E(Y) 0[+.+I(Y)
P,([+, +, + l ) - 3 - - 2 e
1 xF
f
3 -2e
dY (6~ + q~2(Y))tPl+,+l(Y)
l/2-e
xF
+ 21-+~ 1/2
f
l e Ii+,+1+ dy ~[+,+I(Y) + 3 Z 2e e
G(e2). (56)
The first term is equal to 1
(57)
3 - 2e P ' ( [ + ' + ] ) '
and the second and third turn out to be zero after changing to the new variable z = f(y), as can be checked readily. Therefore we end up with ps([+, +, +1) _
1
3
2e
[p,([+, +1) + ell+, +l] +
~(~.2)
,
(5s)
In an analogous way one can derive
eA(e)
1
P ' ( [ + ' +' - 1 ) - 3 - 2e P*([+' - ] ) + 2(1 + 2e)
+ e( II+'-1
3 --12e I~+'-]) + ~(e2) ,
(59)
with 1/2-e
II +'-1---
f
[~I(Y) -
'Pl(Y)lq'[+,-1 d y ,
(60)
-I xF
I~+'-l~-le
f
dy ~o2(y) ~0[+,-I(Y),
(61)
l/2-e
and 1
1
A ( e ) - 3 1+2e
1
1
3 - 2 e 6e "
The integral i~+,-1 and the constant A(e) arise when evaluating
(62)
R. Kluiving et al. / Symbolic dynamics of fully developed chaos 111 xF
xF
f dy (16ee + ~2(Y) )~bI+,-I(Y) + 1 +if2e - -
3 -2e
I/2-e
419
dy -{ ¢I+,-I(Y) 1/2
e
(63)
From eqs. (58) and (59) one arrives at Ps([w,, w2, w3] ) = Q(w 1, w 2 ) [ ( 3 - 2e)-~+'w2((2 - 2e) 8 xP,([w2, w3]) +
,w22d +,w2
(-- ]'l t3 ,Wls l[w2,w3] ]
+,w2',
,
, ,,~ .w,,. ,. [ ½A(e) + k-l) e°+'w2° "w3[l-+2-e
-1
1
1 1~w2.~31)1 3 --2e
+ 6(e~),
(64)
relating the probabilities of level 3 to the probabilities of level 2 just before the critical value. More generally, one is able to derive along the same lines Ps([w~, w2 . . . . .
wk]) = Q(wl, w 2 ) I ( 3 - 2e)-~+'"~2[(2 - 2e)~-.wl~+-w.,
x P , ( [ w 2, w 3 , . . . , w,,,]) + ~+,,,2(-1f "~'ell "2 ...... ~'1)] (*
)(
1A(e)
+ (-1)'-~'e.,=~[I a(_}, ~, (1 + 2e) *-2 + tT(e 2)
1 ,~w:...... ,1)] 3 £ 2e
(k/> 3),
(65)
with 1/2-e
l~W2...... ,1 =-- f
[ ~ l ( Y ) - qh(Y)]O[w 2...... ~I(Y) dY'
(66)
-1
and xF
i[-'2 ...... ,1 = _le f
dy q~z(Y) ~Otw2...... kl(Y) d y ,
(67)
1/2-e
relating the probabilities of level k with the probabilities of level k - 1. Eq. (65) forms the corner-stone of our analysis of the phase-transition in the bungalow-tent map, as will be shown in the following subsections.
420
R. Kluiving et al. / Symbolic dynamics of fully developed chaos II1
4.2. Mean length of the laminar intervals In ref. [3] it was shown that, just before the critical value, the trajectory x shows long intermittent laminar intervals in the neighbourhood of x = ½, which are absent just after the critical value. Closer analysis revealed that a laminar interval consists of a slow spiralling away from the (unstable) fixed point x v. The closer the parameter a is set near ½from below, the longer the trajectory keeps spiralling around x F. Eventually the trajectory escapes, giving rise to a chaotic burst, until it returns again to the close neighbourhood of XF, i.e. the interval [ ~1 - e, ½] (since the map is ergodic on the interval [ - 1 , 1] the trajectory will certainly return to this neighbourhood), giving birth to a new laminar interval. In ref. [3] it was argued, using dynamical considerations, that 1 (nlam) = ~gg -}- G(1),
(68)
with (nla m) the mean length of the laminar intervals (or: the mean spiralling time) and e = 1 _ a. In subsubsections 4.2.1 and 4.2.2 we will present a systematic statistical derivation of (68) based on the probabilities of long alternating sequences, which will be investigated first.
4.2.1. Probabilities of alternating sequences First we will derive an analytical expression for the probabilities P s ( [ ( - , +)J, -1)
( j = 0 , 1, 2 , . . . )
(69)
for a = ½ - e, i.e. just before the critical value. F o r j = 0 we have from eq. (39) P s ( [ - ] ) = ~2 - be.
(70)
For j = 1 one derives from eqs. (65) and eq. (70)
P'([-' +' -])-
2 -2e 3 2e ( ~ - b e )
½eA _l +_2 e + ~
e
( _ i i + , - 1 + i~+,-1)
+ 0(e2).
(71)
For j = 2 we find
Ps([-' +'-'
+'-1)-
2-2e -3 --2 e
+
es([-, +,-1) E
(-/I
(1 + 2e) 3
+ e(d)
R. Kluiving et al. / Symbolic dynamics of fully developed chaos IH
½eA ( 2-2e ) ( l + 2 e ) 3 1+ 3_--7-~e (1+2e)2
( 2 - 2e'~ 2 =\3-2e1 (~2-be) e (2-2e + ----2-~e 3 \3-2e
421
(_i~+_l+i~+._l)+(_i~+._.+_l+I~+_,+_l))
-1"- {~(e2).
(72)
For general j/> 1 one derives
(2-2e~ j
Ps([(--, ~)J, --]) : k 3 - Z e /
(~2 - be)
½eA
(2-2e
(1 + 2 e ) 2j-I
\3--2e
(l+2e) 2 i
e ~, / 2 - 2 e \ j-i i~(+_)q 3ZZe i=, [~3--Z-~e) (+ 1~(+,-)'1) + e(e2).
(73)
c, =--(-l~(+'-)q + l~(+'-)q) ,
(74)
2-2e 3 - 2 e ( 1 + ""ze)z,
(75)
Defining
q
1 ~
[2--2e'~ j-i
q~(e, j ) - 3--2e i=1 cA,3---Z-~e)
(76)
'
and using j-1 E
q i = 1-q____~s ,
i=o
(77)
1 -q
we can write
½eA
_
1 - qJ
(1 + 2e) 2j-1 1 - q + e~(e, j) -
+ 6(e2).
-
(78)
If we further define 4,(~, o) =- o ,
then eq. (78) is valid for all j I> O, as can be checked easily.
(79)
422
R. Kluiving et al. / Symbolic dynamics of fully developed chaos 111
4.2.2. The laminar intervals A laminar interval with length 2j is, in the (+-)-symbolic dynamics, described by j pairs of steps down and up, i.e. by [ ( - , +)~]. On the other hand, a series of 2j alternating symbols, i.e. [ ( - , +)J], does not necessarily correspond to a laminar interval since an alternating going up and going down behaviour could well take place outside the interval [½ - e, ½]. But since long alternating intervals situated outside this interval become more and more improbable for increasing j, we expect that eq. (78) expresses the probability of finding a laminar interval [ ( - , + ) J , - ] if j is chosen large enough. Consider now an orbit with as given initial condition that this orbit has been in a laminar state for J successive pairs of steps down and up, i.e. [ ( - , +)J] (J>> 1), in the (+ -)-symbolic dynamics. Then given this initial condition we ask for the average length of the laminar interval, i.e. the time that the spiralling around the fixed point will continue after this initial condition. The probability that after the initial condition the laminar phase is immediately ended is given by P~([(-, +)J]--~ [+]) =
Ps([(-, +)J, +1) P , ( [ ( - , +)J])
(80)
and the probability that the laminar phase will extend for precisely 2n steps consisting of n turns of spiralling is Ps([(-, + )J] --) [(- , +)", +]) =
Ps([(-' +)J+" +]) Ps([(-, +)J])
(81)
The average length of the laminar interval after the given initial condition is then (nlam) =
~ 2nPs([(-, +)'1--*[(-, +)', +]) n=l
= ~ 2n Ps([(-, +)J+,-l, _ ] ) _ p~([(_, +)J+,, _]) n=l Ps([(-, + ) J - ' , - ] ) = 2 Z~:l Ps([(-, +)J+" ' , - ] ) Ps([(-, +)J-l, _]) (J>l).
(82)
Now we apply eq. (78) for the probabilities in eq. (82). Neglecting the first term with [(2 - 2e)/(3 - 2e)] j as well as the term qJ in the r.h.s, of eq. (78) for j t> J, one derives for the average length of the laminar phase:
423
R. Kluiving et al. / Symbolic dynamics of fully developed chaos IH
(n,am) =
2Z2= o [(1 +2e) -2" --2A-l(1 +2e)2'-~(1 -- q) cla(e, J + n)l 1 - 2 A - ' ( 1 + 2 e ) 2 ' - ' ( 1 - q)q~(e, J - 1)
(83)
For q~(e, j) we have the estimate [2 -2e\ j q~(e, j) <<-(H + e K ) ~ ) - H[1 + 2e + ~(e2)]-zJ - eK[(3 - 2e)(1 + 2e)] -i + 19(e2),
(84)
with H and K being uniformly bounded for small values of e. This estimate is derived in appendix A. Using this estimate in the expression (83) for (nla m) it is clear that (nlam)
=
2 Z~=o(1 + 2 e ) -2" + U(1) 1 1 + U(e) = 2-7+ ~(1).
(85)
Therefore the average length of the laminar interval after the initial condition is independent of J, provided that J is large enough. Eq. (82) gives a more systematic statistical derivation of the result derived in ref. [3] which was based on dynamical considerations for orbits staying close to the fixed point. In appendix B an alternative way of deriving eq. (85) is presented by using a slightly different version of the (+ -)-symbolic dynamics taking into account that laminar sequences are confined to the interval [I - e, ~].
4.3. The correlation functions In this subsection we derive an expression for the correlation function y+,+(n) just before the critical value. The other correlation functions then follow simply from eq. (22). Our starting point is again eq. (65). From this equation one derives straightforwardly 1
P,([+, J , + l ) -
3 - 2e P*([+' j - 1, +1) + P , ( [ - , j - 1, +]) +~e
+e P,([-,
2-2e 3-2e
J, +l) = - -
i~+, j-l,~_,+t
(l+2ey
3--2e
p,([+ J - l , + ] ) '
½a - ~((1¥2~1;
1 3--2e
' e 3-2e
(86)
tI+,J-l,+] -1 ~-'
i[2+,(_,+)J/21)-+-~(82 )
(87)
424
R. Kluiving et al. / Symbolic dynamics of fully developed chaos 111
forj=2,4,6,...,
and 1 3 - 2e Ps([+' j - 1, +]) + P , ( [ - , j - 1, +])
Ps([+' J' +])-
g i[+, j t, +l + 3-2------e'1 ~ +if(e2), p,([_,j,+l)_2-2ep,([+
3-2e
'
j - 1, +1) ~
(88) e
3-2e'1
][+, j-l,
'-~
+]
+G(e2),
(89)
for j = 1, 3, 5 . . . . . The underbrace denotes the summation over the intermediate symbols, i.e.
Ps([Wl, j,wd) - Z l[w,,,w21 1 '--"--'
E
--~
Vl=+,--
"
'
P s ( [ w , , v,, v 2 . . . . .
"
Ol=+,--
vj=+,--
...
llW,.V~.v2...... ,.w2].
E
v~, w2] ) ,
(90)
(91)
Vj=+, -
Let us concentrate on the case of even j. From eqs. (86), (88) and (87) and inserting the expression (62) for A one finds Ps([+, j , + ] ) -
7 - lOe 1 ( 3 - 2 e ) 2 Ps([+' ~j - 2, + ] ) + ~
P s ( [ - , ~ J - 2 ' +])
1 1 1 3 - 2e 12 (1 + 2e) j + e R ( e , j) + "~'~u~,e2),
(92)
with 2e it+, j-z, +l R ( e , j ) - (3 -- 2 - +28) 2 "1 ~ -
1 1 + 6 (1 + 2e) j+l
qt-
e ii+. 3 - 2e
j - - l , +]
1 i~+,(_ +)j/21 3 --2e
(93)
Using eqs. (20) and (22), one derives y+,+(j + I ) -
4-8e (3_2e) 2 y+.+(j-1)
1 3-2e
+ C(e) + e R ( e , j) + G(e 2)
1 1 12 ( l + 2 e ) j (j=2, 4.... ),
(94)
with C(e) =
7-lOe ) p + 2 1 - 1 + ( ~ - - ~ e ) 2 [ s([ ])] + 3 - 2~ Ps([+l) P A [ - I ) .
Solving the recurrence relation (94) one obtains
(95)
R. Kluiving et al. / Symbolic dynamics of fully developed chaos I11
3,+,+(i+1)=(
425
4 - 8 e ~j,2 ( ~ - ~ e ) 2 ] ",/+, + (1) 1-( 1 3-2e
1 1 12 ( l + 2 e ) j 4-8e
+ c(~)
1
4-8e ( 3 - 2 e ) 2 (1 + 2e) 2
) j/2
1 -- ( (3 - 2e) 2 1
4-8e "1 2e)2) j/2 (3 - 2e) 2 [" +
+ ~A(~, j) + e(d)
4 8e (3 - 2e) 2
(] = 2 , 4 , . . . ) ,
(96)
with 4-8e A(e, ]) = R ( e , ]) + - - z "z R ( e , ( ] - 2 ) ) + " " (3
~j/2-1
[ 4-8e • • • + t-(~--
~-e)2 )
(97)
m e , Z).
Inserting y+.+(1) = P,([+, +]) - [p,([+])]2 _
~4 + O(e),
(98)
cf. eqs. (38) and (42), into eq. (96) and changing to the variable n = j + 1, one eventually ends up with /
1
\n-I
-/+,+(,,) = [7+o + 6(~)1 + [ - ~ + e(~)](, 1--757) / 4-8e
+
\(,,-1)/2
( )ltd-ff)2 )
+ e( 2)
(n = 1, 3 , 5 , . . . ) .
(99)
Using eqs. (88), (86) and (87) an expression for y+,+(j + 1) in casej is odd can now be derived. Changing again to the variable n = j + 1 yields y+,+(n) = [ 7 + •(e)] + [7-~0 + e(e)]
+ [ ~ + e(~)]( (n = 2, 4, 6 . . . . ) .
4-8~ (3 - 2~) ~ /
+ e(d)
(100)
426
R. Kluiving et al. / Symbolic dynamics of fully developed chaos III
O.9.
~a
~2
Fig. 5. The correlation function 3',.+(n) just before the critical value, i.e. for a = ½ - e e = 0.01.
with
Finally, for n = 0 one obtains using eqs. (19), (38) and (39) =
(101)
The third term in the formulas (99) and (100) is of order (2), and therefore decays rapidly and independently of the value of e. In contradistinction, the second term is of order exp(-2en) and will have decreased considerably only for values of n greater than the average length of the laminar intervals ( n l a m } = 1/2e. Notice that lim T+.+(n)= 7~ 7 + 6(e).
(102)
Thus the correlations do not die out, indicating the existence of long-range ordering just before the critical value, cf. fig. 5.
5. On the critical value from below
5.1. Infinite memory Taking the limit e - + 0 in cq. (65) yields
R. Kluiving et al. / Symbolic dynamics of fully developed chaos 1H
P,([w,,...,wk]
) = Q ( w a , w 2 ) \ ~(r, ,
~a+,w=2n-,-,~+*~p,,t 2,
k + ( - 1 ) ~-'W~ 1-I
t[
~(-1)iwi(-1)
w
.
.
427
.,w,])
)
i=2
(k~3),
(103)
and, of course we have
Ps([+]) = ~ ,
P~([-]) = P , ( [ - , +]) = Ps([+, - ] ) = ~2 ,
(104) p , ( [ + , +1) = 1 ,
P~([-,-])=0,
cf. eqs. (38)-(43). From the above one derives (105)
P,([(-, +Y,-l) = 5 +
Obviously, the same result in obtained by taking the limit e--+ 0 in eq. (78). From eq. (105) one easily derives P s ( [ - , ( + , -)J, +1-+ [ - ] ) =
P s ( [ ( - , +)J+', - ] )
1 + 4 ( 2 ) j+l
P,([(-, +)J,-])
1 +4(2) j
(106)
and P~([+, ( + , - ) ' , + ]----) [ - ] ) =
p , ( [ ( _ , +)j, _]) _ p , ( [ ( _ , +)~+l, _]) p , ( [ ( _ , + ) j - , , _]) _ P s ( [ ( - , +)J, - ] )
2 3 ' (107)
i.e. Ps([w,, ( + , - ) J , +]--+ [ - ] ) depends on the symbol w I. Thus the ( + , - ) sequence s has an infinite memory on the critical value, i.e. there does not exist an integer m such that s is order-m Markovian. 5.2. The characteristics o f s
Taking the limit e--+ 0 in eqs. (99)-(101) one finds the following expressions for the correlation function 3'+.+ (n): For n = 0: y+,+(O) =~-~4. For n is odd:
(108)
R. Kluiving et al. / Symbolic dynamics of fully developed chaos I11
428
(n) =
29 720
(
A n d for n = 2, 4, 6,
•
)n-I
•
(109)
4 [2_~n-2
"y+, + ( n ) = ~ + 45\3/
(110)
'
cf. fig. 6. One sees that the correlation function, after a short "set-in" time, remains oscillating forever between the values 7~0 and -7~0. This indicates a very strong communication over large distances since the difference between odd and even is always felt, in contradistinction with the case of finite e, cf. fig. 5. The mixing measure of s on the critical value from below is tz(s) = 2Ps([+,
-])
-
(111)
~o
which is bigger than the mixing measure on the critical value from above (/z = 4) in accordance with the fact that on the critical value from below laminar plus-minus intervals are far more probable, i.e. the mixing is better. The order-k Boltzmann entropies (15) follow from the average Shannon entropies (16) which in turn, follow from the probabilities {Ps(wk)}, which can recursively be determined from eq. (103). In table I the Boltzmann entropies h (k) as well as the average Shannon entropies S (k) are presented for k-values up 0.2
....
, ....
, ....
, ....
, ....
, ....
, ....
, ....
, ....
, ....
,
,
,
,
.
,
, ....
, ....
, ....
.
,
,
, ....
T° -
. , . ,
0.'2.
. . . .
,
.
.
,
,
p
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
.
.
.
.
.
.
.
.
. . . .
.
.
.
.
.
.
.
.
.
9o
0 ---
Fig.
6.
The
correlation
"11
function
y+,+(n)
in
the
limit
a~'½.
R. Kluiving et al. / Symbolic dynamics of fully developed chaos IH
429
Table I The Boltzmann entropies htk)(s) and the average Shannon entropies S
0.97986875 0.74167787 0.66166845 0.62166374 0.59717817 0.58085446 0.56876201 0.55969268 0.55223880 0.54627570 0.54103048 0.53665947 0.53263489 0.52918524 0.52591579
0.97986875 0.50348699 0.50164961 0.50164961 0.49923589 0.49923589 0.49620736 0.49620736 0.49260775 0.49260775 0.48857836 0.48857836 0.48433983 0.48433983 0.48014345
-0.47638 -0.00183 0 -0.00241 0 -0.00302 0 -0.00359 0 -0.00402 0 -0.00423 0 -0.00419
to 15. F r o m this table o n e observes that the B o l t z m a n n entropies h~k)(s) c o n v e r g e m o r e rapidly than the average S h a n n o n entropies s~k)(s), in agreem e n t with ref. [9]. Nevertheless the B o l t z m a n n entropies c o n v e r g e slowly: even at k = 15 the asymptotics is not clear, apart f r o m h(s) = limk__,= h~k)(s) 0.4. This m e a n s that o n the critical value f r o m below the E n t r o p y h is smaller t h a n the E n t r o p y on the critical value f r o m above, cf. eq. (35), in a c c o r d a n c e with the e x p e c t a t i o n that relatively large probabilities of laminar intervals lead to a d e c r e a s e o f the uncertainty a b o u t the ( + - ) - s e q u e n c e s.
6. Multifractal description
6.1. The Fibonacci multifractal W i t h every admissible w o r d w k = [wl, w~ . . . . , w~] in the half-infinite ( + - ) s e q u e n c e s associated with the b u n g a l o w - t e n t m a p (1) a point X(w k) of the unit interval can be associated in the following way:
X(w k) =_
3 i=1
w i
2i
(112)
T h e collection of all points C = lim U ' X ( w k) k ~zv
(113)
w k
is called [1] the Fibonacci " d u s t " or Fibonacci fractal, cf. fig. 7. T h e p r i m e in
430
R. Kluiving et al. / Symbolic dynamics of fully developed chaos I11
ii k=J.
•
k=2
+
k=:~
+
k=4
+-
+,
:
,
•
+
:
+-
+-
-_
+= ,_
_
+,
;
Fig. 7. The first five steps in constructing the Fibonacci fractal. the r.h.s, of eq. (113) denotes the restriction to admissible words, i.e. words which do not have two or more neighbouring minuses. Thus with each admissible word w k a point of the Fibonacci fractal is associated. The fractal dimension Df of the Fibonacci fractal was shown [1] to be equal to the capacity of s.
Df = d(s) = l o g 2 y ,
(114)
with 3' = (1 + X/5)/2. Each point X(w k) of the Fibonacci fractal has a probability P+(wk), depending on the value of the control parameter a. A fractal support which has a non-homogeneous probability distribution is called a multifractal [10]. In the following we will investigate the properties of the Fibonacci multifractal associated with the bungalow-tent map for a-= lim~0(½ - e) (the critical value from below) and a = ½ (the critical value from above).
6.2. Box counting and the generalized dimensions D( q) In order to describe the properties of the Fibonacci multifractal introduced in the previous subsection, we study the partition function [11]
Xq(l) = ~ Pq ,
(115)
i
where q ranges from - ~ to + ~ . The number l indicates the size of the boxes constituting a uniform grid which is laid on the unit interval. Pi denotes the probability of words corresponding to the points X(w k) inside the ith box. For empty boxes Pq vanishes. Note that q acts as a magnifying glass: for large q mainly the regions with largest probability will contribute to the summation in (115) and for large negative q regions with smallest probability will contribute. The partition function Xq is found to follow a power law behaviour as l---~0 [12, 11]:
R. Kluiving et al. / Symbolic dynamics of fully developed chaos 111
431
(116)
Xq(l ) ~ l (q-1)o(q) "
In order to distinguish between different words of length k, one introduces a grid with 2 k boxes (intervals), iX(w,), X ( w ~) + (½)k),
(117)
with size lk = (½)k.
(118)
The probabilities Pi in eq. (115) can now be replaced by the probabilities Ps(wk): )(q(lk) = E t pq(wk) wk
(119)
,
where the prime denotes the restriction to the admissible words. For q = 0 this equation reduces to Xo(l,) = ~".' = F k + 2 ,
(120)
wk
cf. ref. [1]. Here F/ denotes the ith Fibonacci number F i = Fi_ ~ + F 1 = F 2 = 1. Combining eqs. (116) and (120) yields 1 In Fk+ 2 k In 2
D(0)
fi_ 2
with
(121)
From Binet's formula
F, =
(1 + V ~ ) k - (1 - V ~ ) k 2%'3
(122)
it then follows by taking k--->oo in eq. (121) D(0) = l o g F / = D f ,
(123)
i.e. D(0) is the fractal dimension of the support of the Fibonacci multifractal, i.e. of the Fibonacci "dust". Performing the limit q - + 1 in eqs. (116) and (119) yields the result D(1)--
~ ~
t
P,(w k) log2P,(w* ) ,
Wk
i.e. D(1) is equal to the Entropy of s, cf. eqs. (16) and (17).
(124)
R. Kluiving et al. / Symbolic dynamics of fully developed chaos HI
432
M o r e generally, the entire the multifractal.
D(q) spectrum is a complete characterization of
6.3. The D( q) spectra and the phase transition In fig. 8 the D(q) spectra for a = ½ (the critical value from above) and for a = lim~0( ½ - e ) (the critical value from below) are depicted. These spectra were evaluated using eqs. (116) and (119) with l = Ik = (½)k for k = 11. For a = ½the probabilities Ps(wk) were given by eq. (32) and for a = lim~+0( ½ - e) these were determined from eqs. (103) and (104). T h o u g h for k = 11 we cannot expect to have reached convergence (cf. table I), the resolution is high enough to see that for a = ½ the D(q) curve differs everywhere from the D(q) curve for a =lim~+0( ½ - e ) (except, of course, at q = 0 where D equals the fractal dimension of the support of the multifractal). Thus the phase transition shows up as a discontinuous change on all probability scales. That the difference in fig. 8 between these two D ( q ) curves is not an artifact of the numerics, can be shown by performing an analytical evaluation of D(q) in both cases at the limiting cases q = _+2. The limiting values D(_+~) can be determined directly from the formulas: we first consider the case a = ½, i.e. the probabilities (32). Since
%
. . . .
,
. . . .
,
. . . .
,
. . . .
,
. . . .
'
.
'
.
,
. . . .
,
. . . .
,
. . . .
,
. . . .
,
. . . .
,
. . . .
,
. . . .
i
.
i
.
,
.
,
.
,
.
,
.
Dq)
O
.
-30
.
.
.
'
.
.
.
.
'
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
30
Fig. 8. The k = I1 approximation to the D(q) spectra of the Fibonacci multifracta| associated with the bungalow-tent map for a = ½ (solid line) and a = ]im~0(½ - e) (dotted line).
R. Kluiving et al. / Symbolic dynamics of fully developed chaos HI Z ' [es(wk)] q ~ ( l )k(q-1)D(q) ,
433 (125)
wk
only the smallest probabilities will give the dominant contribution for q---> - ~ . These contributions are of order ( ½)k corresponding to long intervals of pluses. Thus for q---> - ~ we find
(126)
( I )kq ~ ( 1 )kqD(q) or
In 3 D(-oo) = In 2
1.5849 . . . .
(127)
For q---> ~ only the largest probabilities will give the dominant contributions. These contributions are of order ( ~ )k/2 corresponding to alternating sequences, and thus 1 In 3 D ( + ~ ) - 2 In----2- 0.2924 . . . .
(128)
Next we turn to the case lim,~0(½ - e), i.e. the probabilities determined by eqs. (103) and (104). For q---> - ~ only the smallest probabilities contribute and again we find In 3 D ( - ~ ) - In 2 "
(129)
For q---> + ~ the dominant probabilities are the probabilities
Ps([(+, _)k,2])_
(130)
cf. eq. (105). Thus ~ ( ~ )kot+~),
(131)
from which follows that D ( + ~) = 0
(132)
on the critical value from below. Instead of the D ( q ) spectrum one may consider the f(ot) spectrum given by [lll
434
R. Kluiving et al. / Symbolic dynamics of fully developed chaos 111
(q - 1)D(q) = q a ( q ) - f ( o / ( q ) ) ,
(133)
d O/(q) : 5-2 [(q - 1)D(q)l • uq
(134)
with
The f ( a ) curve is a concave function on the interval
[o/rain, O/max]
with
O/min = D ( q ' - ° ° ) '
(135)
O/max = D ( - ~ ) .
(136)
For the Fibonacci multifractal associated with the bungalow-tent map we see that
In 3 In 2
O/max
for a = ½ and a = l i~o m (½ - e),
1 In 3 2~n2
fora=
0
for a = l i m (½ - e).
~,
O/min =
(137)
(138)
~$0
Acknowledgement This investigation is part of the research program of the Strichting voor Fundamenteel Onderzoek der Materie (FOM) which is financially supported by the Nederlandse Organisatie voor Wetenschappelijk Onderzoek (NWO).
Appendix A. Estimation of the function @(e, j) We first derive analytical expressions for the integrals l/2-e
A,~-
f
&[(+, ) q ( y ) d y ,
(A.1)
O{~+_)q(y)dy,
(A.2)
-1 xF
B~------ f l/2-e
R. Kluivinget al. / Symbolic dynamics of fully developedchaos 111 i=0,1,2
435 (A.3)
.....
up to order e. Let us start with unraveling the integral A i: l/2-e
l/2-e
qJ[(-.+)'-~,-l(f(Y)) dy ,
Ai= -1
(A.4)
-1
where again we have deleted in f the subscript 1 _ e indicating the value of the control p a r a m e t e r . Changing in eq. (A.4) to the new variable x = f ( y ) we arrive at 1
Ai=~-~2
f
1
1
qJ[¢_+),-l._l(X)dx--~3 f ~O[(_+)i-1 l(x) dx+U(e2)
1/2-e
if
(A.5)
1/2
1!
1
1
XF
1
with d 2 = 1 + 2e,
(A.7)
d 3 = -(1 +2e).
(A.8)
Changing again to the new variable y =
f(x) one finds
XF
Ai
l/2--e
- __ d2d 3 1 / 2f- e q*[(+,-)i-'l(Y'dY--dTl(~-~3
q~I(,.-)'- q(Y) dy
d2 -1
+ C(e:),
(A.9)
with d I =3-2e. Using d 3 =
(A.IO)
- d 2 we have, with the help of eqs. (A.1) and (A.2),
1
A i =---~Bil +~ d2
2
A i-i + ~ ( e 2) "
In an analogous way one derives
(A.11)
436
R. Kluiving et al. / Symbolic dynamics o f fully developed chaos 111
1 Bi = ~ Bi 1 + 0~(e2) • d2
(A.12)
Eqs. (A.11) and (A.12) can be solved, yielding A i=
Bi
d_~ [(d2) i ~
1
(~-e)+
l-(d2/dl)il] f--d---~
2 e +6(e2),
1
(A.14)
2--~-- e + G(e 2) 2 2
=
d
(A.13)
which can be rewritten as A i= D
+C(e2),
(A.16)
eE
(A.17)
e
B i=~
(A.15)
+ ~'(e 2) ,
+ eE
where D = (~ - e ) -
and 1
1
)
(A.18)
E = -~ ( l _ d 2 / d 1
go to a constant value if e--~0. From the above results we can establish estimates for Icil and q~(e, j) as we shall see next. From eqs. (74), (66) and (67) we have
l/2-e Icil ~
f
XF I~l(Y) -
'p,(y)lOt(+,-)q(Y)
l
de + le f 1/2
1
1~2(Y)lOt(+,-)q(Y) dy ~
t~l~)'--.'""
e
(A.20)
131 = Sup{I~,(Y) - q~(Y)l},
(A.21)
<< f l l A i + - fl2B, '
with
R. Kluiving et al. / Symbolic dynamics of fully developed chaos HI
437
(A.22)
/32 -- Sup(l~2(y)l}, i.e.
IciL ~ (/3~o + ½/32)
+ ~/3,e
+ 6(~2).
(A.23)
F r o m eqs. (76) and (A.23) it then follows for ~ ( e , j):
q~(e, j) ~<
/3'D+ /3z/2 ~ ( 1 ' ] i ( 2 - 2 e ] -3-~e i=, \ d ~ / \ 3 - 2 e / + 372e
i=,
d-~
\3-2e/
i-l
+(7(e2)"
(A.24)
P e r f o r m i n g the summations one arrives at 2-2e
- eg
J
+ ~(e2),
(A.25)
where H = /31D +/32/2 3 - 2e 3-2e 2-2e
z d2
1 3-2e 2 -2e
(A.26)
and
~1E K-
3-2e
3-2e 2-2e
1
(A.27) 3-2e
did2
2 - 2e
go to a constant value for e ~ 0.
Appendix B. Alternative derivation of (nja m) ~ 1/2e We consider the bungalow-tent m a p for a = ½ - e. Let us denote by I~ the interval [ ½ - e, ½]. If n consecutive iterates x i are located in I, then we speak of a laminar interval of length n [3]. T h e probability of finding one iterate in I~ is
R. Kluiving et al. / Symbolic dynamics of fully developed chaos HI
438
1/2
1/2
Pl =
f 1/2--e
l/2-e
= ~ + T (1) ,
(B.1)
with 1/2
T (~) =
(B.2)
q~z(x) d x ,
f 1/2
e
cf. eq. (4). T h e probability of finding two consecutive iterates in I~ is 1/2
1/2
e XF
__1
-1
6e l + 2 e
(f
1/2
ot_l(y)dy + f
1/2
e
i/li+l(Y)dy )
XF
1/2
f
+
q~z(X) [0[+, l(x) + 01 ,+l(X)] dx
1/2-e
1
1
+ T (2)
6 l+2e
(B.3)
with 1/2
T (2) =
f
~(x) [~,i÷ _t(x) + q't ,+j(x)l dx,
(B.4)
l/2-e
and w h e r e we have neglected terms of order e 2. For the definition of tkf+. I and qJl-,+l see eq. (25). T h e probability of finding j consecutive iterates in I, is d e n o t e d by pj. As we will see we do not n e e d explicit expressions for Pi if j/> 3. If x N = [x 1, x 2, . . . , xN] is a (chaotic) orbit with length N >> 1, then we have
Np~ = # iterates in I~,
(B.5)
N(Pl - P2) = # visits to I~ ,
(B.6)
N(Pi - Pi+l) = # visits to I, with length/> i .
(B.7)
R. Kluiving et al. / Symbolic dynamics of fully developed chaos I11
439
The probability of a laminar interval of length n is then given by t',..,(.) =
N ( p , - p , , + , ) - N ( p , + 1 - P,+2) = P, - 2p,+~ +P,,+2 N(p~ - Pz) P~ - P2
(B.8)
Thus
(,,1..~) = r L ' n(p. - 2 p . + , + p . + 2 ) =
p~
Pl - P2
PJ - P2
1
(B.9)
2e + 6(T (1) - T (2)) + ~(e 2) As a final step we show that T (~)- T (2)~ s~: 1/2
1/2
l/2-e
l/2-e
1/2
=
1/2
~o2(x) dx -
~
~(g~
I/2-e
(g~ (x)) d~
XF XF
i
f
+ 1 + 2e
,.viov
~°2(g3 (g3 (x)) dx
)
1/2-e 1/2
=
f
1/2
~o2(x) dx
lt2-e
irlv
f
1 + 2e
ilav
~o2(g3 (g3 (x))dx,
(B.10)
1/2-~
with g3V(x) the inverse of g3(x) = 1 - (1 + 2e)x.
(B.11)
cf. eq. (1). One easily checks that inv
inv
g3 (g3 (x)) = 2 e + ( 1 - 4e)x + (~(e2).
(B.12)
Changing to the variable y = 2e + ( 1 - 4e)x in the last integral of eq. (B.10) does not affect the integral boundaries and we find 1/2
1
T ( 1 ) - T (2)= (1
1
(1~ 2e)(1 --4e))
f
~°z(y) dy
1/2-e 1/2
- - ( - 2 e + t ? ( e z) f l12-r
q~E(yldy.
(K13)
440
R. Kluiving et al. / Symbolic dynamics of fully developed chaos II1
Since 1/2
f
~P2(Y) dy
~ e,
(B.14)
1/2-e
we have the estimate T ( 1 ) _ T ( 2 ) ~ 8 2,
(B.15)
from which it follows that 1 ~nlam) = ~
+ C(1).
(B.16)
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12]
R. Kluiving, H.W. Capel and R.A. Pasmanter, Physica A 183 (1992) 67, part 1. R. Kluiving, H.W. Capel and R.A. Pasmanter, Physica A 183 (1992) 96, part 2. R. Kluiving, H.W. Capel and R.A. Pasmanter, Physica A 164 (1990) 593. P. Sz6pfalusy, in: Proc. Conf. On Synergetics, Order and Chaos, Madrid 1987, Manuel G. Velarde, ed. (World Scientific, Singapore, 1988) pp. 685-697. A. Csordfis and P. Sz6pfalusy, Phys. Rev. A 38 (1988) 2582. P. Sz6pfalusy, Physica Scripta T25 (1989) 226. A. Csordfis and P. Sz6pfalusy, Phys. Rev. A 39 (1989) 4767. C.E. Shannon, Bell Syst. Techn. J. 27 (1948) 379. G. Gy6rgyi and P. Sz6pfalusy, Phys. Rev. A 31 (1985) 3477. For a review see: T. Tel, Z. Naturforsch. 43a (1988) 1154. T.C. Halsey, M.H. Jensen, L.P. Kadanoff, I. Procaccia and B.I. Shraiman, Phys. Rev. A 33 (1986) 1141. H.G.E. Hentschel and I. Procaccia, Physica D 8 (1983) 435.
~
[~1
Symbolic dynamics of fully developed chaos III. Infinite-memory sequences and phase transitions R. Kluiving and H.W. Capel lnstituut voor Theoretische Fysica, Universiteit van Amsterdam, Valckenierstraat 65, 1018 X E Amsterdam, The Netherlands
R.A. Pasmanter Koninklijk Nederlands Meteorologisch Instituut, Postbus 201, 3730 A E De Bilt, The Netherlands
Received 2 March 1992 A special type of dynamical phase transitions which arises in a particular one-dimensional fully developed chaotic iterated map is studied by means of a symbolic dynamics. Exact analytical expressions for the probabilities of words are found on both sides of the critical value, making it possible to give a statistical description of the critical behaviour at the phase transition in terms of Boltzmann entropies, correlation functions and generalized dimensions.
1. Introduction This is the third and last part of a series of articles on the application of symbolic dynamics to fully developed chaotic ( F D C ) motion in one dimension. In part 1 [1] the statistics and characteristics of general two-symbol sequences were studied. These characteristics include Boltzmann entropies and correlation functions which are functions of the probabilities of subsequences of consecutive symbols. In part 2 [2] three algorithms to extract two-symbol sequences out of a F D C iterated o n e - h u m p m a p x n+l = f ( x n ) were presented. F r o m this, three different symbolic descriptions of fully developed chaotic motion in one-dimensional o n e - h u m p maps were obtained, namely the L R - , the ( + - ) - and the AS-symbolic dynamics. All three of t h e m turned out to have their own advantages. In this third part we study the phase-transition in the bungalow-tent m a p [3] by m e a n s of the ( + - ) - s y m b o l i c dynamics. The reasons why this particular choice of symbolic description is preferred above the LR-symbolic dynamics are the following: 0378-4371/92/$05.00 (~) 1992- Elsevier Science Publishers B.V. All rights reserved
406
R. Kluiving et al. / Symbolic dynamics of fully developed chaos 111
The bungalow-tent map is labeled by a control parameter a. For a = lim~+0(½ - e ) one finds a dynamical behaviour of the iterates which differs dramatically from the dynamical behaviour for a = lim~+0(½ + e), thus showing a phase transition at a = ½. To understand this phase transition in terms of the statistics of half-infinite symbol sequences, one needs to know the probability of occurrence of words of finite length k, i.e. of subsequences of k consecutive symbols in this half-infinite symbol sequence. For a =lim,,+0(½ + e) these probabilities can be determined relatively simply in the ( + - ) - s y m b o l i c description, since we will show that the half-infinite ( + - ) - s y m b o l sequence is order-1 Markovian, i.e. the memory extends over only one time step. The half-infinite LR-sequence, on the other hand, has a more complicated m e m o r y structure and therefore the evaluation of the probabilities would be much more cumbersome. But for a = lim~0( ½ - e ) things are different. The half-infinite ( + - ) - s e q u e n c e as well as the LR-sequence have an infinite memory, i.e. there does not exist a finite integer m such that the sequences are order-m Markovian. It turns out that here it is extremely difficult if not impossible to find closed analytical expressions for the probabilities of words in the LR-description, whereas in the ( + -)-symbolic description these expressions can actually be determined for a = lim~0(½ - e) (section 5). This is due to the fact that the origin of the phase transition lies in the existence of infinitely long laminar intervals of the iterates x i in the close neighbourhood of an unstable fixed point x v for a = l i m ~ ; 0 ( ½ - e ) , which are absent for a = l i m ~ 0 ( ½ + e ) . Since the ( + - ) - s y m b o l i c description uses a partition of the interval [ - 1 , 1] into regions left ( + ) and right ( - ) of the fixed point x F, it is possible to solve r e c u r s i v e l y the integrals of the probability density of the iterates over certain parts of the interval [ - 1 , 1]. This provides immediately the desired probabilities of occurrence of words in the ( + - ) - s y m b o l i c description. Thus the ( + - ) - s y m b o l i c dynamics provides exact analytical expressions for the probability of occurrence of ( + - ) - w o r d s on both sides of the critical point a = ½. In the LR-symbolic dynamics on the other hand, the fixed point x F is not specified by the partition left ( L ) and right (R) of a reference point ~ and therefore this does not lead in a natural way to a recursive scheme of the integrals. As concerned to the AS-symbolic dynamics, the following can be noted: It is not possible to derive the probabilities directly in the AS-symbolic dynamics without knowing the results in the ( + - ) - s y m b o l i c dynamics. But since it was shown in ref. [2] that there is a one-to-one correspondence between the ( + - ) - s y m b o l i c dynamics and the AS-symbolic dynamics, one may, in principle, derive expressions for the probabilities of AS-words left and *1 This point is the so-called critical point of a one-hump map.
R. Kluiving et al. / Symbolic dynamics of fully developed chaos Ili
407
right of the critical point a = ½ from the corresponding expressions in the ( + -)-symbolic description, in this way creating an alternative exact symbolic description of the phase transition. However, since both ( ( + - ) - and ASsequences have an infinite memory at a--lim,~0( 1 - e ) , this derivation is technically complicated and therefore the ( + - ) description is more convenient than the A S description. It is worthwhile noticing that the phase transition studied in this paper is of a different nature from those usually discussed in the dynamical-systems literature [4-7]. In particular, this phase transition is caused by the variation of an external control parameter and therefore reminiscent of the more usual phase transitions in statistical mechanics. This paper is organized as follows: In section 2 the features of the bungalow-tent map are shortly reviewed. For more details about this interesting map we refer to ref. [3]. Also in section 2 we explain how to create a half-infinite ( + - ) - s e q u e n c e from the iterates of a one-hump map and how to determine the probabilities of occurrence of words in this sequence. Furthermore we enumerate the characteristics of such a sequence. Since these characteristics depend on probabilities of occurrence of words, one must know these probabilities. In section 3 these probabilities are analytically evaluated for a = limes0(½ + e), i.e. on the critical value from above. Section 4 is devoted to the ( + -)-symbolic dynamics of the bungalow-tent map for a = 1 _ e with 0 < e ~ 1, i.e. just before the critical value. In subsection 4.1 we derive an analytical expression up to order e from which the probabilities of occurrence of words can recursively be investigated. In subsection 4.2 this expression is used to give a statistical derivation for the mean length of the laminar intervals, thereby corroborating an earlier result which was based on dynamical considerations. This illustrates the power of the ( + - ) - s y m b o l i c dynamics. In subsection 4.3 we investigate the correlation function of the ( + - ) - s e q u e n c e associated with the bungalow-tent map for a = 1 _ e. It turns out that the correlations do not decay significantly for time intervals which are of the order of the mean length of the laminar intervals. Eventually the correlations decay towards a small but nonzero value, which shows that the ( + - ) - s e q u e n c e has an infinite memory. In section 5 we consider the case a = lim~0(½ - e) and compare the characteristics of the ( + - ) - s e q u e n c e with the characteristics of this sequence for a = lim,$0( ½ + e). The differences provide the statistical description of the critical behaviour at the phase transition. Section 6 presents the results of another way of characterizing the phase transition, namely by a discontinuous change in the spectrum of generalized dimensions. These generalized dimensions describe the properties of the
R. Kluiving et al. / Symbolic dynamics of fully developed chaos Ill
408
Fibonacci multifractal associated with the ( + - ) - s y m b o l i c bungalow-tent map.
d y n a m i c s of the
2. Preliminaries
2.1. The bungalow-tent map T h e b u n g a l o w - t e n t m a p [3] is defined on the interval [ - 1 , 1] as follows:
gl(x) = 1 + 2a + 2(a + 1)x
for-l~x~<-½,
g2(x) = 1 + 2(1 - a)x
for - ½ ~ x ~ < 0 ,
g3(x) = 1 + 2(a - 1)x
for 0 ~ < x ~ ½ ,
g4(x) = 1 + 2a - 2(a + 1)x
for½~x~l,
L(x) =
(1)
cf. fig. 1, w h e r e - 1 < a < 1. It is a s y m m e t r i c piece-wise-linear o n e - h u m p m a p . T h e height of the e x t r e m e left and right kink is tuned by the control p a r a m e t e r a. T h e i t e r a t e d system
Xn+1
= ]ca (Xn)
(2)
is F D C , i.e. chaotic and ergodic on the entire interval [ - 1 , 1], for - ½ < a < 1, w h e r e a s for - 1 < a < - ½ the fixed point 2 = - 1 is stable.
1
Ii 8
t
...........
O
~(x)
T
-1 -1
-0,5
8 )
B,5
X
Fig. 1. The bungalow-tent map.
1
R. Kluiving et al. / Symbolic dynamics of fully developed chaos 111
409
In this article the bungalow-tent map is studied for values of a near ½. In particular, use will be made of the expressions derived in ref. [3] for the probability density pa(x) of the iterates for a = ½ and a = ½ - e with 0 < e ~ 1: p~,2(x)= ~o(: 2 , -x)+
~O(x- ½)
(3)
and
+ e~pl(x) + C(e2), Pl/2_e(X)=
I -1 + ~o2(x)+ 6 ( e ) , ge
+ e%(x)+ 6~(f),
-l
- e,
21
½,
e
(4)
½< x < l ,
where ~ol, ~o2 and % are unknown functions. Taking the limit e~0 in eq. (4) yields lim E~O
p~/2_,(x)= 10(½-x)+ 20(x- ~)+ ~6(x- 1),
(5)
which differs from eq. (3). Thus the probability density changes discontinuously at the transition (a~'½)--~ (a = ½). This explains quantitatively the discontinuity [3] in the Lyapounov spectrum, 1
A. = f p.(x) loglf'(x)l d x ,
(6)
-1
at a = ½. In fact there is an infinite number of discontinuities in this Lyapounov spectrum, as shown in fig. 2, their locations ag being determined by the equation [2(a, + 1)]*-~(1- a , ) = ½
(k = 1,2 . . . . ).
(7)
This equation was derived using an exact scaling procedure which showed that the jump phenomena at a 2, a 3 are rescaled copies of the phenomenon at a = ½ [3]. In this article the following terminology concerning values of a close to ½will be used: . . . .
a= ½-e
withO
just before the critical value,
a = l i m (½ - e):
the critical value from below,
a= ~
the critical value from above,
e$O
cf. fig. 3.
(= lifo (½ + e)):
R. Kluiving et al. / Symbolic dynamics of fully developed chaos IH
410
o.~
0
-0.5
i
i
o
o,5
1
m_ O.
Fig. 2. The Lyapounov spectrum of the bungalow-tent map. The jumps occur at values a k determined by eq. (7).
1
a
Fig. 3. Visualization of the phase transition in the bungalow-tent map in terms of the jump in its Lyapounov spectrum. 1 refers to just before the critical value, 2 denotes the critical value from below and 3 refers to the critical value from above.
R. Kluiving et al. / Symbolic dynamics of fully developed chaos I11 2.2.
411
The (+ - ) - s y m b o l i c dynamics
If x = x , , , x l , x2 . . . .
(8)
is a trajectory of a F D C iterated o n e - h u m p map, then the half-infinite ( + - ) - s e q u e n c e s is defined as follows:
S = [S|,S2,. " .],
Si=
+ __
if xi > xi_ 1, ifxi
(9)
A n equivalent way of forming s is
s = [s 1, s 2 , . . . ] ,
si =
+ -
ifxi_ 1
(10)
where x v is the extreme right fixed point of the o n e - h u m p map. The probability of finding a particular word or subsequence w ~ = [ w l , w 2. . . . .
wk],
wi=+
of k successive symbols w 1, w 2 . . . . .
or-,
k=l,2
....
,
(11)
w k in s is given by
N-k+l
P s ( W k ) = l!m= N -
1k + l
~, i=l
6w,.,i6w>,i+l...6wk.si+~ .
(12)
This probability can be shown to be equal to 1
Ps(w k) : J d x p(x) 0[sgn(w,) [f(x) - x]] 1
× 0[sgn(w2) [f(Z)(x) - f ( x ) ] ] ' ' '
0[sgn(wk)[f(k)(x)
-- f(k-l)(X)]]
,
(13)
where p(x) is the probability density of the m a p and {10 0(y) =_
if y > 0 , if y < 0 .
(14)
D u e to the nature of a o n e - h u m p m a p , a minus in the ( + - ) - s e q u e n c e s is always followed by a plus, or, equivalently, one cannot find words w k in s which possess one or m o r e pairs of neighbouring minuses. T h e symbol sequence s gives a description of the trajectory x in terms of
R. Kluiving et al. / Symbolic dynamics o f fully developed chaos 1H
412
ascending and descending behaviour and, due to the FDC-aspect of the map, the statistical properties of s are independent of the starting position x 0 (as long as x 0 does not belong to an unstable cycle), as follows from eq. (13). The level-k (Boltzmann) entropies h (k) [1] of a half-infinite sequence s are given by
h~g)(s) = kS(k)(s) - ( k -
1)5(k-')(8),
(15)
where 1
S(k)(s) =_ _ ~ ~ es(w k) logzPs(w k)
(16t
wk
is the average Shannon entropy [8, 9, 1] associated with the given probabilities of words with length k. Words which have a zero probability do not give a contribution to the summation in the r.h.s, of eq. (16), in accordance with limx+ 0 x l o g x = 0 . The level-k entropy htk)(s) is a measure for remaining uncertainty about s after being informed of all the probabilities Ps(W k) of words with length k, i.e. it is the logarithm of the number of questions per symbol that should be asked to know which particular sequence is under consideration. The limit
h(s) =
lira
k~
h~k)(s)= kl i~m S(k)(s)
(17)
is called the Entropy of the sequence s [1]. It measures the uncertainty about s after being informed of all possible statistical information about s. If s is order-m Markovian, i.e. the memory extends only over the last m symbols, then one can derive [1] that
h(s) = h
+')(s).
(18)
The four symbolic correlation functions y+,+(n), 7+, (n), 7_ +(n) and 7_ _(n) of two symbols at distance n of a sequence s are defined in the following way [1]. If n = 0 then
yw~.w2(O) = 6w~,w2ps([wt]) - Ps([Wt]) Ps([w2]),
(19)
and if n/> 1 then
y,,,.w2(n) = es([w , , n - 1, w2] ) - Ps([w,]) Ps([Wz]),
(20)
where the underbrace denotes the summation over the intermediate n - 1 symbols:
R. Kluiving et al. / Symbolic dynamics of fully developed chaos 111 P,([w,,n-l,
~
w2])-==
"'"
o,=+,-
~ P,([w,,v,,v v._,=+,-
2. . . . .
413
v._,,w2]).
(21)
In ref. [1] it was shown that (22)
= -y_+(n),
%.+(n) = %._(n) = -y+_(n)
i.e. one needs only one of the four correlation functions to know the correlations between two symbols over a distance n. Another characteristic of s is the mixing measure/z defined by /*(s) = 2Ps([+,-])
(23)
E [0, 1].
It measures the degree of mixing of the symbols + and - . If/z(s) = 0, we say that the symbols + and - are not mixed. If/x(s) = 1, then we say that the mixing is maximal.
3. The critical value from above
We will apply the symbolic dynamics presented above to the bungalow-tent map with a = ½. For this map the fixed point x F is equal to ½, and the probability density is given by eq. (3). The probability of finding a plus in the ( + -)-symbol sequence s associated with the map is then XF
P,([+])
=
f
Pl/2(X) dx =
(24)
3. 2 = 3 .
-1
More generally, omitting in f the subscript a -- ~ indicating the value of the control parameter and using the short-hand notation
~Owk(X) : q'tw,,w, ...... ,j(x) = 0[sgn(w,) ...
0[sgn(wk
If(x)
- x]] 0[sgn(w2)
[f(2)(X) --f(x)]] • " "
) [f(k)(x ) _ f(k-')(x)]],
one derives 1/2
PAl+, +
, w 3 , . . . ,wkl ) = f -1
~01+,w3...... kl(f(x))dx
(25)
R. Kluiving et al. / Symbolic dynamics of fully developed chaos 111
414
f(I/2)
=5
l(f
f(1/2)
f(O)
f
+
f(I)
f(1/2)
i)2[~4~÷,w~
-
J
. . ..,. ~I(Y)
dY]
f(O)
I/2
1 (
: 5
2
J
1
5~Ol+.w3...... , ( x l d x = 5 Ps([+' w3 . . . . .
(26)
w,]),
f(1)
from which one obtains the recursion relation wk]) : Q ( W l , w2)(2.J.&.',rl , . 3 ,~+.,.', - ~P.([w2 . . . . , . , . ] ) , (27)
P s ( [ W l , w2 . . . . .
relating the probabilities of level k to the probabilities of level k - 1. Here Q, being a function of two neighbouring symbols w i and wi+l, is defined by (~
ifwi=-andwi+l otherwise.
Q(wi' wi+I)=
=-,
(28)
Relating level k to level l, i.e. applying the recursion formula eq. (27) k - 1 times, yields e s ( [ w l , w2 . . . . :
, w,])
O ( w i , wi+i
2)
-."'~t!',-'-' (6+.w-6 "' ')Ps([wkl). ,.3;
(29)
Using P,([wk] ) = 3(32-)6 -,',,
(30)
cf. eq. (24), and k
k
(31)
2 8 .... = k - E & . . , , j=l
j:l
we arrive at the final result: p s ( w * ) = _5
Q ( w i , wi+l ) (2
. ,(~)
. ~
.
-.,
(32)
for the probability of finding a particular word wk in the ( + - ) - s y m b o l sequence s associated with the bungalow-tent map with a = ½.
R. Kluiving et al. / Symbolic dynamics of fully developed chaos 111
415
F o r the conditional probability
es(Iw, . . . .
-=
, wA--, [w~+ll)
e,([w,,..., wk+d) es([14dl,...,
(33)
14"k] )
o n e finds
Ps([Wl
. . . .
,
w,,]---~ [Wk+l] =
Q(wk, Wk+,)(-~)'~-''~+'(-{) ~+'w~+'-~-'''
= P,([wA--" [w~+,]).
(34)
T h u s the ( + - ) - s y m b o l s e q u e n c e s of the (a = 1)_bungalow_tent m a p is order-1 M a r k o v i a n , i.e. the m e m o r y extends only o v e r one time step. In this case the E n t r o p y of the s e q u e n c e reduces to h(s) = h(2)(s) = 2S(2)(s) - S ~' ~(s)
(35)
= _ 3[½ log2~ + 21og2~ ] = 0 . 5 5 0 9 7 7 . . . .
cf. eqs. ( 1 5 ) - ( 1 8 ) . Since the m a x i m u m possible value for the E n t r o p y of a half-infinite ( + - ) - s e q u e n c e is equal to the capacity d ( s ) = log2[½(1 + X/5)] = 0.6942419 . . . . cf. refs. [1, 2], the ( + - ) - s e q u e n c e s of the (a = ½)-bungalowtent m a p does not c o r r e s p o n d to m a x i m a l uncertainty.
O.3
o
-
o3
,
0
,
,
20
Fig. 4. The correlation function y+ +(n) of the (+-)-sequence s associated with the (a = ½)bungalow-tent map.
416
R. Kluiving et al. / Symbofic dynamics of fully developed chaos 111
For the correlation function 7+,+(n), cf. eqs. (19) and (20), one derives from eq. (32) 74.+(n) = ( - 1 ) " 9 ( 2 ) "41 ,
(36)
cf. fig. 4. The other three correlation functions follow using eq. (22). Notice that the correlations decay exponentially.
4. Just before the critical value 4.1. The probabilities Ps(w ~)
For a = ½ - e, with 0 < e ~ 1, the probability density of the iterates is given by eq. (4). With the fixed point 1 _ ½ _ ½e + 6(e z) xv - 2(1 + e)
(37)
the following expressions for the probabilities up to level 2 are easily derived: xF
Ps([+]) = f p,/z_~(x) dx = 7 + be + G(e2),
(38)
1
P~([-])
= 1 - P~([+])
(39)
= ~ - be + 6(d),
P s ( [ - , +1) = P ~ ( [ - ] ) ,
(40)
Ps([+, -1) = P s ( [ - , +1),
(41)
Ps([ +, +]) = P ~ ( [ + ] ) - Ps([ +, -1) = ~6+ 2 b e + O(e2),
(42)
Ps([-, -])=0.
(43)
Here b is an unknown constant. We now turn to the probabilities of level 3. Omitting in f the subscript ~ - e indicating the value of the control parameter and using again the short-hand notation (25), one has 1
XF
/*
P~([+, +, +]) = J p,~-~(~) ¢[+.+.+j(x)dx = J p,,~ ~(x)¢[+.+~(f(x))dx -I
= Jl + J2 + J3 + J4,
-I
(44)
R. Kluivinget al. / Symbolic dynamics of fully developedchaos 11I
417
with -1/2
f dx [-~ + ~,p,(xll~,~+, +l(f(xl) + ~(d),
J1
(45)
-1 0
dx [½ +
J2 = f
E~I(X)]I~/[+" + ] ( f ( x ) ) +
~(2),
(46)
-1/2 l/2-e
dx [½ + eq~l(X)]qJ[+' +](f(x)) + ~'(e2) ,
J3
(47)
0 xF 14
dx(1
f
+q~2(x))4jl+,+l(f(x))+
~(2).
(48)
l/2-e
Introducing the new variable y = f(x) and using qJ[+.+l(y) = 0
ify>xv,
(49)
dy [1 + e~,(y)l~Oi+,+l(y) + ~(82) ,
(50)
dy ½OI+,+I(Y) + ~7(e2) ,
(51)
one finds l/2-e
1
(
Jl - 3 - 2 e
J -1
xF
1 f
J2- l+2e
1/2-e
J3=J4=O,
(52)
~l(g,(Y)) = q~l(Y),
(53)
gl(y) = 1 +2(½ - e ) + 2 ( { - e)y,
(54)
with
cf. eq. (1). With the help of the notation 1/2-e
1~+'+j=- f -1
[q~l(Y) --
~o~(Y)]q'i+,+] dy
(55)
R. Kluivinget al. / Symbolicdynamics of fully developedchaos IH
418
we can write xF
,f
dy Pl/2-E(Y) 0[+.+I(Y)
P,([+, +, + l ) - 3 - - 2 e
1 xF
f
3 -2e
dY (6~ + q~2(Y))tPl+,+l(Y)
l/2-e
xF
+ 21-+~ 1/2
f
l e Ii+,+1+ dy ~[+,+I(Y) + 3 Z 2e e
G(e2). (56)
The first term is equal to 1
(57)
3 - 2e P ' ( [ + ' + ] ) '
and the second and third turn out to be zero after changing to the new variable z = f(y), as can be checked readily. Therefore we end up with ps([+, +, +1) _
1
3
2e
[p,([+, +1) + ell+, +l] +
~(~.2)
,
(5s)
In an analogous way one can derive
eA(e)
1
P ' ( [ + ' +' - 1 ) - 3 - 2e P*([+' - ] ) + 2(1 + 2e)
+ e( II+'-1
3 --12e I~+'-]) + ~(e2) ,
(59)
with 1/2-e
II +'-1---
f
[~I(Y) -
'Pl(Y)lq'[+,-1 d y ,
(60)
-I xF
I~+'-l~-le
f
dy ~o2(y) ~0[+,-I(Y),
(61)
l/2-e
and 1
1
A ( e ) - 3 1+2e
1
1
3 - 2 e 6e "
The integral i~+,-1 and the constant A(e) arise when evaluating
(62)
R. Kluiving et al. / Symbolic dynamics of fully developed chaos 111 xF
xF
f dy (16ee + ~2(Y) )~bI+,-I(Y) + 1 +if2e - -
3 -2e
I/2-e
419
dy -{ ¢I+,-I(Y) 1/2
e
(63)
From eqs. (58) and (59) one arrives at Ps([w,, w2, w3] ) = Q(w 1, w 2 ) [ ( 3 - 2e)-~+'w2((2 - 2e) 8 xP,([w2, w3]) +
,w22d +,w2
(-- ]'l t3 ,Wls l[w2,w3] ]
+,w2',
,
, ,,~ .w,,. ,. [ ½A(e) + k-l) e°+'w2° "w3[l-+2-e
-1
1
1 1~w2.~31)1 3 --2e
+ 6(e~),
(64)
relating the probabilities of level 3 to the probabilities of level 2 just before the critical value. More generally, one is able to derive along the same lines Ps([w~, w2 . . . . .
wk]) = Q(wl, w 2 ) I ( 3 - 2e)-~+'"~2[(2 - 2e)~-.wl~+-w.,
x P , ( [ w 2, w 3 , . . . , w,,,]) + ~+,,,2(-1f "~'ell "2 ...... ~'1)] (*
)(
1A(e)
+ (-1)'-~'e.,=~[I a(_}, ~, (1 + 2e) *-2 + tT(e 2)
1 ,~w:...... ,1)] 3 £ 2e
(k/> 3),
(65)
with 1/2-e
l~W2...... ,1 =-- f
[ ~ l ( Y ) - qh(Y)]O[w 2...... ~I(Y) dY'
(66)
-1
and xF
i[-'2 ...... ,1 = _le f
dy q~z(Y) ~Otw2...... kl(Y) d y ,
(67)
1/2-e
relating the probabilities of level k with the probabilities of level k - 1. Eq. (65) forms the corner-stone of our analysis of the phase-transition in the bungalow-tent map, as will be shown in the following subsections.
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R. Kluiving et al. / Symbolic dynamics of fully developed chaos II1
4.2. Mean length of the laminar intervals In ref. [3] it was shown that, just before the critical value, the trajectory x shows long intermittent laminar intervals in the neighbourhood of x = ½, which are absent just after the critical value. Closer analysis revealed that a laminar interval consists of a slow spiralling away from the (unstable) fixed point x v. The closer the parameter a is set near ½from below, the longer the trajectory keeps spiralling around x F. Eventually the trajectory escapes, giving rise to a chaotic burst, until it returns again to the close neighbourhood of XF, i.e. the interval [ ~1 - e, ½] (since the map is ergodic on the interval [ - 1 , 1] the trajectory will certainly return to this neighbourhood), giving birth to a new laminar interval. In ref. [3] it was argued, using dynamical considerations, that 1 (nlam) = ~gg -}- G(1),
(68)
with (nla m) the mean length of the laminar intervals (or: the mean spiralling time) and e = 1 _ a. In subsubsections 4.2.1 and 4.2.2 we will present a systematic statistical derivation of (68) based on the probabilities of long alternating sequences, which will be investigated first.
4.2.1. Probabilities of alternating sequences First we will derive an analytical expression for the probabilities P s ( [ ( - , +)J, -1)
( j = 0 , 1, 2 , . . . )
(69)
for a = ½ - e, i.e. just before the critical value. F o r j = 0 we have from eq. (39) P s ( [ - ] ) = ~2 - be.
(70)
For j = 1 one derives from eqs. (65) and eq. (70)
P'([-' +' -])-
2 -2e 3 2e ( ~ - b e )
½eA _l +_2 e + ~
e
( _ i i + , - 1 + i~+,-1)
+ 0(e2).
(71)
For j = 2 we find
Ps([-' +'-'
+'-1)-
2-2e -3 --2 e
+
es([-, +,-1) E
(-/I
(1 + 2e) 3
+ e(d)
R. Kluiving et al. / Symbolic dynamics of fully developed chaos IH
½eA ( 2-2e ) ( l + 2 e ) 3 1+ 3_--7-~e (1+2e)2
( 2 - 2e'~ 2 =\3-2e1 (~2-be) e (2-2e + ----2-~e 3 \3-2e
421
(_i~+_l+i~+._l)+(_i~+._.+_l+I~+_,+_l))
-1"- {~(e2).
(72)
For general j/> 1 one derives
(2-2e~ j
Ps([(--, ~)J, --]) : k 3 - Z e /
(~2 - be)
½eA
(2-2e
(1 + 2 e ) 2j-I
\3--2e
(l+2e) 2 i
e ~, / 2 - 2 e \ j-i i~(+_)q 3ZZe i=, [~3--Z-~e) (+ 1~(+,-)'1) + e(e2).
(73)
c, =--(-l~(+'-)q + l~(+'-)q) ,
(74)
2-2e 3 - 2 e ( 1 + ""ze)z,
(75)
Defining
q
1 ~
[2--2e'~ j-i
q~(e, j ) - 3--2e i=1 cA,3---Z-~e)
(76)
'
and using j-1 E
q i = 1-q____~s ,
i=o
(77)
1 -q
we can write
½eA
_
1 - qJ
(1 + 2e) 2j-1 1 - q + e~(e, j) -
+ 6(e2).
-
(78)
If we further define 4,(~, o) =- o ,
then eq. (78) is valid for all j I> O, as can be checked easily.
(79)
422
R. Kluiving et al. / Symbolic dynamics of fully developed chaos 111
4.2.2. The laminar intervals A laminar interval with length 2j is, in the (+-)-symbolic dynamics, described by j pairs of steps down and up, i.e. by [ ( - , +)~]. On the other hand, a series of 2j alternating symbols, i.e. [ ( - , +)J], does not necessarily correspond to a laminar interval since an alternating going up and going down behaviour could well take place outside the interval [½ - e, ½]. But since long alternating intervals situated outside this interval become more and more improbable for increasing j, we expect that eq. (78) expresses the probability of finding a laminar interval [ ( - , + ) J , - ] if j is chosen large enough. Consider now an orbit with as given initial condition that this orbit has been in a laminar state for J successive pairs of steps down and up, i.e. [ ( - , +)J] (J>> 1), in the (+ -)-symbolic dynamics. Then given this initial condition we ask for the average length of the laminar interval, i.e. the time that the spiralling around the fixed point will continue after this initial condition. The probability that after the initial condition the laminar phase is immediately ended is given by P~([(-, +)J]--~ [+]) =
Ps([(-, +)J, +1) P , ( [ ( - , +)J])
(80)
and the probability that the laminar phase will extend for precisely 2n steps consisting of n turns of spiralling is Ps([(-, + )J] --) [(- , +)", +]) =
Ps([(-' +)J+" +]) Ps([(-, +)J])
(81)
The average length of the laminar interval after the given initial condition is then (nlam) =
~ 2nPs([(-, +)'1--*[(-, +)', +]) n=l
= ~ 2n Ps([(-, +)J+,-l, _ ] ) _ p~([(_, +)J+,, _]) n=l Ps([(-, + ) J - ' , - ] ) = 2 Z~:l Ps([(-, +)J+" ' , - ] ) Ps([(-, +)J-l, _]) (J>l).
(82)
Now we apply eq. (78) for the probabilities in eq. (82). Neglecting the first term with [(2 - 2e)/(3 - 2e)] j as well as the term qJ in the r.h.s, of eq. (78) for j t> J, one derives for the average length of the laminar phase:
423
R. Kluiving et al. / Symbolic dynamics of fully developed chaos IH
(n,am) =
2Z2= o [(1 +2e) -2" --2A-l(1 +2e)2'-~(1 -- q) cla(e, J + n)l 1 - 2 A - ' ( 1 + 2 e ) 2 ' - ' ( 1 - q)q~(e, J - 1)
(83)
For q~(e, j) we have the estimate [2 -2e\ j q~(e, j) <<-(H + e K ) ~ ) - H[1 + 2e + ~(e2)]-zJ - eK[(3 - 2e)(1 + 2e)] -i + 19(e2),
(84)
with H and K being uniformly bounded for small values of e. This estimate is derived in appendix A. Using this estimate in the expression (83) for (nla m) it is clear that (nlam)
=
2 Z~=o(1 + 2 e ) -2" + U(1) 1 1 + U(e) = 2-7+ ~(1).
(85)
Therefore the average length of the laminar interval after the initial condition is independent of J, provided that J is large enough. Eq. (82) gives a more systematic statistical derivation of the result derived in ref. [3] which was based on dynamical considerations for orbits staying close to the fixed point. In appendix B an alternative way of deriving eq. (85) is presented by using a slightly different version of the (+ -)-symbolic dynamics taking into account that laminar sequences are confined to the interval [I - e, ~].
4.3. The correlation functions In this subsection we derive an expression for the correlation function y+,+(n) just before the critical value. The other correlation functions then follow simply from eq. (22). Our starting point is again eq. (65). From this equation one derives straightforwardly 1
P,([+, J , + l ) -
3 - 2e P*([+' j - 1, +1) + P , ( [ - , j - 1, +]) +~e
+e P,([-,
2-2e 3-2e
J, +l) = - -
i~+, j-l,~_,+t
(l+2ey
3--2e
p,([+ J - l , + ] ) '
½a - ~((1¥2~1;
1 3--2e
' e 3-2e
(86)
tI+,J-l,+] -1 ~-'
i[2+,(_,+)J/21)-+-~(82 )
(87)
424
R. Kluiving et al. / Symbolic dynamics of fully developed chaos 111
forj=2,4,6,...,
and 1 3 - 2e Ps([+' j - 1, +]) + P , ( [ - , j - 1, +])
Ps([+' J' +])-
g i[+, j t, +l + 3-2------e'1 ~ +if(e2), p,([_,j,+l)_2-2ep,([+
3-2e
'
j - 1, +1) ~
(88) e
3-2e'1
][+, j-l,
'-~
+]
+G(e2),
(89)
for j = 1, 3, 5 . . . . . The underbrace denotes the summation over the intermediate symbols, i.e.
Ps([Wl, j,wd) - Z l[w,,,w21 1 '--"--'
E
--~
Vl=+,--
"
'
P s ( [ w , , v,, v 2 . . . . .
"
Ol=+,--
vj=+,--
...
llW,.V~.v2...... ,.w2].
E
v~, w2] ) ,
(90)
(91)
Vj=+, -
Let us concentrate on the case of even j. From eqs. (86), (88) and (87) and inserting the expression (62) for A one finds Ps([+, j , + ] ) -
7 - lOe 1 ( 3 - 2 e ) 2 Ps([+' ~j - 2, + ] ) + ~
P s ( [ - , ~ J - 2 ' +])
1 1 1 3 - 2e 12 (1 + 2e) j + e R ( e , j) + "~'~u~,e2),
(92)
with 2e it+, j-z, +l R ( e , j ) - (3 -- 2 - +28) 2 "1 ~ -
1 1 + 6 (1 + 2e) j+l
qt-
e ii+. 3 - 2e
j - - l , +]
1 i~+,(_ +)j/21 3 --2e
(93)
Using eqs. (20) and (22), one derives y+,+(j + I ) -
4-8e (3_2e) 2 y+.+(j-1)
1 3-2e
+ C(e) + e R ( e , j) + G(e 2)
1 1 12 ( l + 2 e ) j (j=2, 4.... ),
(94)
with C(e) =
7-lOe ) p + 2 1 - 1 + ( ~ - - ~ e ) 2 [ s([ ])] + 3 - 2~ Ps([+l) P A [ - I ) .
Solving the recurrence relation (94) one obtains
(95)
R. Kluiving et al. / Symbolic dynamics of fully developed chaos I11
3,+,+(i+1)=(
425
4 - 8 e ~j,2 ( ~ - ~ e ) 2 ] ",/+, + (1) 1-( 1 3-2e
1 1 12 ( l + 2 e ) j 4-8e
+ c(~)
1
4-8e ( 3 - 2 e ) 2 (1 + 2e) 2
) j/2
1 -- ( (3 - 2e) 2 1
4-8e "1 2e)2) j/2 (3 - 2e) 2 [" +
+ ~A(~, j) + e(d)
4 8e (3 - 2e) 2
(] = 2 , 4 , . . . ) ,
(96)
with 4-8e A(e, ]) = R ( e , ]) + - - z "z R ( e , ( ] - 2 ) ) + " " (3
~j/2-1
[ 4-8e • • • + t-(~--
~-e)2 )
(97)
m e , Z).
Inserting y+.+(1) = P,([+, +]) - [p,([+])]2 _
~4 + O(e),
(98)
cf. eqs. (38) and (42), into eq. (96) and changing to the variable n = j + 1, one eventually ends up with /
1
\n-I
-/+,+(,,) = [7+o + 6(~)1 + [ - ~ + e(~)](, 1--757) / 4-8e
+
\(,,-1)/2
( )ltd-ff)2 )
+ e( 2)
(n = 1, 3 , 5 , . . . ) .
(99)
Using eqs. (88), (86) and (87) an expression for y+,+(j + 1) in casej is odd can now be derived. Changing again to the variable n = j + 1 yields y+,+(n) = [ 7 + •(e)] + [7-~0 + e(e)]
+ [ ~ + e(~)]( (n = 2, 4, 6 . . . . ) .
4-8~ (3 - 2~) ~ /
+ e(d)
(100)
426
R. Kluiving et al. / Symbolic dynamics of fully developed chaos III
O.9.
~a
~2
Fig. 5. The correlation function 3',.+(n) just before the critical value, i.e. for a = ½ - e e = 0.01.
with
Finally, for n = 0 one obtains using eqs. (19), (38) and (39) =
(101)
The third term in the formulas (99) and (100) is of order (2), and therefore decays rapidly and independently of the value of e. In contradistinction, the second term is of order exp(-2en) and will have decreased considerably only for values of n greater than the average length of the laminar intervals ( n l a m } = 1/2e. Notice that lim T+.+(n)= 7~ 7 + 6(e).
(102)
Thus the correlations do not die out, indicating the existence of long-range ordering just before the critical value, cf. fig. 5.
5. On the critical value from below
5.1. Infinite memory Taking the limit e - + 0 in cq. (65) yields
R. Kluiving et al. / Symbolic dynamics of fully developed chaos 1H
P,([w,,...,wk]
) = Q ( w a , w 2 ) \ ~(r, ,
~a+,w=2n-,-,~+*~p,,t 2,
k + ( - 1 ) ~-'W~ 1-I
t[
~(-1)iwi(-1)
w
.
.
427
.,w,])
)
i=2
(k~3),
(103)
and, of course we have
Ps([+]) = ~ ,
P~([-]) = P , ( [ - , +]) = Ps([+, - ] ) = ~2 ,
(104) p , ( [ + , +1) = 1 ,
P~([-,-])=0,
cf. eqs. (38)-(43). From the above one derives (105)
P,([(-, +Y,-l) = 5 +
Obviously, the same result in obtained by taking the limit e--+ 0 in eq. (78). From eq. (105) one easily derives P s ( [ - , ( + , -)J, +1-+ [ - ] ) =
P s ( [ ( - , +)J+', - ] )
1 + 4 ( 2 ) j+l
P,([(-, +)J,-])
1 +4(2) j
(106)
and P~([+, ( + , - ) ' , + ]----) [ - ] ) =
p , ( [ ( _ , +)j, _]) _ p , ( [ ( _ , +)~+l, _]) p , ( [ ( _ , + ) j - , , _]) _ P s ( [ ( - , +)J, - ] )
2 3 ' (107)
i.e. Ps([w,, ( + , - ) J , +]--+ [ - ] ) depends on the symbol w I. Thus the ( + , - ) sequence s has an infinite memory on the critical value, i.e. there does not exist an integer m such that s is order-m Markovian. 5.2. The characteristics o f s
Taking the limit e--+ 0 in eqs. (99)-(101) one finds the following expressions for the correlation function 3'+.+ (n): For n = 0: y+,+(O) =~-~4. For n is odd:
(108)
R. Kluiving et al. / Symbolic dynamics of fully developed chaos I11
428
(n) =
29 720
(
A n d for n = 2, 4, 6,
•
)n-I
•
(109)
4 [2_~n-2
"y+, + ( n ) = ~ + 45\3/
(110)
'
cf. fig. 6. One sees that the correlation function, after a short "set-in" time, remains oscillating forever between the values 7~0 and -7~0. This indicates a very strong communication over large distances since the difference between odd and even is always felt, in contradistinction with the case of finite e, cf. fig. 5. The mixing measure of s on the critical value from below is tz(s) = 2Ps([+,
-])
-
(111)
~o
which is bigger than the mixing measure on the critical value from above (/z = 4) in accordance with the fact that on the critical value from below laminar plus-minus intervals are far more probable, i.e. the mixing is better. The order-k Boltzmann entropies (15) follow from the average Shannon entropies (16) which in turn, follow from the probabilities {Ps(wk)}, which can recursively be determined from eq. (103). In table I the Boltzmann entropies h (k) as well as the average Shannon entropies S (k) are presented for k-values up 0.2
....
, ....
, ....
, ....
, ....
, ....
, ....
, ....
, ....
, ....
,
,
,
,
.
,
, ....
, ....
, ....
.
,
,
, ....
T° -
. , . ,
0.'2.
. . . .
,
.
.
,
,
p
. . . .
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.
.
.
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.
.
.
.
. . . .
.
.
.
.
.
.
.
.
.
9o
0 ---
Fig.
6.
The
correlation
"11
function
y+,+(n)
in
the
limit
a~'½.
R. Kluiving et al. / Symbolic dynamics of fully developed chaos IH
429
Table I The Boltzmann entropies htk)(s) and the average Shannon entropies S
0.97986875 0.74167787 0.66166845 0.62166374 0.59717817 0.58085446 0.56876201 0.55969268 0.55223880 0.54627570 0.54103048 0.53665947 0.53263489 0.52918524 0.52591579
0.97986875 0.50348699 0.50164961 0.50164961 0.49923589 0.49923589 0.49620736 0.49620736 0.49260775 0.49260775 0.48857836 0.48857836 0.48433983 0.48433983 0.48014345
-0.47638 -0.00183 0 -0.00241 0 -0.00302 0 -0.00359 0 -0.00402 0 -0.00423 0 -0.00419
to 15. F r o m this table o n e observes that the B o l t z m a n n entropies h~k)(s) c o n v e r g e m o r e rapidly than the average S h a n n o n entropies s~k)(s), in agreem e n t with ref. [9]. Nevertheless the B o l t z m a n n entropies c o n v e r g e slowly: even at k = 15 the asymptotics is not clear, apart f r o m h(s) = limk__,= h~k)(s) 0.4. This m e a n s that o n the critical value f r o m below the E n t r o p y h is smaller t h a n the E n t r o p y on the critical value f r o m above, cf. eq. (35), in a c c o r d a n c e with the e x p e c t a t i o n that relatively large probabilities of laminar intervals lead to a d e c r e a s e o f the uncertainty a b o u t the ( + - ) - s e q u e n c e s.
6. Multifractal description
6.1. The Fibonacci multifractal W i t h every admissible w o r d w k = [wl, w~ . . . . , w~] in the half-infinite ( + - ) s e q u e n c e s associated with the b u n g a l o w - t e n t m a p (1) a point X(w k) of the unit interval can be associated in the following way:
X(w k) =_
3 i=1
w i
2i
(112)
T h e collection of all points C = lim U ' X ( w k) k ~zv
(113)
w k
is called [1] the Fibonacci " d u s t " or Fibonacci fractal, cf. fig. 7. T h e p r i m e in
430
R. Kluiving et al. / Symbolic dynamics of fully developed chaos I11
ii k=J.
•
k=2
+
k=:~
+
k=4
+-
+,
:
,
•
+
:
+-
+-
-_
+= ,_
_
+,
;
Fig. 7. The first five steps in constructing the Fibonacci fractal. the r.h.s, of eq. (113) denotes the restriction to admissible words, i.e. words which do not have two or more neighbouring minuses. Thus with each admissible word w k a point of the Fibonacci fractal is associated. The fractal dimension Df of the Fibonacci fractal was shown [1] to be equal to the capacity of s.
Df = d(s) = l o g 2 y ,
(114)
with 3' = (1 + X/5)/2. Each point X(w k) of the Fibonacci fractal has a probability P+(wk), depending on the value of the control parameter a. A fractal support which has a non-homogeneous probability distribution is called a multifractal [10]. In the following we will investigate the properties of the Fibonacci multifractal associated with the bungalow-tent map for a-= lim~0(½ - e) (the critical value from below) and a = ½ (the critical value from above).
6.2. Box counting and the generalized dimensions D( q) In order to describe the properties of the Fibonacci multifractal introduced in the previous subsection, we study the partition function [11]
Xq(l) = ~ Pq ,
(115)
i
where q ranges from - ~ to + ~ . The number l indicates the size of the boxes constituting a uniform grid which is laid on the unit interval. Pi denotes the probability of words corresponding to the points X(w k) inside the ith box. For empty boxes Pq vanishes. Note that q acts as a magnifying glass: for large q mainly the regions with largest probability will contribute to the summation in (115) and for large negative q regions with smallest probability will contribute. The partition function Xq is found to follow a power law behaviour as l---~0 [12, 11]:
R. Kluiving et al. / Symbolic dynamics of fully developed chaos 111
431
(116)
Xq(l ) ~ l (q-1)o(q) "
In order to distinguish between different words of length k, one introduces a grid with 2 k boxes (intervals), iX(w,), X ( w ~) + (½)k),
(117)
with size lk = (½)k.
(118)
The probabilities Pi in eq. (115) can now be replaced by the probabilities Ps(wk): )(q(lk) = E t pq(wk) wk
(119)
,
where the prime denotes the restriction to the admissible words. For q = 0 this equation reduces to Xo(l,) = ~".' = F k + 2 ,
(120)
wk
cf. ref. [1]. Here F/ denotes the ith Fibonacci number F i = Fi_ ~ + F 1 = F 2 = 1. Combining eqs. (116) and (120) yields 1 In Fk+ 2 k In 2
D(0)
fi_ 2
with
(121)
From Binet's formula
F, =
(1 + V ~ ) k - (1 - V ~ ) k 2%'3
(122)
it then follows by taking k--->oo in eq. (121) D(0) = l o g F / = D f ,
(123)
i.e. D(0) is the fractal dimension of the support of the Fibonacci multifractal, i.e. of the Fibonacci "dust". Performing the limit q - + 1 in eqs. (116) and (119) yields the result D(1)--
~ ~
t
P,(w k) log2P,(w* ) ,
Wk
i.e. D(1) is equal to the Entropy of s, cf. eqs. (16) and (17).
(124)
R. Kluiving et al. / Symbolic dynamics of fully developed chaos HI
432
M o r e generally, the entire the multifractal.
D(q) spectrum is a complete characterization of
6.3. The D( q) spectra and the phase transition In fig. 8 the D(q) spectra for a = ½ (the critical value from above) and for a = lim~0( ½ - e ) (the critical value from below) are depicted. These spectra were evaluated using eqs. (116) and (119) with l = Ik = (½)k for k = 11. For a = ½the probabilities Ps(wk) were given by eq. (32) and for a = lim~+0( ½ - e) these were determined from eqs. (103) and (104). T h o u g h for k = 11 we cannot expect to have reached convergence (cf. table I), the resolution is high enough to see that for a = ½ the D(q) curve differs everywhere from the D(q) curve for a =lim~+0( ½ - e ) (except, of course, at q = 0 where D equals the fractal dimension of the support of the multifractal). Thus the phase transition shows up as a discontinuous change on all probability scales. That the difference in fig. 8 between these two D ( q ) curves is not an artifact of the numerics, can be shown by performing an analytical evaluation of D(q) in both cases at the limiting cases q = _+2. The limiting values D(_+~) can be determined directly from the formulas: we first consider the case a = ½, i.e. the probabilities (32). Since
%
. . . .
,
. . . .
,
. . . .
,
. . . .
,
. . . .
'
.
'
.
,
. . . .
,
. . . .
,
. . . .
,
. . . .
,
. . . .
,
. . . .
,
. . . .
i
.
i
.
,
.
,
.
,
.
,
.
Dq)
O
.
-30
.
.
.
'
.
.
.
.
'
.
.
.
.
.
.
.
.
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.
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.
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.
30
Fig. 8. The k = I1 approximation to the D(q) spectra of the Fibonacci multifracta| associated with the bungalow-tent map for a = ½ (solid line) and a = ]im~0(½ - e) (dotted line).
R. Kluiving et al. / Symbolic dynamics of fully developed chaos HI Z ' [es(wk)] q ~ ( l )k(q-1)D(q) ,
433 (125)
wk
only the smallest probabilities will give the dominant contribution for q---> - ~ . These contributions are of order ( ½)k corresponding to long intervals of pluses. Thus for q---> - ~ we find
(126)
( I )kq ~ ( 1 )kqD(q) or
In 3 D(-oo) = In 2
1.5849 . . . .
(127)
For q---> ~ only the largest probabilities will give the dominant contributions. These contributions are of order ( ~ )k/2 corresponding to alternating sequences, and thus 1 In 3 D ( + ~ ) - 2 In----2- 0.2924 . . . .
(128)
Next we turn to the case lim,~0(½ - e), i.e. the probabilities determined by eqs. (103) and (104). For q---> - ~ only the smallest probabilities contribute and again we find In 3 D ( - ~ ) - In 2 "
(129)
For q---> + ~ the dominant probabilities are the probabilities
Ps([(+, _)k,2])_
(130)
cf. eq. (105). Thus ~ ( ~ )kot+~),
(131)
from which follows that D ( + ~) = 0
(132)
on the critical value from below. Instead of the D ( q ) spectrum one may consider the f(ot) spectrum given by [lll
434
R. Kluiving et al. / Symbolic dynamics of fully developed chaos 111
(q - 1)D(q) = q a ( q ) - f ( o / ( q ) ) ,
(133)
d O/(q) : 5-2 [(q - 1)D(q)l • uq
(134)
with
The f ( a ) curve is a concave function on the interval
[o/rain, O/max]
with
O/min = D ( q ' - ° ° ) '
(135)
O/max = D ( - ~ ) .
(136)
For the Fibonacci multifractal associated with the bungalow-tent map we see that
In 3 In 2
O/max
for a = ½ and a = l i~o m (½ - e),
1 In 3 2~n2
fora=
0
for a = l i m (½ - e).
~,
O/min =
(137)
(138)
~$0
Acknowledgement This investigation is part of the research program of the Strichting voor Fundamenteel Onderzoek der Materie (FOM) which is financially supported by the Nederlandse Organisatie voor Wetenschappelijk Onderzoek (NWO).
Appendix A. Estimation of the function @(e, j) We first derive analytical expressions for the integrals l/2-e
A,~-
f
&[(+, ) q ( y ) d y ,
(A.1)
O{~+_)q(y)dy,
(A.2)
-1 xF
B~------ f l/2-e
R. Kluivinget al. / Symbolic dynamics of fully developedchaos 111 i=0,1,2
435 (A.3)
.....
up to order e. Let us start with unraveling the integral A i: l/2-e
l/2-e
qJ[(-.+)'-~,-l(f(Y)) dy ,
Ai= -1
(A.4)
-1
where again we have deleted in f the subscript 1 _ e indicating the value of the control p a r a m e t e r . Changing in eq. (A.4) to the new variable x = f ( y ) we arrive at 1
Ai=~-~2
f
1
1
qJ[¢_+),-l._l(X)dx--~3 f ~O[(_+)i-1 l(x) dx+U(e2)
1/2-e
if
(A.5)
1/2
1!
1
1
XF
1
with d 2 = 1 + 2e,
(A.7)
d 3 = -(1 +2e).
(A.8)
Changing again to the new variable y =
f(x) one finds
XF
Ai
l/2--e
- __ d2d 3 1 / 2f- e q*[(+,-)i-'l(Y'dY--dTl(~-~3
q~I(,.-)'- q(Y) dy
d2 -1
+ C(e:),
(A.9)
with d I =3-2e. Using d 3 =
(A.IO)
- d 2 we have, with the help of eqs. (A.1) and (A.2),
1
A i =---~Bil +~ d2
2
A i-i + ~ ( e 2) "
In an analogous way one derives
(A.11)
436
R. Kluiving et al. / Symbolic dynamics o f fully developed chaos 111
1 Bi = ~ Bi 1 + 0~(e2) • d2
(A.12)
Eqs. (A.11) and (A.12) can be solved, yielding A i=
Bi
d_~ [(d2) i ~
1
(~-e)+
l-(d2/dl)il] f--d---~
2 e +6(e2),
1
(A.14)
2--~-- e + G(e 2) 2 2
=
d
(A.13)
which can be rewritten as A i= D
+C(e2),
(A.16)
eE
(A.17)
e
B i=~
(A.15)
+ ~'(e 2) ,
+ eE
where D = (~ - e ) -
and 1
1
)
(A.18)
E = -~ ( l _ d 2 / d 1
go to a constant value if e--~0. From the above results we can establish estimates for Icil and q~(e, j) as we shall see next. From eqs. (74), (66) and (67) we have
l/2-e Icil ~
f
XF I~l(Y) -
'p,(y)lOt(+,-)q(Y)
l
de + le f 1/2
1
1~2(Y)lOt(+,-)q(Y) dy ~
t~l~)'--.'""
e
(A.20)
131 = Sup{I~,(Y) - q~(Y)l},
(A.21)
<< f l l A i + - fl2B, '
with
R. Kluiving et al. / Symbolic dynamics of fully developed chaos HI
437
(A.22)
/32 -- Sup(l~2(y)l}, i.e.
IciL ~ (/3~o + ½/32)
+ ~/3,e
+ 6(~2).
(A.23)
F r o m eqs. (76) and (A.23) it then follows for ~ ( e , j):
q~(e, j) ~<
/3'D+ /3z/2 ~ ( 1 ' ] i ( 2 - 2 e ] -3-~e i=, \ d ~ / \ 3 - 2 e / + 372e
i=,
d-~
\3-2e/
i-l
+(7(e2)"
(A.24)
P e r f o r m i n g the summations one arrives at 2-2e
- eg
J
+ ~(e2),
(A.25)
where H = /31D +/32/2 3 - 2e 3-2e 2-2e
z d2
1 3-2e 2 -2e
(A.26)
and
~1E K-
3-2e
3-2e 2-2e
1
(A.27) 3-2e
did2
2 - 2e
go to a constant value for e ~ 0.
Appendix B. Alternative derivation of (nja m) ~ 1/2e We consider the bungalow-tent m a p for a = ½ - e. Let us denote by I~ the interval [ ½ - e, ½]. If n consecutive iterates x i are located in I, then we speak of a laminar interval of length n [3]. T h e probability of finding one iterate in I~ is
R. Kluiving et al. / Symbolic dynamics of fully developed chaos HI
438
1/2
1/2
Pl =
f 1/2--e
l/2-e
= ~ + T (1) ,
(B.1)
with 1/2
T (~) =
(B.2)
q~z(x) d x ,
f 1/2
e
cf. eq. (4). T h e probability of finding two consecutive iterates in I~ is 1/2
1/2
e XF
__1
-1
6e l + 2 e
(f
1/2
ot_l(y)dy + f
1/2
e
i/li+l(Y)dy )
XF
1/2
f
+
q~z(X) [0[+, l(x) + 01 ,+l(X)] dx
1/2-e
1
1
+ T (2)
6 l+2e
(B.3)
with 1/2
T (2) =
f
~(x) [~,i÷ _t(x) + q't ,+j(x)l dx,
(B.4)
l/2-e
and w h e r e we have neglected terms of order e 2. For the definition of tkf+. I and qJl-,+l see eq. (25). T h e probability of finding j consecutive iterates in I, is d e n o t e d by pj. As we will see we do not n e e d explicit expressions for Pi if j/> 3. If x N = [x 1, x 2, . . . , xN] is a (chaotic) orbit with length N >> 1, then we have
Np~ = # iterates in I~,
(B.5)
N(Pl - P2) = # visits to I~ ,
(B.6)
N(Pi - Pi+l) = # visits to I, with length/> i .
(B.7)
R. Kluiving et al. / Symbolic dynamics of fully developed chaos I11
439
The probability of a laminar interval of length n is then given by t',..,(.) =
N ( p , - p , , + , ) - N ( p , + 1 - P,+2) = P, - 2p,+~ +P,,+2 N(p~ - Pz) P~ - P2
(B.8)
Thus
(,,1..~) = r L ' n(p. - 2 p . + , + p . + 2 ) =
p~
Pl - P2
PJ - P2
1
(B.9)
2e + 6(T (1) - T (2)) + ~(e 2) As a final step we show that T (~)- T (2)~ s~: 1/2
1/2
l/2-e
l/2-e
1/2
=
1/2
~o2(x) dx -
~
~(g~
I/2-e
(g~ (x)) d~
XF XF
i
f
+ 1 + 2e
,.viov
~°2(g3 (g3 (x)) dx
)
1/2-e 1/2
=
f
1/2
~o2(x) dx
lt2-e
irlv
f
1 + 2e
ilav
~o2(g3 (g3 (x))dx,
(B.10)
1/2-~
with g3V(x) the inverse of g3(x) = 1 - (1 + 2e)x.
(B.11)
cf. eq. (1). One easily checks that inv
inv
g3 (g3 (x)) = 2 e + ( 1 - 4e)x + (~(e2).
(B.12)
Changing to the variable y = 2e + ( 1 - 4e)x in the last integral of eq. (B.10) does not affect the integral boundaries and we find 1/2
1
T ( 1 ) - T (2)= (1
1
(1~ 2e)(1 --4e))
f
~°z(y) dy
1/2-e 1/2
- - ( - 2 e + t ? ( e z) f l12-r
q~E(yldy.
(K13)
440
R. Kluiving et al. / Symbolic dynamics of fully developed chaos II1
Since 1/2
f
~P2(Y) dy
~ e,
(B.14)
1/2-e
we have the estimate T ( 1 ) _ T ( 2 ) ~ 8 2,
(B.15)
from which it follows that 1 ~nlam) = ~
+ C(1).
(B.16)
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12]
R. Kluiving, H.W. Capel and R.A. Pasmanter, Physica A 183 (1992) 67, part 1. R. Kluiving, H.W. Capel and R.A. Pasmanter, Physica A 183 (1992) 96, part 2. R. Kluiving, H.W. Capel and R.A. Pasmanter, Physica A 164 (1990) 593. P. Sz6pfalusy, in: Proc. Conf. On Synergetics, Order and Chaos, Madrid 1987, Manuel G. Velarde, ed. (World Scientific, Singapore, 1988) pp. 685-697. A. Csordfis and P. Sz6pfalusy, Phys. Rev. A 38 (1988) 2582. P. Sz6pfalusy, Physica Scripta T25 (1989) 226. A. Csordfis and P. Sz6pfalusy, Phys. Rev. A 39 (1989) 4767. C.E. Shannon, Bell Syst. Techn. J. 27 (1948) 379. G. Gy6rgyi and P. Sz6pfalusy, Phys. Rev. A 31 (1985) 3477. For a review see: T. Tel, Z. Naturforsch. 43a (1988) 1154. T.C. Halsey, M.H. Jensen, L.P. Kadanoff, I. Procaccia and B.I. Shraiman, Phys. Rev. A 33 (1986) 1141. H.G.E. Hentschel and I. Procaccia, Physica D 8 (1983) 435.