On mean field fluctuations in globally coupled logistic-type maps

ELSEVIER Physica D 106 0997) 9-38

On mean field fluctuations in globally coupled logistic-type maps

Sergey V. Ershov, Alexei B. Potapov * Keldysh Institute for Applied Mathematics, Moscow 125047, Russian Federation

Received 16 April 1996; revised 9 December 1996; accepted 24 January 1997 Communicated by Y. Kuramoto

Abstract

mean field fluctuations in globally coupled map xn+l(i) = f (x, (i)) + ~ ( l / N ) ~ ;=1 x~ ( j ) We consider when the local

map f has a flat power-law top, e.g. f ( x ) = 1 - aLxl ~ with fl > 1. It is shown that in the thermodynamic limit N -+ oo the amplitude of mean field fluctuations is O(E 1/(#- 1)) for e << 1 which is in good agreement with numerical calculations,

We also demonstrate the relation between this problem and behaviour of averages as functions of the parameter a in lD maps. For the latter, we give both theoretical grounds and experimental evidence that the "mean deviation" of an average value behaves as a power of the deviation of the parameter, e.g. f I (x) (a + 3a) - (x) (a)l da ~ I~a I I/~

1. I n t roduc t i on

Nowadays one of the main directions in the chaotic dynamics is "spatiotemporal chaos", that is, irregular

behaviour in mult ibody distributed systems. The qualitative difference from low-dimensional chaos arises when

the dimension is not just large, which is a loose and indeterminable notion but when it becomes infinite. This means

that the number of "interacting bodies", or "elements", or subsystems N ~ oo. If there exists a convergence of

statistical characteristics as N diverges, one calls it thermodynamic limit. It is the convergence as N -* ~ that

enables one to speak about "infinite size" systems and thus brings sense to the distinction of spatiotemporal chaos

(now thought of as the limiting behaviour) from "low-dimensional" chaos.

There are two extreme classes of spatiotemporal models which differ drastically in their features and behaviour.

The first type is constituted by systems with local (short-range) interactions, of which the simplest example is

coupled map lattice

xi(n + 1) ----- f ( x i ( n ) ) + E ( f ( x i - l ( n ) ) -- 2 f ( x i ( n ) ) -.F f ( x i + l ( n ) ) ) (1.1)

(here i is a lattice point, n is a discrete time and f is called local map), introduced by Kaneko [ 1 ] and then studied

extensively (see [2] and references therein for overview). Its main feature is quick decay of spatial correlations

* Corresponding author. E-mail: [email protected].

0167-2789/97/$17.00 Copyright © 1997 Elsevier Science B.V. All rights reserved Pit S0167-2789(97)00026-2

10 S.V Ersho~; A.B. Potapov /Physica D 106 (1997) 9-38

which leads to the convergence of statistical characteristics for N --~ ~ [2]. This also enables one to introduce an "elementary brick" - the conditional probability, on the basis of which it is possible to describe the behaviour with infinite number of degrees of freedom and the convergence to the thermodynamic limit when the phase space itself changes; and even derive a self-consistent master equation for the limiting characteristics [3].

To the best of our knowledge, (1.1) (and its modifications) are the only models of spatiotemporal chaos for which the existence of the invariant measure, its properties and behaviour at least for ~ --~ 0 were studied rigorously [4-6].

The second class is constituted by systems with global interactions covering all the domain and decaying very slowly (or not at all). Here the basic model is globally coupled map (GCM)

1 N xi(n + 1) = f(xi(n)) + ~ ~ x j ( n ) ,

s=l

¢1.2)

introduced by Kaneko [7,8] apparently as a discrete-time version of the popular in statistical physics model of

coupled oscillators

1 N dXidt = f (x i ) + E--N j~=l xj

(see [9,10]). This system was studied rather thoroughly and extensively, e.g. to solve the problem of "synchronization", i.e. of cooperative global behaviour, and some fruitful asymptotic approaches were proposed [9,10]. The GCM retains the phenomenon of cooperative behaviour; it was found by Kaneko that for smooth local map and

small E the mean field hn ------ ( l / N ) Y~=l xj(n) fluctuates, large as N is. The reason for such a behaviour is that there is no stable equilibrium probability measure; and the simultaneous distributions recorded never converge but reveal an irregular oscillation in contrast with what is seen in systems with local coupling [7]. In case of a piecewise linear local map f the fluctuations are much weaker than for a GCM with a smooth local map, but the absence of equilibrium probability measure can be proved nearly rigorously [11]. It is the absence of relaxation towards statistical equilibrium that is one of the principal differences from the case of local couplings such as (I. 1) where under the action of the dynamics (at least for expanding local maps and small couplings) the probability measure

quickly converges to the equilibrium one [4-6]. That such a behaviour is the result of large coupling range was convincingly justified by the work of Chat6

and Manneville [12] on coupled maps on high-dimensional lattices. They have found that when this dimension becomes large enough, and so the effective number of neighbours coupled gets very large too, there seems to be no

stationary probability distributions because the statistical characteristics they recorded revealed steady and complex

oscillations. However, quantitative characteristics of this effect can differ drastically. Say, for ( 1.2) with piecewise local map f

the fluctuations of the mean field (and other statistical quantities) are only tiny, with characteristic amplitude of the order o f e - C / ' 2 , which was proved with the help of the self-consistent Frobenius-Perron operator (SCFPO) approach

in 111]. At the same time, the fluctuations are much greater for smooth local maps, such as, e.g., f ( x ) = 1 - alxl f with fl > 1. According to Kaneko, for fl ----- 2 they exhibit fluctuations as large as O(~), and this is a huge contrast with piecewise linear local dynamics. As to the question concerning the dependence upon fl, it seems to be untouched even experimentally. So it would be natural to further the approach of SCFPO to globally coupled "power-law top" maps, and find if they are so particular that this approach fails - or, if it does not, what is the main difference from

piecewise linear maps. In the lbllowing we shall show that

- the SCFPO approach applies to this case;

S. V. Ershov, A.B. Potapov/Physica D 106 (1997) 9-38 11

- the fluctuations, as well as in the case of piecewise linear local maps, are due to singularities of 1D distributions

pn(x); - for logistic-type maps these singularities are of power-law type, much stronger than discontinuities (steps) for

piecewise linear maps, and thus mean field fluctuations are much stronger too: O(~ I/l~ t i) in contrast with O(e -c/~2) for the former case.

- In view of this estimate, the amplitude of fluctuations quickly decreases as fl --~ 1.

These statements are in good agreement with numerical calculations, see Figs. 1 and 2.

This problem has deep connection with a problem of how the average values depend upon the parameter in 1D

maps. It is known that for piecewise linear smooth maps this dependence is continuous (though not differentiable,

see [13]), while for smooth maps it is rather wild; and the extremely ragged "parameter curve" had become really

famous. As far as we know, the first who guessed that this drastic difference has some relation to large fluctuations

was Kaneko, see e.g. [8]. Encouraged by his belief, we have studied this connection and found that the "mean

deviation" of an average value behaves as a power of the deviation of the parameter:

f l(x)(a +~a) - (x)(a)lda ~ 16al I/t~,

which is in good agreement with numerical calculations, see Figs. 3 and 4. It appears that the deviation of averages

contains terms of the same nature as those entering the equations for mean field fluctuations. In conclusion we shall

discuss how this connection may appear helpful in numerical detection of mean field fluctuations in multidimensional

systems.

log106h

I d I I i

'2 3 ~, 5 6

log]0(1/e)

Fig. 1. The amplitude (Ih - (h)l) of mean field fluctuations for f ( x ) = 1 1.95rxl/~, fl = 3. The lattice size N = 200 000. Straight line

of the slope 1/(fl - 1 ) is a guide for eyes.

-0.5 -

- - 1 , 0 -

1.5-

-'2.0 ! I

logl0 3h

O O

©

Iogl0(l/,)

Fig. 2. The same as Fig. 1 but for fl = 4

12 S. V. Ershov, A.B. Potapov/Physica D 106 (1997) 9-38

0 . 0 1 0 -

0 . 0 0 8

0.006

0.004

0.002

0.000 0.00

00°~ o.b2 0.'o4 o.'o6

Fig. 3. The mean (with respect to a) deviation of (x) (as defined in (7.6)) vs. the deviation of the parameter a for the logistic map (fl = 2). The pivot value of a and the averaging interval corresponding to different markers are: a0 = 1.8247, Aa = 0.247 x 10-1 ("o"); a0 = 1.80453, Aa = 0.53 x 10 -3 ("o"). Straight line is a guide for eyes. The individual deviations (x)(a + ~a) - (x)(a), had they been plotted, would have covered almost all the panels with an irregular "fog".

0.010

0.0c)8

0.006

0.004

0.002

0.000 0.00

. / (I (~,) (,, + ~,) - (x)(,)1)

! o ~ " . . . .

oOO

O.bl 0.b2 0.'O:I O.b4

Fig.4. The same as Fig. 3but for the logistic type map f ( x ) = 1-a[xl3. Hereao = 1.870216, Aa = 0.216× 10 -3 ('%");a 0 = 1.8700118, Aa = 0.118 x 10 -4 ("e"); a0 = 1.8700096, Aa = 0.460 × 10 -5 ("A"); a 0 = 1.871004, Aa = 0.402 x 10 -5 ("+").

2. Self-consistent master equation

This approach originates to Kaneko [7] and (at least formally) applies to infinite size system (N --+ oo) like (1.2) with any 1D local map f . So up to Eq. (2.3) this section coincides with Sections 2 and 3 of [11], to which the reader can refer for more detail that we omit here.

For infinite lattice the mean field is defined as

1 U hn =-- l im ~"~ xn(i)

N---~OO 2 N l "-"u

and so it can be der ived f rom the p robab i l i ty dis t r ibut ion 1 Pn (x (i)) induced by the random field {Xn ( i ) } ~ _ ~ as

1 In fact, even the distance between elements never enters the dynamic equation, so the distributions are the same for any ]i - J l unless i and j coincide.

S. V. Ershov, A.B. Potapov/Physica D 106 (1997) 9-38 13

= f xp.(x, hn dx.

Notice that the value of the mean field is independent of spatial correlations. 2 This is the key feature, and

enables one to derive a reduced master equation, using only ID projection of the full distribution, pn (x). Indeed,

the dynamics is

xn+l (i) = f (xn( i ) ) + Ehn, (2.1)

that is, at each iteration the value of hn is the same for all xn (i), i.e. it is a time-dependent constant (within the

ensemble). Therefore the evolution of Pn (x) is governed by the FPO (Frobenius-Perron operator, see [14-16] for

overview) of the effective map Fn (x) -- f (x) + Ehn:

p,,+I(X)----(E.F, P n ) ( X ) - - f S ( x - F n ( x ' ) ) p . ( x ' ) d x ' = f S ( x - f ( x ' ) - ~ h n ) p . ( x ' ) d x ' .

This reduced master equation, based on 1D distributions, was originally introduced by Kaneko [7] and called

SCFPO (self-consistent Frobenius-Perron operator).

One easily finds that the effective map (2.1) is a composition of the local mapping and the shift by ehn. Hence

the corresponding FPO £F, is also a composition £F, = AEh.£f, where £ f is the FPO of the local map f

(2,2) f

(•f pn)(X) J 3(x - f ( x ' ) )pn (x ' )dx '

and A~h is a shift:

(a~hp)(x) = p(x -- Eh)

and so, as well as in [ 11 ], we can write the SCFPO as

Pn+l (X) = (AEh. Ef pn)(X) = (£.f +Eh. Pn)(X), hn = f xp.(x) dx . (2.3)

However, the value of hn itself cannot characterize the fluctuations. Indeed, there are no fluctuations at all if all

hn are the same, large as they are. Instead of hn we should therefore use something like hn - (h) or hn' - hn". The latter opportunity enables more straightforward application of the SCFPO formalism. Indeed, let us take two

arbitrary distributions, Pn~ (x) and Pn2 (x) with n I and n2 so large that all transients have died out, and consider their

subsequent evolution, i.e. pn~ +n (x) and Pn2+n (X). It is convenient to write Pn (x) instead of Pnl +n (x), and fin (x) instead of Pn2+n (x). In these notations, we deal with two trajectories, {Pn} and {fin }, starting from different initial conditions P0 and/30, respectively. Also, we denote the corresponding values of the mean field as hn -- f Pn (x) dx,

hn --- f fin (x) dx. Now the evolution of the two distributions obeys

Pn+l(X) = (A~hnff~f pn)(X), /~n+l (X) = (A~n~f pn)(X).

Using our notation Fn(x) =-- f ( x ) + Ehn and the composition property of the operator A:

Ah+h, = AhAh,

the above equations can be written as

Pn+l(X) = (I~.F. Pn)(X), / gn+ l (X) ----- (A,8~n£F. Pn)(X),

2 Even if they existed in initial configuration, they apparently decay as n ~ cx~. because to this point the analysis of [ 11 ] seems to be valid for any local maps with continuous invariant distribution.


where 8hn ~ [~n -- hn; it is of the same order of magnitude as hn - (h). Subtracting the first equality from the

second one, we see that the difference 8pn = fin -- Pn obeys

~Pn+l = (AEshn -- 1)ff-F, pn + ~.Fn~Pn.

It leads by iteration to

n-- I

~Pn = ~ E.F~_] "'" £ F n l(A~Shn_l_l -- 1)l~Fn_l_ I P n - l - I + £Fn_l "'" I?'Fo(~Po (2.4) l=O

and, since (Ah) - j = A_h,

n - I

Spn = Z £Fn-] ..... Fn_l (1 - - A_Eshn_l_j )A~hn_l_] ff.,F_l_ ) P n - l - I + £Fn_ I ..... Fot~Po / = 0

n - I

= Z ff-'Fn-I ..... Fn_l(Pn-I -- A-~hn_l_ t Pn- l ) ~- ff~Fn_ I ..... Fo3PO, 1=0

where (Fn, o Fn,,)(x) -- Fn'(Fn"(x)). Multiplying by x and integrating, one obtains

n - 1

8 h n - - - - l ~ o f ( F n - l O " ' o F n - l ) ( X ) [ f i n - l ( x ) - - P n - l ( X + e 6 h n - l - 1 ) l d x + f x ' ( ~ F , _ , = ..... FoSPO)(X)dX,

because (2.2) implies that

dx dx. (2.5)

One can expect that if the mean effective mapping F is mixing, then at least for e small enough its slight time-

dependent perturbation, Fn, is mixing too. Since now invariant measure is absent, we mean by "mixing" that under

the action of the composition map Fn-I o . . . o FO the difference of two distributions 8po vanishes:

n ----~ ~K) £F,_~ ..... FoSPO O. (2.6)

If this is true at least in the sense of a weak convergence,

n - I / ,

n--,__~ Z ] (Fn-1 o . . . o Fn-z) (x)[~n-z(x) - ~n-z(x + EShn-t-I)] dx. (2.7) 8hn J 1=0

Unfortunately, the ergodic properties of logistic-type maps (to say nothing about time-dependent ones!) are not

understood at the necessary extent as yet to prove (or reject) this assumption rigorously. Nevertheless, some results,

especially those of Lasota et al. [ 17] who proved the convergence for Markov operators which expand supports, can

be considered as an evidence in favour of this assumption, because the mechanism of expansion and overlapping of supports seems to work for maps which are "expanding on average", i.e. have positive Lyapunov exponents. Some additional discussion as to why we can expect the convergence can be found in Section 3 of [ 11 ].

3. The role of singularities

In Section 5 we shall show that for a logistic-type local map (i.e. if f has the only extremum x = Xc and near it admits power-law approximation f ( x ) = f ( X c ) - a Ix IS), the distribution Pn (x) is a sum of singularities and the "regular", or rather "less singular" part, p,] (x):

p~ (x) = 2.. , sk,~ k >_0

where

S.V. Ershov, A.B. Po tapov /Phys i ca D 106 (1997) 9 - 3 8

O(dk,n • [x -- Xk,n])

IX -- Xk, n lot + p~(x),

15

(3.1)

-- 1 - 1/fl,

sk,n is the "weight" of the singularity and dk,n = ±1 its "direction", showing whether its "tail" lies to the left (d = - 1 ) or to the right (d = 1) of the singularity point Xk,n. All these values refer to the kth singularity at time n. As usual, O(x) is the step function (1 for x > 0 and 0 otherwise).

In Section 5 it is also shown that if the initial distribution is of the form (3.1), then the same will be true for all

its images Pn; the same naturally applies to fin. Now let us see whether this form of distribution is important and what comes of it. Substituting the corresponding

expansion in (2.7) and denoting by Xk.n, gk,n and Jk,~ the corresponding characteristics of singularities of/3~ (x),

we have

cx~ n - I

k=O 1=0

x r O ( ~ - t [ x - - - - 2 k , n - l ] ) _ O(dk,_2n-l[X---~k,n-__~t+_e~hn-____.__l-I])] dx + o((e3h)l- '~). [. IX -- X k , n - I [or IX -- X k , n - I "~ 6 ~ h n - I - 1 1 °t

The term o((e3h) I- 'r) is due to the contribution of the "regular part", and 3h denotes the characteristic amplitude of fluctuations.

According to (5.5) gk.,,-t = Sk + O(x - k /~E3h) , while (5.6) gives Sk = O(X-k/~), where )~ is the Lyapunov exponent of "mean" local map F ( x ) -- f (x) + • (h). Therefore

sk.n-I = Sk • (1 q-O(~3h)),

so we can approximate g,.n-t by Sk. Since the integral is obviously bounded by O((~3h) l -u) , this replacement perturbs ~hn by _< O ( ( ~ h ) 2 - a ) , and so

oo n - 1 / ,

= ~ ~ s, I(F~_~ o. . . o G-t)(~) ~hn k = 0 I = 0

, ]

[ o (d~ ,~_~ - ~k,~-d) o(d~,~_&c - ~k,.-t + E~h._~_l])] ]_ ]x - 2k,~-tl ~ -- Ix -- xk,n-t + E3hn- l - j 14 ~ dx + o((~3h)1-~). (3.2) x

Therefore, 3ha is a sum

oo n- - I

3h,, = E Z 3h(nk'l} q- °((E3h)l-c~) (3.3) k=O I = 0

whose terms are the contributions due to the kth singularity as it was I iterations before:

S k f ( F . _ , o . . . o Fn- l ) (X) r~ h (n k , I )

x [ O(ak'-~n-t--[xz 2k'---2-t]) -- O(dk,~-Z[x-- 2.__k,±- d + e3h_______n-d-~l) ] dx. k I x - Xk,n-ll '~ [x - 2k,n-I + e3hn- I -1 ['~ J


Since ]Sk • (Fn-l o . . . o fn -D(x) l < O(1), one immediately finds that

]Sh~k't)l < O((~:Sh) I -u ) ~ O((~6h)U#), (3.4)

so it is not impossible that there are terms in the sum (3.2) as large as O((e~h) 1/~) (here 3h is the characteristic

amplitude of fluctuations), in contrast with the case of piecewise linear maps when they could not exceed O(E3h). The potentially large terms are only those with "proper" l.

Indeed, using obvious inequality

I(Fn-1 o . . . o Fn-l)(X + tz) - (F,-1 o . . . o Fn-t)(x)l < 1#1" (max I f ' l ) ~

one easily obtains the estimate

f (Fn-I o . . . o Fn-l)(x)[pn-l(X) -- Pn-l(x + E~hn-I-1)] dx

: / [ ( F n - 1 o . . . o Fn_l)(X ) -- (Fn-1 o . . . o Fn_l)(X -- EShn- l -1)]" Pn--l(X) dx

< e l S h n - l - 1 1 ( m a x , f ' J ) ' f Pn-l(X) dx : ElShn-t-1 ](max I f ' l ) / ,

meaning that only the terms with l > -or log ESh/log max I f ' [ can be as large as O((eSh)U~). Then, "far" terms, i.e. those with I >> - l o g ESh/log ~. where Z is the Lyapunov exponent of the local map, are

also small, since using (2.5) one can write

f (g,-1 o . . . o F n - l ) ( x ) [ f i n - l ( X ) - f in- l (X + E S h n - l - 1 ) ] d x = f x ~'Fn-i ... . . Fn_lAfin-I dx ,

where A,fin-Z (x) - - f in-I (x) - Pn- I (x q- ESh n_l_ 1 ). Since we assumed mixing (2.6) of the time-dependent map Fn,

1--~ ¢x~ ,., £Fn-I .... . Fn_lAfin-I > IJ.

Thus we conclude that in (3.3) some terms with "proper" l (their number depends upon the mixing properties of the local map which are not quite understood as yet) can be as large as O((~3h) l/#) where Sh is some characteristic amplitude of mean field fluctuations. I f this does occur, they dominate in the sum, so 8h --~ (~Sh)l/t~ and

Sh ~ E l/(fl-1),

which agrees with numerical results of Kaneko [7,8] and ours (see Figs. 1 and 2).

4. The amplitude of fluctuations

Therefore, to calculate the amplitude of fluctuations we should estimate the term 3h(n k't). It is time-dependent, because the characteristics of singularities differ irregularly with time. Though they deviate from their unperturbed values only slightly, see (5.5), the value of Sh(n k't) varies substantially (of the order of its mean value). Indeed, rewriting it in the form

= Sk . f [ ( F n - 1 o . - - o Fn-l ) (Xk ,n- I + ~) -- (Fn- I o . . . o Fn_l)(Xk,n_ t -- f ~ h n - I - I + ~) ] 8h(k, 1)

× d~, (4.1)

s. v. Ershov, A.B. Potapov / Physica D 106 (1997) 9-38 17

one can see that this integral is as sensitive to Yk,n-I as ( F n - i o . . . o Fn- l ) (Xk ,n- l ) . For large l the latter (due to the positive Lyapunov exponent) changes drastically in response to slight variations of Yk.n-Z. Besides, the

"displacement" eShn-t-1 also fluctuates. o,(k t) Therefore, the terms ann ' can be regarded as random (with respect to n) numbers; and so is their sum Shn, As

a result, any of its particular value is of only negligible importance; instead, we should use something like mean

square deviation, or the most probable value, or something like it. Unfortunately, this way one encounters strong

difficulties because these terms are all mutually correlated, the decay of correlations being determined by the mixing

properties of the local map. These are not quite understood yet; we cannot even prove that the decay is exponential to say nothing about its rate. So it appears nearly impossible to estimate, e.g., the mean square deviation (to say

nothing about the most probable value) neatly. Instead, we would have had to use some doubtful and inaccurate assumptions, and this would have produced the result of a very little rigour.

Therefore, we just assume that the "typical" value of the sum 8hn, which we denote 8h, is of the order of the typical magnitude 8h {k,t) of its largest term (recall that those with either too small or too large I could not make any

substantial contribution):

8h ~ max 8h (k't). (4.2) k,l

Hence we are going to study 8h(n k't).

It will be useful to introduce the "mean" local map F ( x ) =-- f ( x ) 4- ~ (h), where (h) is the average value of hn.

Obviously, Fn = F + ~(hn - (h)), and the deviation hn - (h) is of the same order of magnitude as 8h. Now let us transform the composition Fn-1 o . . . o Fn- t which enters the definition of 8h(,, k'l)"

8h(n k'l) -- Sk f (F. , t o l;n_l)(X )

x [ 0 ( dk ,._.~n-J_[x __-- .Xk ,.___.nn - / ] ) _ 0 ( d,{:., n - t [ x _~ -~__.k ,~-_t 4- ~ 8 h..___.~n - / - 1 ]) ] d x L Ix - Xk,n-ll ~ Ix -- x*,n-I 4- ESh, , - l_ I I a J

(4.3)

in the following way. First, we replace the "leftmost" l - m mappings by the time-independent F, only the "tail"

remaining time-dependent. Denoting the Lyapunov exponent of the map F ( x ) as ~., the error of this substitution

can be estimated as

( F n _ 1 o . . . o F n _ l ) ( X ) ~ ( F n _ 1 o . . . o F n _ l + m ) ( ( F n _ l + m _ I o . . . o F n _ l ) ( X ) )

and therefore

6h(~ k't) = Sk

= F l - m ( ( F n _ l + m _ 1 o . . . o Fn- l ) (X) ) 4- O(X l-m " ~ h )

f F l - m ( ( F n _ l + m _ l o . . . o Fn_l)(X))

x [ O(dk'--n-l[----~XZ~---k'--n-l]) -- O(dk'--n-t[xz£---k'n----L+---eShn-t-l])] d x + O()~l-rn(f~h) 2-°t) Ix -- ffk,n_ll et IX -- ffk,n-I 4- f ~ h n - l - 1 let .j

= Sk f F l - m ( ( f n - l + m - I o . . . o Fn_l)(fffk,n_ l 4- ~))

X [O(dk ,n - l~ ) O(dk,n-I[~__ 4- " S h n - l - l ] ) ] d~ 4- O()Ll-m( , t~h)2-e) . L ~ V I~ +~h.-~-~l ~ J

(4.4)


In this integral the principal contribution (of the order of O((ESh) l - a ) ) can only come from the O(ESh) neigh-

bourhood of the singularity point Xk,n-t; the outside parts (far from the singularity) make only a small contribution.

In this neigbourhood, i.e. for [~1 < O(eSh), we can linearize (Fn-~+m-~ o " " o Fn- t ) ( x ) :

(Fn_l+m_ I o . . . o Fn_l)(Xk,n_ l At- ~)

(Fn_l+m_ 1 o . . . o Fn_l)(.~k,n_l) -a t- ( fn_ l+m_ 1 o . . . o Fn_l)" (Xk,n_l) .

~" (Fn- l+m-I o . . . o Fn- l ) (Xk ,n- l ) + (Fm) ' (Xk,n- l ) , (4.5)

for ~ ~ ESh it is valid while ~ m . eSh << 1. Denoting

Yk,n,m =- (Fn+m-~ o . . . o Fn)(Yk,,,), 8On =-- (Fm)'(.rk,n) " 8hn-~

and substituting the approximation (4.5) in (4.4) we after some simple algebra arrive at

Sh(nk,l) .~ Sk re_,. [l~(Clk,n-lSign(Fm)'(.~k,n-l) " [y -- Yk,n-l,m]) [(fm)t(Yk,n_l)[1-a (Y) × - ( f ~ ~ d

_ O(dk,n-I sign(Fm)'(YCk'n-l)lY -- Yk,n-l,m ""}'- [yeSrln-tl Yk,n-l,m + ESrln-l]) ] dy + 0(~. l-m (e6h)Z-a). (4.6)

Introducing the notation

Z>.(Xs, U,d)=- F"(x)\ ~---Xsl" ~ - - X s + # l '~ 1

and recalling that 1 - ot - 1/13 one can transform (4.6) to

Sh(nk,l) ~ Sk 7)l-m(Yk n-l,m, EfOn--l, dk n-I sign(Fmf (xk,n-l)) + O()~l-m(ESh)Z-ot) . [(Fm),(Yk,n_l)[l /~ "

Now let m = 11. S(nce the linearization (4.5) is valid for ~m . e6h << 1, l should satisfy

x~ << 0(1) E-~ ' (4.8)

and if this holds,

8h(k,l ) ~ Sk ~ DI/2(Yk n- t t/2, 66l'ln--l, dk,n-I s ign(Ft /2) ' (Xk,n- l ) ) + o((e6h)1-,~). ( Ft /2) ' (yc,~,n_l) l/tJ

Then I(Ft/2)%fk,n_t)[ "" )J/2 and Sk ~ ) - k / ~ (see (5.6)), so

8h(nk,O ,.,. )-(k+l/2)/~ . Dl/2(Yk,n-l, l /2, ESrln-l, dk,n-I s ign(Fl /2) ' (Xk,n- t ) ) + o((e6h) l -~) . (4.9)

Our consideration above concerning the dependence of the integral in (4.1) upon the location of singularity also applies to Dn (Xs, # , d): for large n it depends upon Xs rather irregularly. Therefore, 79~ can take on large values for some "favourable" values Of Xs and nearly vanish for others. In Section 6 we calculate the mean square value (with respect to Xs, for /z fixed!)

l Xs+fAx s 1 DnZ(Xs, # , d) dxs (Dn2( ", # , d)) --= 2Ax----~

Xs-Axs

S. V Ershov, A.B. Potapov/Physica D 106 (1997) 9-38 19

and show that if Axs >__ C4/z, then there exists such n . , that this mean square value satisfies

f Xs+f Axs ~ ct)t 1 _ ~ D2, (Xs , /x ,d)dxs > O(1) . I/_t] l-c~ Axs 2Ax~ - '

Xs- Axs

where y > 0 is the (unknown) function from Conjecture 2. This n , can be asymptotically estimated as

n . = - l o g [#l / log ;~ + O(1). Moreover, the typical value of Dn. over the interval Axs is of the same order of magnitude:

IDn.(Xs, U , d ) [ ~ [ U [ j - a ~ • (4.10)

Indeed, were it not so (i.e. had D,,, achieved this value only at exceptional points Xs), it should have taken on much

higher values at other Xs to have the average value of that order, which is impossible since its upper bound is of

nearly the same order of magnitude, see (6.20).

Therefore, if both # and Xs fluctuate, and unless they are so strongly correlated that for each # the location

of singularity Xs takes the "unfavourable" value for which Dn, is "exceptionally" small, then the typical value of Dn.(x~, # , d) is O(uJ- '~) .

In other words, the estimate (4.10) holds if # and Xs are independent to some extent; and Xs runs over the interval that is wide enough, as determined by the condition (6.22): Axs > C4/z (so that it contains not a few "favourable"

points).

This is the case with the arguments of Dt/2 in (4.9). Indeed, now the condition Axs > C4/z becomes

Ayk.n-l.l/2 >_ C4~6rl. (4.11)

In Section 5, where the singularities and their evolution are studied, it is shown that Xk, n (and thus ffk,n tOO) fluctuates

with the amplitude of O(Lk~3h), see (5.7). Therefore,

Yk,n l,l/2 ~ (Fn-l/2-1 o... o Fn-l)(Xk,n-l) ~ Fl/Z()fk,n-I)

fluctuates too, the amplitude of deviations being Ayk,t/2 ~ Xt/2)~k~3h. Further, the typical value of the "displacement", ~30,,, is now E30 ~Xt/2~3h. Thus, (4.11) is satisfied already for k = O(1).

Now let us check whether the displacement 3r/n-t and the position of singularity Yk,n-l,l/2 are strongly correlated.

The value of Yk,~-t is determined by hn-t/2-1 . . . . . h~-i through Fn-t/2-1 o . •. o Fn-t and by hn- I - I . . . . . h n - l - l - k

through ~k,~-t, the greatest being the influence of the "oldest"/~n_t_ l-k (see (5.5) and the concluding paragraph of

Section 5.5)). Meanwhile 3rln_ I is determined mainly by h n - t - l - hn - l - I , and therefore is almost independent of

/~n-t-L-k. Thus, one can expect that for k large enough, ~3rln_ l and Yk,n-t are not strongly correlated but fluctuate more or less independently.

Therefore, there exists such l. that the typical value of Dr.~2(...) as n changes is (~30) 1-'~. So,

DI./2(Yk,n-1,,I./2, ~3~ln-l,, dk,n-l, sign(Fl*/2)'(Xk,n-l.)) ~ (e3r/) l - a : 0(()~.1/263h)1/l~) •

Certainly, this can only be true if (4.8) holds for this l., and this is the case. Indeed, by (6.2) the "favourable" value of ½l can be estimated as ½1. = - log(~6r / ) / log ~. + O(1). Recalling that 60 ~ ~J/Z3h we have

log(e3h) / , - - - + o ( 1 ) ,

log X

or X t* = O(1)(1/~3h), so (4.8) is satisfied.

20 S.V. Ershov, A.B. Potapov/Physica D 106 (1997) 938

Returning to (4.9), we see that since k = O(1), the typical value of 8hn (k'l) for "favourable" l is

8h(k't) = O((ESh) l/t~). (4.12)

According to (3.4), 18h~nk'l) l < O((eSh) I/E) for any l, k and n, and thus, with the help of our assumption (4.2),

we obtain the estimate for 8h:

8h -.. ( ~ h ) l/~

from what it immediately follows that

8h ~ E U(~-I). (4.13)

5. T h e s i n g u l a r i t i e s o f d i s t r i b u t i o n s

The integral in the Frobenius-Perron operator (2.2) is actually the sum

Pn(Y) (ff-.fpn)(X) = Y ~

yEf_l(x) I f ' (Y ) l (5.1)

taken over all points for which f ( y ) = x. Now it is obvious that the transformed distribution ~-fPn is singular at the point x if either this is the image of the critical point Xc where fP(Xc) = 0, or if Pn was singular at one of the preimages ofx. This means that, to put it loosely, there is the "mother" singularity at f (Xc) , directly determined by the behaviour of f ' near Xc; and the "daughter" singularities, located at the trajectory of this point. They are just the images of the "mother" singularity, and their properties can be derived linearising f near the trajectory of the critical point.

So in case of one isolated map xn+l = f ( x n ) one can calculate the locations, amplitudes and directions of singularities. The case of GCM can be treated quite similarly, just replacing f by the effective time-dependent mapping

Xn+l = f (Xn) + ehn =-- F ( x ) + e(hn - (h)).

So for E << 1 the coordinates and amplitudes of at least the first singularities can be obtained using usual perturbative approaches and linearization of the mapping around the trajectory of the critical point.

5.1. Action o f the local map

Let f ( x ) be a unimodal map with unique critical point Xc, the maximum, near which it behaves like f ( X c +~) =

f (Xc) - al~r ~ with fl > 1. Consider its action on a distribution of the form (3.1)

O(dk,n • Ix -- Xk,n]) Pn (X) = Z Sk,n .~_ pr (X),

Ix Xk,n I" k>_>O

assuming that xk,n ~ Xc. For this map, the Frobenius-Perron operator (5.1) becomes

p n ( f + l ( x ) ) pn (f_-I (x)) - - + , (5 .2 )

(~fpn)(X) [ f , ( f+l (x ) ) l i f t ( f _ l ( x ) ) l

where f + l ( x ) > Xc and f - - l ( x ) < Xc are the two possible preimages (there are two of them for any x < f ( X c ) and none otherwise).

S. V. Ershov, A.B. Potapov / Physica D 106 (1997) 9-38 21

Substituting for pn(x) the distribution of the form (3.1) we see that 12fp n can have singularities if either

f , ( f :~ l (x)) = O, that is, for x = f ( X c ) (the "mother" singularity), or if f :~l(x) hits one of the points where

p~ is singular (the "daughter" singularities), that is, if f21 (x) = xk,n, or, which is the same, if x = f(xk,n). So

(£fpn)(X) has singularities at the points f ( X c ) and f(Xk.n), and nowhere else; these are again x- '~-type power-law singularities.

Indeed, for x = f (Xc) - ~ (5.2) becomes

1

(F-.fpn)(f(Xc) - ~) = flal/#~ d[pn(Xc + (~/a)l/l~) + pn(Xc - (~/a)l/l~)]

0

forO < ~ << 1,

for~ < 0 ,

which can be easily derived using approximation of the top of the map f ( x ) = f (Xc) - a Ix - Xc I t~. Since, as we

have assumed, xk,n ~ Xc, the distribution Pn is continuous at the point x -- Xc, and so the "mother" singularity is

( g f p n ) ( f (Xc) - ~ ) = - - 2p , (Xc) 0(~)

- - + less singular part; (5.3) t6al/# I~1 ~

here by the "less singular part" we mean any function growing slower than ~-~ when ~ ~ O. Its particular form

depends upon the smoothness of Pn (x) at the point x = Xc.

Similarly, near the point f(xk.n), i.e. for x = f(xk,~) + ~ with I~1 << 1 (5.2) implies

( £ f p n ) ( f (Xk,n) -t- ~ ) = Pn ( f+ l ( f (Xk.n)) -}- f,(f+l'(-f (xk,n))) )

I f ' ( f + 1 ( f (xk,n) ) )l

pn (f_-l(f(Xk,n))-b ~ ) • f,(f-I (f(Xk.n))) +

I f ' ( f / - 1 (f(xk.n)))l

Since we assumed that xk,~ ¢ Xc, the denominators do not vanish a n d f + l ( f (Xk,n)) ~ f 2 L ( f (xk.~)), thus either f+ l (f(Xk,n)) : Xk,n or f - - I (f(Xk,n)) = Xk,n, that is, only one of the terms is singular. I fxk, , > Xc, this is the first

one and so f+ l (f(Xk,n) = xk,~, otherwise this is the second one and so f_-I (f(Xk,n) = xk,n. With Pn of the form

(3.1), both cases lead to

(~-.f p n ) ( f (xk.n) "q- ~) = Sk,n O([dk,n sign f '(Xk,n)l~)

If'(Xk.n)l 1-~ [~ I ~ + less singular part,

which together with (5.3) gives us all the singularities of Efp, , . It is therefore

0 (dk,. Ix - ~k,. ]) (Ef pn)(X) = E Sk'n

k>O Ix - ;ck.~ I ~ + less singular part,

where

xo'n ---- f (Xc) , Xk+l,n = f(Xk,n), k = O, 1,2 . . . .

2 $ O , n - - pn(Xc), gk+l,n = Sk,nlf'(Xk,n)[ -1/13, k = O, 1,2 . . . . (5.4) ~al/~

clo,n = --1, clk+l,n = dk,n sign f ' (xk,n), k = O, 1, 2 . . . .

So, if pn(x) is of the form (3.1), then such will be its image (£fPn) (x), etc.


5.2. Complete action o f the effective map

Even if the dynamics is time-dependent, the distribution evolving under its action also remains of the form (3.1)

if such was the initial distribution. The characteristics of singularities of pn+l (x) = (£FnPn)(X) , denoted as Xk,n+J, dk,n+l, sk,n+l, can be obtained

by replacing f in (5.4) with Fn(x) = F ( x ) + e(hn - (h)):

X0,n+l = F(Xc) + E(hn - (h)), Xk+l,n+l = F(Xk,n) + E(hn -- (h)), k = 0, 1, 2 . . . .

2 so,n+1 -- P "~'al'" pn(Xc) , Sk+l,n+l = Sk,n[F'(Xk,n)[ -1/[~, k = 0, 1,2 . . . .

d0,n+l = - 1 , dk+J.n+l = dk,n sign Fr(Xk,n), k = O, 1, 2 . . . .

Applying these recursive relations k times one obtains explicit expressions for Xk,n, Sk,n, and dk,n. In case of small

E(hn -- (h)) these can be linearized, at least for several first singularities (which are the major ones), and we get

hn-k -- (h) hn-I (h) xk,n = Xk + ~ • (Fk) ' (Xo) (hn-k-J - (h)) + + " " + j ,

F' (Xo) (5.5) S~,n = Sk + O(Erh)~-k/~),

dk,n = Dk,

where )~ > 1 is the Lyapunov exponent of the map F ( x ) , 6h is the amplitude of mean field fluctuations h~ - (h)

and

Xk = F k+l(Xc),

2P(Xc) . [(Fk) , (Xo) l_ l /~ = O(,k_k/~), (5.6) & = ~a~/~

Dk = sign(Fk) '(X0)

are nothing but the singularities of the invariant distribution P (x) of the map F (x). Since dk,,, = + l , the last equality in (5.5) holds unless Xk,,n, and Xk, lie to the different sides of Xc for some

n r < n - k and k' < k. According to Section 5.3, this can only occur for very large k ~ - log(~3h).

Then as I(Fm)'(Xc)[ ~ )~m we conclude that xk,n moves around with the amplitude

AXk,n ~ Ixk,n - Xkl "~ E~h~ ~. (5.7)

Moreover, since I(F m)'(Xc)[ ~ U n while )~ > 1, (5.5) implies that Xk,n depends mainly upon "early" values of the

mean field, i.e. h n - k - I . . . . . h n - k - l + M , with M = O(1), and so for large k is naturally expected to be more or less

independent of hn.

5.3. New tiny singularities due to fluctuations

Our conclusion that the leading singularities are Ix - xk,n [-a is only valid if the distribution is not singular at

the critical point Xc. Since the singularities are located at the trajectory of this critical point, this means that we assume it is never closing. For an isolated local map this is quite reasonable: were the trajectory of the critical point

in which f ' = 0 closed, there would be no chaos but only a superstable periodic attractor. Unfortunately, for a GCM, according to (5.7) the singularity points Xk,n are perturbed by the mean field and

deviate from Xk by ~ ~.kESh; so even if the unperturbed trajectory never comes close to the critical point, the perturbed one can hit it all right. This does occur, originating some new, strong singularities - but with tiny weights.

S. V Ershov, A.B. Potapov/Physica D 106 (1997) 9-38 23

Indeed, let Xk.,~ = Xc at some time n. Substituting this in (5.7) gives I Xk, -- Xcl "~ ~k. ~3h. Then, since the points

Xk --= Fk+l (Xc) of the unperturbed trajectory are distributed over all the attractor with the invariant density P (x),

they hit the neighbourhood [Xc - A, Xc + A] at the average frequency 79A = f~c+~ P(x ) dx and it is reasonable

to estimate the length of the trajectory before it hits this neighbourhood for the first time as k > O( 1/79A). As the

invariant density P (x) has power-law singularities Ix - Xkl -'~, its integral satisfies T'A < O(A l ~), and therefore k >_ O(A-l+a). In our case A = IXk, -- Xc[ ~ )~k*~gh, so k, > O([Xk*E3h] - I+~) which for ~ << 1 gives

)k. > O ( 1 / ~ h ) . (5.8)

Since Xk.,n = Xc, the distribution Pn is singular near the critical point:

O(dk,,n[X - Xc]) Pn (x) = sk.,n

Ix - X c l

Substituting this into (5.2) we see that near the point f ( X c ) the image £fPn behaves like

( £ f p n ) ( f ( X c ) - ~) = O(1) • Sk.,n~ -~' -F O(1)~ -c~ H- less singular part,

so near x0,n+l = f ( X c ) + ~hn the distribution Pn+l ( x ) = ( l ~ f p n ) ( X -- f h n ) is

! - - J

P~+1 (x0m+l - ~) = So,n+l~ ~ + s0,,~+l~ -~ + less singular part,

where a ' = 1 - fl-2 is the power and Som+l~ = O(l)sk.,n is the weight of the new singularity. Its contribution to

the deviation of the mean field is given by an equation similar to (4.3) integral

, f So.n+ I j (Fn-i o . . . o F n - I ) ( x ) ~ ( X O , n + l _ x)Ot, (X0,n+ 1 - x - - ~¢~hn)° t ' J d x .

which does not exceed

21s~'),n+l I max I Fnl (~3hn) l - a ' < O(1)[Sk,[(E6hn) 1+~-2 < O((E~h)l/l~+l/l~2),

1 --OF

where we estimated S'o,n+ 1 using (5.5) and (5.6) as

t

S0.n+ i = O(1)Sk,,n "~ O(1)Sk, = O(X -k*l~)

and derived k -k* from (5.8).

Under the action of the effective map, this singularity propagates along the perturbed trajectory, its weight

decreasing similarly to that of the main singularities:

S~,n+ k = S~,n+ 1 - O ( X - ( I - E ) k) = S~,n+ 1 • O()~-k/¢ ~2)

so its contribution to the mean field only decreases, being always < o([~h]l/¢~). Meanwhile, the contribution of the "main" Ix - Xk,n [-a singularities which have O(1) weights is O((E6h)~/~).

So in the limit e --* 0 the influence of the "new" singularities is negligible. In fact it is even smaller than that which

follows from the above estimate, for we have ignored that it is a rare occasion when the fluctuating Xk,,n falls very

near the critical point.


6. The mean square value of 7)n(Xs, ft, d)

In this section we shall study the integral

f - - xs +

n(Xs, U , a ) - - j f " ( x ) \ 7 /

in case of 1D local map and show that there exists such n , that for Axs > C41/~1

x~+/x, ( I l z l ~ 2ot}, I 2 2Axs Dn*(Xs'#'d)dxs > Cll/Zl2-2'~ \ ~ X s ] '

Xs - Axs

(6.1)

where y is some unknown function that vanishes when IAxs/iZl ~ oo; it comes from conjecture 2. The value of

n , can be asymptotically estimated as

log I/z] n , ~ - - - + O(1) (6.2)

log ~.

(here ~. is the Lyapunov exponent of the map f ) . The integrals Dn with either too large or too small n are much smaller and do not satisfy this estimate.

Here and below we shall use the notation Ra(x) = O(x)lxl -~ where 0(x) = 1 f o r x > 0 and 0 otherwise, so

B

Dn(Xs, Iz, d) =-- [ fn(x)(Ra(d[x - Xs]) - R~(d[x - Xs + tt])) dx, (6.3)

A

where A and B are the left and the right ends of the attractor; we assume that Xs - Axs > A, Xs + Axs < B. The idea of this section is the following. We introduce a "modification" of (6.3):

Xs+AXs

Dn(Xs, It, d) =-- [ fn(x)(kot(d[x - X s ] ) - Ro~(d[x - Xs + / z J ) ) d x , (6.4) I /

X s - Axs

where Ra(x) is 2Axs-periodic extension of R~(x) beyond [0, 2Axs]:

Ra(x + 2Axs) for x e [ - 2 A x s , 0], Ra(x) = Ra(x) f o r x e [0, 2Axs]. (6.5)

Ra(x -2Axs ) for x e [2Axs, 4Axs],

Fig. 5 shows graphically the second factor in the integrand in (6.3) and (6.4). Owing to the periodicity of Re, (x) (6.4) is a convolution, and we shall estimate its Lz norm

Xs+Axs

1 f --2 (~2 (. , /z, d)) ----- 2Axs 7) n (Xs,/z, d) dxs (6.6)

Xs-Axs

with the help of Fourier transform. This will be done in Section 6.1; in Section 6.2, we shall show that this is a good estimate for (732( ., # , d)) (6.1) if Axs/l/zl is large enough.

S.V. Ershov, A.B. Potapov / Physica D 106 (1997) 9-38 25

I

l L ~ . / ~ ( x - x, + ~t)

~t x~ X, + Ax~

~ ( x - xs + ~t) =/~(x - x, + rt)

= Ra(x- x, +g + 2Ax~) I I l I

Fig. 5. The original functions R~ (x - Xs) and Ra (x - Xs + / z ) (top panel) and their periodic extensions k~ (x - xs) and k~ (x - xs + tl) (bottom panel) within [Xs - AXs, Xs + AXs] where the integration in (6.4) is taken. Shown is the case xs 6 [Xs - Axs + l~. Xs + Axsl.

6.1. The modi f i ed in tegral

As an L2 norm, (6.6) is the sum of squares of the Fourier coefficients of Dn (Xs,/z, d) (as a function of Xs). The latter are the products of the Fourier coefficients of fn (x) and k,~ ( d [ x - Xs]) - ka ( d [ x - Xs + #]), because (6.4)

is a convolution. Using the periodicity of ka (x) one obtains after some simple algebra that

26

o~

<~2(',/z, d)) : 4 Y~ k=-~x~

where

S. V. Ershov, A.B+ Potapov / Physica D 106 (1997) 9-38

i J r ~i i,, Jl'~ntk~'2'~c~(kd~'2 sin 2 rrk/z 2Axs'

Xs+AXs 2Axs

i ' f " A 1 fn(x)eizrkx/Axs dx, Rot(k) =- - - R , ~ ( x ) e ' ~ k x / a x ~ dx sn(k) =-- 2./7 s

X s - A x s 0

are the corresponding Fourier coefficients. Then, since fn and Rs are real functions and d =- 4-1,

oo (~](-,/z, d ) ) 8 Z [ y ~ ( k ) 1 2 l ? ~ ( k ) l e s i n 2rrklz kz = > 8 Z IJ~(k)12lR'~(k)12 sin 2 7rk/l. (6.7)

2Axs - 2Ax~ k = 0 k=kl <

for any k2 > k l > 0. Now let us calculate R~(k). By definition, k,~(x) = x -c~ forx ~ [0, 2axs], so

2Axs

Ru(k) ~- -- ~ 1 f Ro~(x)e iJrkx/Axs dx

0

- - ~ \ ~Xs / y-C~eiJrY dy - y-C~ eiJrY dy •

2k

Integrating by parts one obtains

l! I z -a 2 y-aei:ry dy < - - + e i~ry oey -c~-l dy = --z -a. 7g Jr

z

Thus, denoting 7~ =_ I f o Y-aeirrY dyl, we arrive at

> _ _ ~ c ~ ( k "]°~-' ( 21-'~ \ _ k-U iR (k)l tX;gsJ • 1 )

and (6.7) becomes

47~ ( 21-~ ) ,2 ( k "] 2a-2 : rk# . ( D 2 ( - , u , d ) ) > ~ - X s 1 - J rT~ak?~ ~- - -~[P(k) le \~Xs / sin e (6.8)

k=kj 2Axs

Now let ki and k2 be the bounds of the spectral domain which contains an essential part (e.g. ¼) of the spectrum of fn :

Xs+Axs k2 1 ~ 1 f ~_, [fn(k)12 > ~ Z Ifn(k)l 2 -- ~ [fn(x)]2 dx"

k=k l k : 0 X s - Axs

Quite obviously, at least the upper bound k2 grows with n because the greater n, the more ragged is fn (x). So there exists n, = n,(I/zl, Axe) such that ke(n) _< Axs/I/zl for n < n,. On the other hand, f~+l (x) cannot be more than max I f ' l times "more ragged" than f~ (x), which means that

ke(n+ 1) <max ] f ' ] . k2(n)

S. V. Ershov, A.B. Potapov /Physica D 106 (1997) 9-38 27

or k2(n,) > k2(n, + 1 ) /max I f ' l . Our definition of n , implies that k2(n, + 1) >_ Axs/I/~l, thus

1 ~ lAxs < k2(n,) < ~ - - . (6.9) max I f ' l - -

Therefore in (6.8) Isin zrk#/2AXsl > k l t t l /Axs using which inequality (6.8) becomes

--~ 4 T ~ 2 ( 21-~ ) k2 ( k "] 2~

• k=kl

> 4 ~ 2 1 - k7 a I/~[ 2 y~ [ f '~(k) l 2 - A x ~ s r T ~

• k=kl

Xs+ Axs ( 2'-~ ) ('~tlkl']2~ ' f >7-¢ 2 1 7r7¢~ k~-C' 1/*12-2c' \ - ~ X s / Axs [fn*(x)12dx"

Xs-AX~

Xs+AXs n 2 Concerning the last integral, it seems quite natural to assume that there exists f = O( 1 ) such that f x~- ~-~ [ f (x) 1 dx

> 2AXs • f 2 thus

( 2,-° . _ -- k - a 1#12-2a {D~.(., # , d ) ) >_ 2['R.af] 2 1 ~ ! ) \ ~ 2 , (6.10)

Now we should estimate the lower bound of the essential spectral domain, kl (n.) . Unhappily the necessary

statements cannot be proved, being merely conjectures, reasonable as they are.

For most points x the averaged contraction rate converges to the Lyapunov exponent:

1 lim - log I ( f~) ' (x) l = X.

So ( f " ) ' (x) satisfies

, ~ ( I - F ) n < I( fn) ,(x)[ <_ ~ . ( l + F ) n (6.1 1)

for some (unknown) F(n) n ~ O. This implies that the interval of scales in f n ( x ) is no wider than [X ( l+ r ) , ,

X- ( l -F)" ] . Since the wave number k in the interval [X - AX, X + AX] corresponds to the scale ~ A X / k , most

"mass" of the spectrum of fn (x) calculated over this interval is contained in the domain

C I ~ ( I - F ) n . • X "< k < cX (l+r)n • AX,

where c > 1 is a constant. So we arrive at:

Conjecture 1. If the interval [X - A X , X + AX] is fixed and only n changes, then the bounds of the essential spectral domain of fn (x) satisfy

kl(n) > c - I Ax )v ( l - r )n , k2(n) < c A X X O+r)n

n---+ (x3 for some F(n) ---+ O. The constant c can depend upon AX.

However, the interval [Xs - AXs, Xs + Axs] over which the Fourier transforms in (6.7) are calculated is not of the fixed length; rather AXs ~ [/zl. To treat this case, let us take AX << 1 but such that always Axs < AX. Denote by n t the nearest integer for which

I ( fn ' ) ' (Xs)[ =-- A = A X / A x s .

28 S.V. Ershov, A.B. Potapov /Physica D 106 (1997) 9-38

Since AXs < AX << 1, fn l (x) admits linearization fo rx e [Xs - Axs, Xs + AXs] and therefore:

f n ( x ) ,~ f n 2 ( f n l (Xs) + A[x - Xs])

where n2 --= n - n l. Due to this linear relation, f n (x) in the interval [Xs - AXs, Xs + Axs] has the same spectrum

as fnz (x) in the interval

[ fn i (Xs ) _ A A x s , f n i ( X s ) + AAxs] ~- [ fnJ(Xs) _ A X , f n i ( X s ) + AX].

For the latter the bounds of the essential spectral domain can be estimated by means of Conjecture 1, and since they

coincide with the bounds of the essential spectral domain of f n ( x ) in [Xs - Axs, Xs + Axs], we conclude that the

latter satisfy

kl(n) _> c - I A X ) ~ (l-F(n2))n2, k2(n) < cAX)~ (l+F(n2))n2

from what it follows that

kl(n) > (c - I A X F ) 2/(I+F) . [k2(n)] I-2F/(I+F).

For n = n , we can estimate k2 by means of (6.9) which gives

k l (n . ) > C . IAxs/lzl l -× ,

where y -- 2F(n2) / ( l + F(n2)) and F(n2) ~ 0 when n2 ~ cx~.

in

(6.12)

(6.13)

(6.14)

Now let us estimate n2 for n = n. , when (6.9) holds. Substituting the second inequality of (6.12) in (6.9) results

1 I AXs ~ (l+F'(n2))n2 > - c A X m a x l f ' l - 7 "

Similarly, combining the first inequality of (6.12) with (6.9) and using that kl < k2 gives

~(1--F(n2))n2 < - A X

thus

1 log IAxs/~l 1 log IAxs//zl O(1) _< n2 _< + O(1).

I + F'(n2) log )~ 1 - F'(n2) log )~

We do not know the exact form of F(n2), but since it vanishes for n2 ~ 00, the above inequality results in the

asymptotics

log I Axs/#l Ax~/ l~l~ n2 ~ + O(1) > c~ (6.15)

log

and so F ~ 0 when fAxs/#l --+ oo.

Therefore, y ---- 2 F / ( 1 + F ) also vanishes when IAxs/lZl ~ c~.

This enables one to derive an explicit estimate for n. . Indeed, recalling that n I was taken such that I ( f ni )f(Xs) f = A X / A x s , and using (6.11) we have

1 log(AX/Axs ) 1 log(AX/Axs ) < n l <

1 + F ( n l ) log~. 1 -- F ( n l ) log~.

S.V. Ershov, A.B. Potapov/Physica D 106 (1997) 9-38 29

so for Ax~ << 1, when nl is large and F(nl is negligible,

log Axs nl ~ - - + O(1).

log

Therefore if IIxl << Axs << 1 and so both this estimate and (6.15) hold, n , =- nl + n2 satisfies

log IIXl n , ~ - - + O ( 1 ) . (6.16)

log

Recalling (6.14) we arrive at the final conclusion:

Conjecture 2. If one calculates the Fourier transform of fn (x) over the interval [ X s - Axs, Xs + Axs] with Ax~ << 1,

then there exists such n = n , that the upper bound of the essential spectral domain obeys (6.9) and its lower bound

satisfies

I -v kl(n) > C - ~

where y is some unknown function that vanishes when I Axs/IX[ ~ ~ . The value of n , asymptotically obeys (6.16).

Substituting this estimate for kl (n , ) in (6.10), one has

21-~ IX ¢l -v)c~ 12-2'~ IX 2~v (~ . ( . , IX , d)) > 2ITg~C~f] 2 1 rrT-¢~CC ~ ~ j ' I IX ~Xs (6.17) g

6.2. The errors caused by the modification

For the sake of simplicity we assume below that IX >_ 0; the opposite case is quite similar; it even reduces to

the former using obvious identity Dn(Xs, IX, d) = -Dn(Xs + IX, -IX, d). Let also d = +1 (the opposite case is

absolutely similar and leads to the same estimates).

Comparing (6.3)

B

IX, +1) =-- I fn(x) (R~(x - Xs) - R~(x - Xs + IX)) D,l(x~, dx t l

A

with its modification (6.4)

Xs+Axs

~n(xs, Ix, +1) =-- I fn(x)(Rc~(X - Xs) - / ~ ( x - Xs + IX))dx * /

Xs Axs

one finds that if x~ 6 [Xs - Axs + IX, Xs + Axs] their integrands coincide for x 6 [Xs, Xs + Axe], as can be easily

seen from Fig. 5. Looking at this sketch more closely, we can write

79,, (Xs, IX, + 1 ) - Dn (Xs, IX, + 1 ) B

=- I fn(x) (Ra(x -- Xs) -- Rc~(x -- Xs + IX)) dx

Xs+ Axs


X s - ~ Xs

- f f ' (x)(k~(x-x~)- k~(X-Xs + lz))dx - f f'(x)R~(x -xs)dx. X s - AXs x s - #

If Xs ~ [Xs - AXs + # , Xs + Axs] the argument o f /~a( . ) is always within [ - 2 A x s , 2AXs], so using (6.5) and

substituting R,~(x) -- O(x)lx] -'~ one has

B

f 79n (Xs, # , +1) - Dn (Xs, # , +1 ) = f n ( x ) ( ( x -- Xs) - a -- (x -- Xs + / z ) -c~) dx

X s + A x s

X s - #

-- f f n ( x ) ( ( x -- Xs + 2Axs) -~ - (x - Xs + / t + 2AXs) -c~) dx

X s - Axs

xs

-- f f n ( x ) ( x -- Xs + 2 A x s ) ~ d x

xs-#

from what it follows

r79n(Xs, # , + l ) - Dn (Xs, # , + l ) J _< - - 2maxlfl

- ((Xs + Ax~ - x~ + iz) 1-~ - (Xs + Axs - x~) l -~) .

Since the derivative of z l - ~ is a decreasing function, finite differences obey

(Z + # ) l - a _ z l - ~ _< (1 - ot)/z • z -ce

so for Xs c [Xs - Axs + # , Xs + Axs]

J~)n (Xs, ~ , 21-1) - ~Dn(xs, lz, + l ) l _</z max ] f l • (Xs + AXs - Xs) -~ . (6.18)

The case Xs 6 [Xs - AXs, Xs - Axs + tz] is quite different and this estimate fails. Instead we shall use obvious

inequality

I /) , (xs, # , +1 ) - D , (xs , # , +1)1 _< max ID, (xs, # , + l ) l + max ID,(xs, # , +1)1, Xs Xs

(6.19)

where, as readily follows from definition (6.3),

/ [~Dn(xs, # , + l ) ] _< max ]fJ - [ / (x -

i Xs

/.tl -or _< 2 max If]

1- -or

Xs --t- #)-or dx -t- f [ ( x - Xs) -~ - (x - Xs + # ) - ~ ] dx

Xs

(6.20)

Then

X s - x s + A X s

[D, (xs, /z, +1)] _< max I f ] . / Ika (x ) -/~c~(X + #) l dx.

X s - x s-- Ax s

S. V. Ershov, A.B. Potapov / Physica D 106 (1997) 9-38 31

The integral of a periodic function over its period is obviously independent of the position of the interval of integration, thus

X,-xs+Axs 2Xs

f I /?c, (x ) - /}c, (x + # ) l d x = f [ / ~ ( x ) - / ~ c , ( x + /z ) l dx

X~-.r,- A.rs 0

and calculating the last integral using (6.5), one arrives at

SO

_ _ /Z 1-u [Dn(Xs,/z, +1)1 _< 2max I f l ' 1 - u

__ ~ 1 - ~

]D,(x~. # . + l ) l + ID~(xs, Iz, +1)1 _< 4max If l i - -_~ ,

which together with (6.18) and (6.19) gives

ID. (x.~,/~, +1) - Dn (xs, # , + l ) l

4(max Ifl)2bl 2-~ - (Xs + Axs -- Xs) -~ for Xs ~ [Xs - Axs + #, X~ + Axd < { /~I ~ \ 2

t 4 m a x I f } ~ _ ~ ) forxs E [X~ - Axs, Xs - Axs + #].

Now we can estimate the difference of L2 norms:

{D~ (xs, # , +1)) - (D2(Xs, Iz, +1))

Xs+Axs 1 f --2 2

-- (Dn(X s, # , +1) - Dn (Xs, #, +1)) dxs J 2AXs

Xs- Axs Xs+Axs

<_ max(ID~(xs, # , +1)1 + [Dn(X.~,/z, +1)1)~--7--- J ID~(x~, tz, +1) - D.(x~. # , +1)1 dx~ Xs Z~Xs

Xs- Axs /zl__ u , 2 /Z c~

< ( 2 m a x [ f l l - ~ - - d ] " [ ( 2 - ~ X s ) + 4 2 @ x ~ ] '

which in case/z <_ AXs can be reduced to

--9 (nl_axlS['~2/z2_2oe ( /z )~ I(DT,(x~, / , ,+l) ) - ( D 2 ( x s , # , + l ) ) l < 1 0 \ 1 - u / ~ "

The case when # < 0 and/or d = - 1 is quite similar and leads to the same inequality, and combining it with (6.17) one arrives at

(D2,( .,/z, d ) ) > 2 l ~ C C ' f ] 2- I/.Zl 2-2~ ~ 2a7 - -- Axs

x l jr.R.c~C~ [-~xs [ - 5 (1 _ ~ ) T ~ g u f ] A x---~ "

Taking into account that y > O, we can write in case I~/Axsl <_ 1

32 S. V Ershov, A.B. Potapov/Physica D 106 (1997) 9-38

(792"(" # ' d ) ) -> 2[7"q'ceCCef]2-- " I/zl2-2ce to~x~lZ 2~O'

2Cll/zl 2-2'~ /z 2~y [ 1 ~ (l-2y)al l l 2c2 I (6.21)

Since y vanishes as Axs/ [# l ~ oo, there exists such r that ?' < ¼ for Axs/]#] _> r. Let us denote C3 ~-

max{l , C2} and C4 --- max{r, C2/~}. If Axs is large enough, namely

Axs >__ I ~ l ' C4 (6.22)

then obviously our"technical" requirement I # / A x s l < 1 is satisfied, so (6.21) holds. Besides, (6.22) implies y < ¼, F,I/(I-2F)~ thus '-'3r~2/~ >- ~3r'l/~J-2Y)'~ (because by definition C3 . . . . > 1) and therefore Axs > I#[ • ~3 Substituting the

latter inequality in the term { 1 . . . . } of (6.21) we arrive at the desired estimate (6.1).

7. How the statistical characteristics of 1D map depend upon the parameter

Now let us turn to a different problem, which is nonetheless closely related with the behaviour of mean field in globally coupled maps. Namely, we are going to study how the statistical characteristics of the attractor of an

isolated 1D map depend upon the parameter of this map. That these two problems have some correspondence can be seen from comparison of the following facts:

(1) For logistic-type maps the dependence of averages upon the parameter is rather wild (cf. the well-known plot of the Lyapunov exponent vs. this parameter), while the mean field fluctuations are rather strong, 8h ~ ~1/(~ 1).

(2) For piecewise linear maps the dependence upon the parameter is continuous and Lipshitzian, though not differentiable [13], while the mean field fluctuations for coupled piecewise linear maps are tiny, 8h ~ e -°~1/'2) [11].

(3) Apparently (though this is still an open question) those maps which have continuous invariant distribution and

therefore smooth dependence of averages upon the parameter, do not produce any fluctuations of the mean field at all if globally coupled on a lattice.

So it seems very likely (this assumption originates to Kaneko [7]) that this is not just a coincidence, but the behaviour of averages for an isolated map does determine the amplitude of mean field fluctuations when these maps are globally coupled.

In this section we shall give some additional reasons that may be useful to substantiate this conjecture. Namely, we show that

ao+Aa

2 A a l f [(x)(a + 8a) - (x)(a)[ da ~ 18a] 1/~,

a o - Aa

where (x)(a) denotes the average value of Xn for an isolated logistic-type map Xn+l = 1 - a Jxn I [3. This estimate is in a rather good agreement with numerical results, see Figs. 3 and 4.


7.1. Perturbative approach series

Let us denote the invariant distribution of the mapping xn+l = fa (xn) as Pa (x); it satisfies

P. = £f,, P,,

where £L,, which we shall abbreviate to £,,, is the corresponding Frobenius-Perron operator, see (2.21) or (5.1).

Comparing two such equations for a = a0 and a = a0 + IX, one obtains

P"o+n - f-'ao Pao+n = (Cao+# - Cao) Pao+u,

which we shall write as

Pl, -- CO P . = g . ,

denoting gu ----_ (£~ - £0) P~, and dropping for the sake of simplicity the constant ao, so that £u and P~ are used

instead of £ao+~ and P.o+~,. Applying the operator £~ to both sides of this equality and summing up the results for

k = 0 . . . . . n one obtains

n

k P~' - ~Or"+l Pu = ~ £og/z" k=O

NOW let us multiply this equality by some function ~0(x) and integrate, using (2.5):

where (~0). _= f ~ o ( x ) P . ( x ) d x .

At last, we assume that under the action of the dynamics Xn+l = fO(Xn) the distribution at least weakly converges

to the invariant one: £~)Pu " - ' ~ P0 (relaxation towards statistical equilibrium). This implies

f ~o(x). (£~+' P . ) ( x ) d x = f ~o(x)Po(x)dx = lim (~o)o ~/--)" O0 , / J

and so one can pass to the limit n -+ c~ in (7.1) which gives

(q)). - ((P)O = ( p ( f d ( x ) ) g . ( x ) d x . (7.2)

Now let the dependence of the map upon the parameter be additive (just for the sake of simplicity): f , ,0+. =

IX + f .o , as in another form of the logistic-type maps: f (x) = a - [x l ~. For such a map (L;• p ) (x ) = (£0 p) (x - IX),

as can be easily seen from (2.2), so

g u ( x ) = Pu(x -- IX) - P . ( x )

and lor ~0(x) = x (7.2) becomes

(x) u - (x)o = ~ f ~ ( x ) [ P u ( x - Ix) - Pu(x)] dx. (7.3) n = l


As we have said in the end of Section 5.1, the invariant distribution Pu is a sum of the "regular component" P~ and the sum of power-law singularities with parameters depending upon/z through the map fu, see (5.6):

Pu(x) = P~(x)r + ~ b'k,u- O[Ok,tt(x - Xk,u)] k>_0 Ix - Xk,rt [~ ' (7.4)

where P~ is the "regular part" with weaker singularities or none at all. Substituting this expansion in (7.3) one has

o~ {X)I z -- {X)o = -- ~ Sk, # y ~ ~Dn(Xk,#, - # , Ok,lz ) "~- O(1#11-~), (7.5)

k>0 n=l

where Dn (Xs, #, d) is the same as in (4.7) and the term o(l#l ~-~) is the contribution of the "regular part".

This series can be thought of as a "static" version (no dependence upon time) of (3.2). It is this similarity that relates the problem concerning mean field fluctuations with that concerning dependence upon the "static" parameter.

7.2. Averaging over the singularity locations

The particular values of deviation (x)(a + tz) - (x)(a) depend upon a, and this dependence appears to be completely wild: it can vary several orders of magnitude over a tiny interval, as will be discussed in Section 7.3. So

any particular value seems to be of only small importance, more suitable characteristics being something like the "mean deviation"

ao + Aa

' f (l(x)(a + 8a) - (x)(a)[) --- 2Aa I(x)(a + ~a) -- (x)(a)[ da. (7.6)

ao-- Aa

This averaging makes the locations of singularities

Xk,a = fak(Xc) ~ Xk,ao + (a -- ao)(fako)'(Xc)

move around with the amplitude AXk,a ~ )~kAa; thus (cf. the arguments of Section 4) mean deviation (7.6) is

determined by the "typical" (with respect t o Xk, a ) value of the integral 73n (Xk,a, -~a , Dk). For any a0, Aa and ~a there exists such k that condition (6.22), which now requires AXk,a >_ C46a, is satisfied. Therefore, there exists such n that the typical value of Dn (Xk,a, - r a , Dk) (as a function of a) in the interval [a0 - Aa, a0 + Aa] is (see (4.10))

[~ )n (Xk ,a , -3a , Dk)l ~ 13al 1-c'.

Thus, one can expect that for these values of a the whole sum (7.5) is not smaller (because for other values of a this deviation can be very large, see Section 7.3):

I(x)(a + ~a) - (x)(a)[ > O([Sal I/t~)

and so

([(x)(a + 6a) - (x)(a)[) > O([~all/~),

which is in agreement with numerical calculations (see Figs. 3 and 4).

(7.7)

S. V. Ershov, A.B. Potapov / Physica D 106 (1997) 9-38

7.3. The giant O(1 / l og [3a I) deviations

35

For flat top maps the "periodic" values of the parameter (i.e. those for which the dynamics is periodic) are dense;

moreover, in any neighbourhood of a "chaotic value" a there exists a + 8a corresponding to an n-periodic motion.

Indeed, consider the nth iterate of the critical point, f n (Xc). As one changes the parameter by 3a, this fn (Xc)

moves by ~ ~.~3a, so there exists such 6a = O(~ -n) that f~(Xc) = Xc; obviously n "- - log ]8aJ. This means

that far from a chaotic parameter value one can find rather short periodic trajectories; while close to it they are very

long. The closed trajectory of the critical point is superstable, thus the attractor is just this trajectory. Theretbre,

(~o) (a + 6a) is a sum over these n points, while (~0) (a) was an integral. The latter can hardly be approximated by the

former with better than O(n -1 ) accuracy (cf. numerical integration over n mesh points); and since n ~ - log ]3a ],

this means that

_ O 1 I(~o)(a + 3a) - (~o)(a)[ > - (i-oog-]Sa [ ) '

which holds as long as the attractor remains periodic. At one end of the periodic interval chaos arises via tangent

bifurcation, when the invariant measure is concentrated around periodic attractor and so all averages change contin- uously at this point. Therefore, even when the perturbed value a + 6a is again "chaotic", the deviation of averages

can still be as large as O(1 / l og ]gal l In terms of the sum (7.5), given that [Dn ] < O(1)[6al I/1~ (see (6.20)), these giant deviations can be due to two

reasons: either the relaxation is too slow so that the number of substantially contributing terms is very large; or some

of the prefactors Sk,a diverge as a ~ a0. The latter is quite possible because right after the tangent bifurcation has

occurred, the effective width of the attractor w is very small, while f Pa (x) dx = 1. Given that Pu (x) is of the form

(7.4), this would have been impossible had all the weights satisfied I Sk,a I < w - ~/t~. Therefore, near this bifurcation

point at least some of them exceed O(w-1/t~) and so (~0)(a + 3a) - (~0)(a) can be as large as O(16a/wl 1/t~). If the

width of the peaks scales as w ~ 3a, which is quite possible, then it can even occur that (~0)(a + 3a) - (~p)(a) ~ 1.

It appears that such giant deviations are "rare" in the sense that the corresponding parameter intervals have small

total length, so the "mean deviation" (7.6) shows almost power-law decay as 8a --+ 0, see Figs. 3 and 4. But

sometimes they play an essential role, as was shown by Daido [18] for Lyapunov exponent (which is also a sort of

an average, see [19]).

8. How one can use the dependence on the parameter in a single m a p to es t imate the ampl i tude of

f luctuations in globally coupled maps

The relation between the two problems is far more general than our results (4.13) and (7.7). Say, for multidi-

mensional maps our arguments used when deriving the "basic" estimate for (D2( .,/z, d)) fail completely, while

relations (3.2) and (7.5) still hold, expressing the amplitude of fluctuations 3h and the mean deviation of averages

([ (x)(a + 3a) - (x)(a)b) as sums of terms similar to Dn (Xs, p,, d). Our loose estimates of the "typical value" also

seem to be insensitive to the dimension (so loose they are!). Therefore, apparently even in this case

~h ~ V/(~32( ., e6h, d)),

ao + Z~ a

I(x)(a + ~a) - (x)(a)[ da ~ (Dn2( ., ~a, d)).

ao--Aa

(8.1)


Thus if one is able to calculate mean deviations 1 / (2Aa) fao+Aa jao_Aa [(x) (a + ,~a) - (x) (a)l da and if it appears that

there exists such ~c that

then

a o + A a 1/ 2Aa ](x)(a + 8a) - (x)(a)l da ~ 18a] ~c

ao- Aa

/ _

v / (D .z ( ., u , d ) ) ~ u ~

and substituting this dependence into the estimate for the amplitude of fluctuations (8.1), one arrives at 8h ~ (ESh) x,

so finally

8h ~ E x/(1-r).

This may appear useful when the fluctuations of the mean field are so tiny that one cannot detect them in numerical

simulation, because finite lattice size (bounded by the available computer memory) results in a random noise added

to the "genuine" fluctuations. In the same time, the precision in computation of averages for an isolated map is

bounded by the time spent only, and so one can achieve very high accuracy (if not in a hurry) in calculating (x) (a).

Then, from these data he can derive the exponent K and so make an estimate for mean field fluctuations.

9. Conclusion. Do chaotic oscillations of probabilities contradict linearity of the master equation?

The evolution of probability measure in the dynamical system (1.2) can be described in two ways. First, one can

use the full master equation, which always is linear. Besides, there exists an alternate approach that is based upon

the SCFPO and uses only one-dimensional projections of this measure. They obviously give the same results.

However, the mean field fluctuations are complex and irregular as indicated by numerical experiments; apparently

they are chaotic. For the case of piecewise local map f ( x ) we even have a theoretical evidence: it was shown that

the SCFPO, as a dynamical system Pn ~ Pn+l has positive Lyapunov exponent [ 11 ]. For the case of logistic map

with/3 = 2 the same was demonstrated numerically in [8]. So initially close distributions Pn (x) and Pn (x) diverge

as time goes on.

Therefore, under the action of the full linear master equation the probability measure evolves in such a way that

at least its 1D projections exhibit chaotic oscillations! Isn't that a contradiction?

No, it is not. This is because one cannot prove that the dynamics under the action of a bounded linear operator is

regular (periodic or quasiperiodic) using only its linearity and boundedness. Besides, one needs some properties of

the phase space where it operates. These always hold for finite-dimensional systems, and usually even for infinite-

dimensional. But the phase space of probabilities in the thermodynamic limit (when the SCFPO approach is applied) is just the case when they broke!

Indeed, as argued in [20], correlations between different sites decay as n -+ ¢x~, so in the attractor the full probability distribution is the infinite product

79n( . . . . xn(i - 1), Xn(i), Xn(i + 1) . . . . ) . . . . pn(xn(i - 1))- pn(xn(i) ) pn(xn(i + 1 ) ) . . . , (9.1)

i.e., we have an uncorrelated random field. In [20] it was proved that for random fields with decaying correlations, L2 norm of infinitesimal deviation of m-dimensional projections 79m (Xn (i) . . . . . Xn (i + m)) satisfies

JIT~m - 7~m IJL2 = o ( 4 ~ ) - Ip . - PnlJL2


One can obtain a similar ~ asymptotics for other integral norms, such as L I, etc.

Therefore, for the full (infinite-dimensional) distributions the norm of deviation is always O(1). In a sense, in this

phase space there are no close neighbours! and it is due to this feature that usual arguments that a bounded linear

operator cannot provide divergence of close trajectories and thus chaos fail. It can in our case!

One can argue that there should be "good" norms in which the deviation does not grow as the dimension increases.

But the analysis shows that either they are actually norms only in the subset where (9.1) holds (which is not a linear subspace!), or, if they are norms in all phase space, it appears that the full master (transfer) operator 79,, ~ 79,,+1

is not bounded in them.

In case of finite-size systems the situation is quite different. The behaviour of probabilities cannot be chaotic -

and it is not. As to the value of hn ~ ( l / N ) ~--]~/N_ 1 xn (i), it does oscillate irregularly - but is no longer a statistical

characteristic! It is just a dynamic variable, though close (up to ~ 1/~/-N) to the average f xpn(x) dx. Since the

latter cannot be chaotic, we conclude that its oscillations, though actually periodic or quasiperiodic, look chaotic

at "large scales" (greater than ~ 1 /v 'N) . As N increases, they get more and more complex at smaller and smaller

scales, just as in the Landau scenario of turbulence, though remaining regular for any finite N. Only in the limit

N --+ oo do they exhibit a "genuine" chaos.

An alternate torm of this explanation is the following. For finite N the equation

Pn+l (x) = (A~h, ff.j'Pn)(X)

still holds exactly, while the equality hn = f xpn (x) dx which closed the SCFPO system, is now only approximate; in fact

= f xpn(x)dx + h,,

where ~ = O(1 /v /N) can be thought of as an external chaotic signal. It is external because it can be derived from

neither purely dynamical nor statistical description - one must know both dynamic variables and probabilities to

calculate it.

So the evolution of p~ (x) is governed by a perturbed SCFPO, subjected to the "noise" ~ . The latter "smooths"

the distributions over the scale of O(sen) = O ( 1 / ~ ) . So at those small scales they are no longer singular, and

instead of (4. ! 2) we have

Sh~k.l> = { O([EShn_tl~-~), E~h._l >> O(1/v/~, const! • ~Shn-I, ~Shn-I << O(1/V/-~. (9.2)

Therefore, if N is so large that the amplitude of the mean field fluctuations satisfies ESh >> O ( 1 / v ' N ) , then it is the

upper estimate that holds, and by the arguments of Section 4 one arrives at

8h = O(EI/(/~--I)).

On the contrary, the behaviour of infinitesimal perturbations 8hn, which determines whether the oscillations

are chaotic or not, is now quite different. Indeed, noisy maps are always strongly mixing, the lower bound of the

relaxation rate being determined by the noise amplitude, as proved in [20]. Thus, now in (2.4) tar terms decay exponentially as l --~ ~ ; the effective number of terms in this sum is bounded by some LN. The same is true for its consequence, the sum (2.7). According to (9.2), for small or infinitesimal perturbations 8hn each term 8h(n k'l~ satisfies I~h (k'l) I ~ 0(1) • ~LShn-t[, thus

16h,,I < O(1) • ~ max{18hn_l[ + . . . + lShn-LNI}


and if ~ is small enough, the perturbations decay exponentially. This means that the "noisy" SCFPO is contracting

and the behaviour of probabilities it describes is therefore not chaotic.

One can even call this is a "synchronization" of the SCFPO by the external chaotic signal ~n.

The situation when there is a "genuine" noise fin(i) acting at each site of the lattice:

N E Xn+l(i) = f (Xn( i ) ) + ~ ~-~Xn(j) + fin(i)

j= l

can be treated quite similarly. If the noise is small enough (in comparison with E3h*, where 3h* is the "noiseless"

amplitude of mean field fluctuations 8h* --~ ~l/(t~-l)), the mean field fluctuates, and even the estimate 3h

l /~-T) still holds. Although, even in the thermodynamic limit these oscillations are not actually chaotic, being

at most quasiperiodic at small scales (_< O(fi)). If the noise is strong enough, it can even suppress the fluctuations

completely.

Acknowledgements

One of us (SE) is very grateful to Prof. K. Kaneko who attracted his attention to this problem, and then very

patiently kept on drawing my attention to the relation between the dependence of averages upon the single map's

parameter and the amplitude of mean field fluctuations, in which relation I (SE) could not believe, and would hardly

have believed without him. It was he who proved to be right. The work was partially supported by the Russian

Foundation for Fundamental Studies under the grants # 94-02-3412-a and 93-01-01687.

References

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On mean field fluctuations in globally coupled logistic-type maps

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