ELSEVIER Physica D 103 (1997) 34-50

PHYSICA

Generalized phase transitions in finite coupled map lattices

Michae l B lank 1 Russian Academy of Science, Institute for Information Transmission Problems, B. Karetnij 19, 101447, Moscow, Russian Federation

Abstract

We investigate generalized phase transitions of type localization-delocalization from one to several Sinai-Bowen-Ruelle invariant measures in finite networks of chaotic elements (coupled map lattices) with general graphs of connections in the limit of weak coupling.

1. Introduction

During the last decade a new class of models, so-called networks of chaotic elements or coupled map lattices (CMLs), has been introduced to investigate complex dynamical phenomena in spatially extended systems. One can

find review of various applications of CMLs in [8]. Mathematical investigation of ergodic properties of such systems

was started using methods and ideas of statistical physics in [7] and then continued in [2,3,5,9,13]. The main result obtained in these papers was the stability of statistical properties of CMLs in the limit of weak coupling. In this

paper we discuss ergodic properties of finite CMLs with general graphs of connections among elements in the limit of weak coupling and show that in the general case there may be generalized phase transitions in such systems of type localization-delocalization. These phase transitions correspond to a situation, when trajectories, which should normally be dense, remain confined to a small region (which vanishes, when the coupling constant goes to zero).

We call this "localization phenomenon". The first observation of this type was published in [2] (see also [10] for numerical studies of other phase transitions in the simplest dyadic models of CMLs). We shall give several general

statements about Sinai-Bowen-Ruelle (SBR) measures of such systems and then we shall discuss in detail two particular families of CMLs.

It is worthwhile to discuss the nature of the localization phenomenon. Suppose for a moment that our uncoupled system is a hyperbolic map. Consider a periodic trajectory of a map, in a small neighborhood of which the map is strictly hyperbolic. Then near any point of this trajectory local stable and unstable manifolds are going arbitrarily close to one another. Therefore stable and unstable directions can be mixed by means of arbitrary small perturbations, and the result depends on whether the contraction is stronger than expansion (localization), or not. CMLs that we consider are not hyperbolic, but the same result gives the absence of the local expanding property. This property

1 E-mail: blank@obs-nice.fr.

0167-2789/97/$17.00 Copyright © 1997 Published by Elsevier Science B.V. All rights reserved PII SO 167-2789(96)00251-5

M. Blank/Physica D 103 (1997) 34-50 35

o o o o

(b)

(c)

Fig. I. "Typical" graphs F: (a) linear chain; (b) cyclic chain; (c) Cayley tree.

means that a map expands distances between any close enough points. This property distinguishes the situations

in Theorems 1.1 and 1.2. To show that this phenomenon is not something obscure, specific for only discontinuous

maps, we shall prove its presence for the well-known family of quadratic maps in Section 5.

Let {fi} be a sequence of one-dimensional nonsingular mappings f i : X --+ X = [0, 1] from the unit interval into itself, let X d be the direct product of these intervals, and let us denote a point in this direct product by

-~ (Xl . . . . . Xd) E X d. Remark that the dimension d need not be finite.

Definition 1.1. By a CML we shall mean a map F~ : X d --+ X d defined as follows:

F~(2) = ~s o F(.~),

where the map F is the direct product of the maps fi ( i.e., (F(.~))i ":=

describing the coupling, is defined as follows:

(q~.r)i = (1 - 8)xi + 6, Yijxj, J

and the matrix F = (Yij) is a stochastic matrix, describing the graph of interactions among the maps.

(I.1)

f i ( x / ) ) , and the map ~ : X d ~ X d,

(1.2)

The matrix F need not be symmetric, which means that the connections (couplings) may be oriented. We shall consider only connected graphs, however, there may be "free" nodes (i.e., Yij = 0 for some i and any j), which

does not contradict the connectivity of the graph F. Typical examples of the graphs F are shown in Fig. 1: linear chain, cyclic chain, Cayley tree.

The simplest known type of chaotic maps is the so-called piecewise expanding (PE) maps.

Definition 1.2. We shall say that a map f is piecewise expanding (PE) if there exists a partition of the interval X

into disjoint intervals {Xj }, such that f lclos(X 9 is a C2-diffeomorphism (from the closed interval Clos(Xj) to its image), and the expanding constant of the map

Lf := inf [ f ' (x ) l j , xEXj

is positive and for some integer x the iteration fK of the map f has the expanding constant X U strictly larger than 1.

36 M. Blank/Physica D 103 (1997) 34-50

/// / ,I" / , ' i \

c

Fig. 2. A "typical" PE map.

A typical example of a PE map is shown in Fig. 2. The PE map is only piecewise monotonic and need not

be continuous. As it is well known, starting from the paper of Lasota and Yorke [ 12], these maps have all the

statistical properties that one can demand on chaotic deterministic dynamical system. It has smooth (absolutely

continuous) invariant measure/zf (SBR measure of the map), exponential correlation decay and CLT with respect

to this measure, etc. In this paper we want to discuss stability of these properties with respect to small perturbations

due to the coupling. We shall emphasize some aspects of this problem, because they seem quite counterintuitive, at least from our point of view.

Definition 1.3. An image of a measure # under the action of a map f is a new measure f / z such that f # ( A ) = # ( f - I A) for any measurable set A.

Definition 1.4. Let there exist an open set U in the phase space such that for any smooth measure # with the support

in this set its images fn l z converge weakly to a measure/zf , not depending on the choice of the initial measure #. Then the measure # f is called a SBR measure of the map f .

Definition 1.5. By a tuming point of a map f we shall mean a point, where the derivative of the map is not well defined. We shall denote the set of turning points by Turn[f] . (In Fig. 2 the point c is a turning point.)

Recall that a point x is called periodic for a map f if f n (x) = x for some integer n > 1. Examples of periodic turning points are given in Fig. 3.

Let us fix the sequence of maps fi and the matrix F . Clearly for e > 0 small enough the map F~ is a PE

d-dimensional map. The only difference between the multidimensional and the one-dimensional cases is that one should use the smallest eigenvalue of Jacobi matrix F t of a map F as the expanding constant of the map (see [ 1 ] for

details). If we also assume that )'fi > )~, which is large enough (actually much larger than 1), then all the ergodic properties of the corresponding CMLs follow from general theory of multidimensional PE maps [ 1 ]. However, we

know only that the expanding constants are positive and larger than 1 for some iteration of the map. Therefore general statements could not be applied here and one should use the specific structure of the map FE. In the sequel we shall suppose that each map fi has only one SBR measure/xi.

M. Blank/Physica D 103 (1997) 34-50 37

Theorem 1.1 ([2,911). Let ~.f~ > Z > 2 for any integer i. Then for any e small enough CML FE has a smooth

invariant SBR measure #e. converging weakly as e --~ 0 to the direct product of one-dimensional smooth SBR

measures #i.

Corollary 1.6. Let the assumptions of Theorem 1.1 be valid. Then for e small enough CML F~ also has the only

one SBR measure.

The main results of this paper are the following statements, generalizing Theorem 1.1 and showing that at least

some additional assumptions are necessary for the stability of the direct product of SBR measures/ai.

Theorem 1.2. Let for any i the expanding constants of PE maps ft are strictly positive Zf~ > L > 0, and there are

no periodic turning points. Then for any e > 0 small enough the CML Fe has a smooth invariant SBR measure #~.

converging weakly as e --+ 0 to the direct product of one-dimensional smooth SBR measures #i.

So the only topological obstacle for the stability of statistical properties of CMLs with respect to weak coupling

is the existence of periodic turning points. In fact we use the absence of periodic turning points in the proof of

Theorem 1.2 when we investigate properties of the decomposition (3.5) of the transfer operator corresponding to

the CMLs. The point is that in this case the second part /32 (describing the dynamics close to the set of turning

points) vanishes after a finite number of iterations.

Theorem 1.3. There exists a sequence of PE maps f~ with ~fi > 1 and with periodic turning points such that for

any e > 0 small enough CML Fe has several smooth invariant SBR measures, converging (as e ~ 0) to the direct

product of singular invariant measures of the maps f /w i th supports on the trajectories of the periodic turning points.

Remark 1.7. Under the assumptions of Theorem 1.3 for any e > 0 small enough Lebesgue measure of the support

of every ergodic SBR measure of CML F~ is of order e '1.

This theorem not only states the localization of invariant measures (supports on small sets) but also guarantees

their smoothness and the absence of other SBR measures. Actually the localization phenomenon was firstly shown

in [2,3], where it was proved that it is possible to construct CMLs such that its SBR measure is localized in a small

neighborhood of a fixed point, but the statement about the smoothness of invariant measures, absence of other SBR

measures, and investigation of their properties are new. It is worthwhile to remark that we assumed in Theorem 1.3,

that all the expanding constants ~.fi are greater than one, which is not the general case for CML's, constructed of

Lasota-Yorke type maps. Therefore even more strong localization may take place, when there are no smooth SBR

measures for CMLs for any weak enough coupling.

Theorem 1.4. There exists a sequence of PE maps fi with periodic turning points such that for any e > 0 small

enough CML F~ has only singular invariant SBR measures with supports on periodic trajectories of the map F~,

converging (as e ~ 0) to the direct product of singular invariant measures of the maps ft. with supports on the trajectories of the periodic turning points.

This statement shows that a system of chaotic maps can be stabilized by arbitrary weak coupling. We shall show further that the opposite statement is also true, i.e., a system of stable maps can become a chaotic CML under arbitrary weak coupling.

These results may be considered as some kind of generalized phase transitions, because when the coupling

strength e goes to zero we can have two quite different types of behavior - there are two different "phases" - with

38 M. Blank/Physica D 103 (1997) 34-50

one SBR measure and with several of them.

Remark 1.8. The statements of Theorems 1.3 and 1.4 do not depend on the dimension of the system (being finite)

and we actually prove them for the two-dimensional case (d = 2) with identical piecewise linear maps. The existence of the localized invariant measures can be proved also in the same way for the infinite-dimensional case, but the answer to the question about the convergence to these measures is unknown.

A particular case of Theorem 1.2 for cyclic graph F and identical maps 3~ with expanding constants strictly greater than 1 was proved earlier in a different way in [ 11].

2. Functions of generalized bounded variation

We mainly follow here Refs. [1-3]. Let X d = [0, 1] d C ~d (or d-dimensional unit toms) with the uniform norm

Ix l = max{lxil: x ~ R d, i = 1 . . . . . d} and d-dimensional Lebesgue measure m = rod. Let L := L ( X d, m) be the space of m-integrable functions on X d, and let/~ ~ h C L means a representative of the corresponding equivalence

class h C L. For an arbitrary measure/z on X d and an integrable function h we introduce the following notation:

f Iz(Y, h) = I h(x) d/z(x); /z(Y) = / z (Y , 1),

I t -

Y

Osc(h, Y, x) = sup{Ih(x) - h ( y ) l : y 6 Y},

W (h, Y, t) = m(Y, Osc(h, Bt (x ) n Y, x)) ,

lim sup inf W(/~, Y, t) , vat(h, Y) = ~ t~0 /~ehcL

var(h) = var(h, xd) , Ilhllv ---- var(h) + Ilhll,

Osc(h, Y) = sup{Osc(h, Y, x ) : x ~ Y},

Bt(x) = {y ~ x d : l x - Yl < t},

Ilhll = m ( X d, Ihl).

Definition 2.1. The functional var(h, Y) we shall call the generalized variation (or simply variation) of the function h over the set Y c X d and the functional Osc(h, Y), the oscillation. The Banach space of integrable functions with bounded variation

{h : X d ~ ~ l : v a r ( h ) < o¢},

equipped with the variational norm

Ilhllv = var(h) + Ilhll,

we shall denote by BV(X) (or simply by BV) and shall call the space of functions of bounded variation.

The following statement provides some elementary facts about the properties of these functionals.

Lemma 2.2. Let functions h, h 1, h2 be of bounded variation and let Y, YI, 1"2 be connected closed subsets of X d. Then the following inequalities are valid:

(1) var(hl + h2, Y) _< var(hl , Y) + var(h2, Y).

(2) var(h, Y1 U Y2) _< var(h, YI ) + var(h, 112) and this inequality becomes an equality if Clos(Yl ) n Clos(Y2) = 13. (3) var(ch, Y) = c var(h, Y) for any nonnegative number c.

(4) var(hl x h2, Y) < var(hl , Y)Ilh21]c~ + var(h2, Y)]]hl IIoo.

M. Blank/Physica D 103 (1997) 34-50

(5) Osc(h, Y) < var(h, Y).

(6) For any x c Y

Ih(x)l _< var(h, Y) + Ih(y)[ dy.

Y

(7/ Let Y = [a, b] then

Ih(a) + h(b)] _< var(h, Y) + ~ Ih(x)l dx.

g

(8) For any n u m b e r s 0 < a l < a 2 < " ' < a n _ < 1

i n f ~ I/~(ai) - h(a/+l)l _< var(h, [a l ,an]) . h(~hcL i

39

(2.1)

Proof All the properties here are quite the same as for usual one-dimensional variation. However, to show how

this technic works, let us prove the only nontrivial items (5)-(8). Suppose that the set Y is a segment. Let us fix

0 < t << 1 and choose an integer k > 0 such that

(k - l)t < re(Y) < kt.

Then

t > m ( Y ) / k > t × m(Y) / ( t +m(Y) ) .

Consider a partition of the segment Y to consecutive disjoint open intervals Yi of length m (Y)/k such that the union

of their closures contains Y. Then

W ( h ' Y ' t ) = m ( Y ' O s c ( h ' B t ( x ) A Y ' x ) ) > m ( Y ' Z O s c ( h ' Y i ' x ) l Y i ( x ) ) i

re(Y) = m(Yi) Osc(h Yi) = m(Y__)) Osc(h, Y) > t - - Osc(h, Y).

i ' k t + m(Y)

Hence for every 0 < t << 1

t + m(Y) 1 Osc(h, Y) < W(h, Y, t).

m(Y) t

Now going to the limit as t ~ 0 we obtain the required inequality for the case of the connected Y. The general

case, when the set Y consists of several connected components, is reduced to the considered one by the restriction

of the function h to each of the connected components of Y. The item (5) is provetl.

Property (6) is a simple consequence of (5), because clearly the value of the function could be estimated by the

sum of its oscillation and its mean value.

To prove the item (7) set c = 1 (a + b), e = ½ (b - a), E = [0, e] and consider a function H" E --+ R 1 , defined

via

H ( x ) = l h ( c - x ) l + l h ( c + x ) l , x ~ E.

By the item (5)

Osc(H, E) < var(H, E) < var(h, [a, c]) + var(h, [c, b]) < var(h, Y).

40 M. Blank/Physica D 103 (1997) 34-50

Besides, by the definition of the oscillation we have for all x E E

e

H(x) <_ Osc(H, E) + ~ - a It(x) dx <_ var(h, Y) + ~ Ih(x)[ dx

0 Y

because the second addenda is equal to the mean value of the function Ih[. Thus, setting x = e and using that

It(e) = [h(a) + h(b)l, we obtain the required inequality. The item (7) is proved.

It remains to prove the item (8).

E l h ( a i ) - h ( a i + l ) [ < ZOsc(h , [a i ,a i+l] ) <_ Zvar (h , [a i ,a i+ lD < var(h,[al,an]). [] i i i

Note, that property (8) shows the correspondence between our generalized variation and the usual one-dimensional

variation.

We shall also need a multidimensional analog of inequality (2.1).

Lemma 2.3 ([2,3]). Let I d c X be a direct product o f d intervals: I d = I1 x • • • x Id (i.e., I d is a rectangle). Then

2 f [hi dmd var(hlld ) < 2var(h, I d) + minj IIj-----~l (2.2) J

X

for any function h : X --+ R 1 of bounded variation.

This statement follows from the estimate of the trace of a function on a boundary M a of the rectangle Ia:

f 2 f lhldmd, (2.3) [hi dmd-l < var(h, 1 a) + minj Ilj~l ai d x

where ma-1 is (d - 1 )-dimensional Lebesgue measure on the surface 81 d. It is worthwhile to remark that estimates

of this type can be obtained for arbitrary domains Y c X:

var (h lY) < A(Y)var(h, Y) + B(Y) f [hi dmd,

X

but the coefficients A (Y) and B(Y) depend crucially on the form of the domain and can be arbitrarily large even for

small domains (actually they even need not to be finite in the general case).

3. Operator approach for CMLs. Generic case

Operator approach for CMLs is based on the investigation of the Perron-Frobenius operator (transfer-matrix) P& describing the action of the CML Fe on densities of smooth measures on X d. For e _>_ 0 small enough this operator leaves the space of functions of bounded variation invariant, and the main idea, which was first proposed in [ 12] for the investigation of one-dimensional PE maps, is to obtain for arbitrary integers k estimates of the following type:

var(P~: h) < Cotkvar(h) + A[lh[I, (3.1)

M. Blank/Physica D 103 (1997) 34-50 41

where 0 < a < 1 and C, /~ < ec. Recall, that for the uncoupled map F (e = 0), the analogous inequality

var(PkF h) < C 0 ~ v a r ( h ) + A0 I[h II (3.2)

immediately follows from the well-known properties of the Perron-Frobenius operators for the one-dimensional

PE maps Ji. Now if the operator PFE satisfies inequality (3.1), then a standard technic, based on abstract ergodic

theorem due to Ionescu-Tulchea and Marinescu (see the suitable for this approach variant in [ 1 ]), gives a possibility to obtain the spectral decomposition of this operator as a sum of a contraction and a finite-dimensional projector,

and to obtain all the standard statistical properties.

Remark 3.1. The class of coupling maps ~ should be defined up to a nonsingular (piecewise smooth) conjugation.

Indeed, suppose that f := q)f~p-1, y = ~0x and x ~ ~ o Fx then

y ~ (~o o ~ o ~p-1) o # ,

where b y / " we mean a direct product of the conjugated maps j~:

P : = ~ p o F o ~ p -1.

Let us start from the situation without the periodic turning points. Consider the Perron-Frobenius operator, corresponding to CMLs

PF~ = QePF,

where the operator Qe corresponds to the coupling, and PF is the Perron-Frobenius operator for the uncoupled system. Suppose for a moment that these two operators are commutative. Then

pk k k 1~; = Q~PF"

From the definition of CMLs it follows that the coupling operator Q~ is close to identity in the following strong sense:

IIQ~II,, _< 1 + Be, (3.3)

where the constant B < ec does not depend on the parameter E > 0 for e small enough. Therefore

var(P~F h) _.< IIQk~ IvllekFhllo ~ (1 + Be)~(var(ekF h) + IIP~hl[)

< (1 + Be)k(Coot~var(h) + (Ao + 1)llhll)

=- (1 + Be)kCoot~var(h) + (1 + Be)~(Ao + l)[[hll.

Thus, choosing/~ large enough, such that

:---- C0(I + Be)kot~ < 1,

we shall have the main inequality with

C := C0(1 + Be) '~, ~ := ~l/~, A := (1 + Be)'~(Ao + 1).

Unfortunately, to complete this

these two operators are not commutative in the general case and some additional work is necessary program. We mainly follow here the idea introduced in our paper [4] to investigate random

42 M. Blank/Physica D 103 (1997) 34-50

perturbations of one-dimensional PE maps. Therefore we shall discuss in detail only the steps of the construction, where the difference between the multidimensional case (considered in the present paper) and one-dimensional

(considered in [4]) and between the types of the perturbations really takes place.

Let Jj := [cj, dj] C ffj := [cj, ~lj], j = 1 . . . . . r, be two families of one-sided interval neighborhoods of

the turning points. Here as in the sequel we write [U, v] = {x e X: u < x _< v or v < x < u}, in particular

[u, v] = Iv, u]. The ~ are chosen such that any two of them are disjoint. We write J = Uj Jj and j = U j -~ and

assume that for each j holds either f ( J j ) N J = 0 or f ( J j ) D Ji for some i in such a way that f (c j ) = ci but

f (d j ) q{ ~.. Consider a partition of the phase space X a into rectangles Aj such that the restriction of the map F to zlj is a

diffeomorphism (i.e., every Aj is a direct product of some intervals of monotonicity for the maps J~). Then

PFh = Z PF, jh,

where

PF, jh(x) := h (F j - lx ) l det(l~--lx)']lFAj (X), (3.4)

Fj := FIAj, and (Fx)' is the Jacobi matrix of the map F. In the sequel we shall consider the partitions of the unit interval into the intervals of monotonicity, consisting

of the intervals J and their complements. By our construction any rectangle Aj is a direct product of the intervals of monotonicity. The union of the rectangles Aj such that at least one of the intervals of the corresponding direct

product belongs to J we shall denote by Y. We shall study the behavior of FE on Y and on X d \ Y separately under

the assumption that transitions from X d \ Y to Y are restricted in the following sense: Define/51,/52 : BV ~ BV by

/51h := QePF(h" 1xd\y), /52h := QePF(h" 1y) . (3.5)

A straightforward calculation shows that if there are constants Cl, C2 > 0 and o t e (0, 1) such that

var(/Sjkh) < Clakvar(h) + C2llhll Vk E Z+ (3.6)

and that there is some N e 7/+ such that

/52/5(/52 = 0 Yk = 1 . . . . . N, (3.7)

then

var((QepF)Nh) < (1N (N q- 1)C~ q- C1)otNvar(h) 'b (1 + C1 q- C~)C2½(N -1- 1)2llhll. (3.8)

Given not only one perturbation operator but a whole family Q~, we will have to show that on our decomposition

X a = Y U (X a \ Y) of X depending on e, the corresponding operators/51,~ and/52,~ satisfy (3.6) uniformly in e l (i.e., with constants C1, C2, oe not depending on ~). Then there is N > 0 such that (½N(N + 1)C13 + Cl)ot N < ~,

and therefore

var((QepF)mh) < ½var(h) + C31lhtl

for some constant C2 < O~ and any function h e BV. Note that if there are no periodic turning points, then/52 k = 0 for some integer k that depends only on F (but

not on e), such that property (3.7) holds for j = 2. Therefore it remains to investigate only the properties of the operator/51.

M. Blank/Physica D 103 (1997) 34-50 43

Denote the restriction of the operator dE to the j th rectangle Aj of our partition by dE,j, and let us define a function J(x) := I det(Fj-lx)'l. Then

Oe.jh(x) = J(x)QePFh(F'x).

For the sake of simplicity we change the notation here and drop the index j , i.e., we shall denote Fj by F and A: by A. Let for some point y • A we have

var , var - - _< ¢3

and let BV0 := {h • BV(X), h(u) = 0: u e oxd}. Then Ilhlllc~ _< var(h) for all h • BV0.

Lemma 3.2. Let Q be a (sub)Markov operator satisfying

var(ah) < var(h) + C . Ilhll

for all h • BV0. Let

0 := PFi-' QPFi,

then Pp 0 = QPF and

var(Qh) < (1 +/~)2. var(h, .4) + C(I +/3)11JIIoo. Ilh 1,all

for all h • BV0.

Proof. Let g(x) := J(x) /J(y) .Thenvar(g) < ~, Ilgll~ _< 1 + ½~ and

Qh(x) = g(x) • QPFh(J(y) . h)(Fx)

because QPF is a linear operator. Therefore

var(Qh) < var(g). II QPF(J(Y)" h)ll~ + Ilglloo- var(QPF(J(y) , h))

< ,6.½var(QPF(J(y) . h)) + (1 + ½/3). var(QPF(J(y) , h))

< (1 +/~) . (var(PF(J(y). h)) + C. I[PF(J(Y)" h)ll)

( ( ( ' 7 ) " ) ) _<(1+/~). var .h o . l e a + C - I I J ( y ) . h - l A I I

_<,, ' . h . +

< ( 1 + / ~ ) . var I I h ' l , a [ l ~ + - - v a r ( h . l A ) + C . l l J I l ~ . l l h . l ~ l l

_< (1 + ¢~). (/~. ½var(h. 1A) + (1 + ½/~).var(h. 1~) + C . IlJIl~" IIh' 1AID

<_ (l + g)Zvar(h . lA) + (l + #)CIIJIl~ . llh . lAl[. []

Next we apply this lemma to the sequence { Fi } of branches of F.

Lemma 3.3. Let QI . . . . . QN be (sub)Markov operators with

var(Qjh) < var(h) + Cjl[hl[

44 M. Blank/Physica D 103 (1997) 34-50

for some constants Cj > 0 (j = 1 . . . . . N). Then there is for each k = 1 . . . . . N a (sub)Markov operator Ok such

that

Q1 eF~ Q2 PF2 " " Qk PFk = Pp, PF2 "" PPk Ok

and for each h 6 BV

var(Qkh) < (1 + 3fl)2k. var(h) + (~kllhll,

where

j=l i=j

The proof of this lemma can be done by induction.

Now combining this lemma with the PE property for the maps fi (i.e., there exists an integer N such that

~-N := mini {3.f,y } > 4), we obtain for each h 6 BV that

var(QIPF, Q2PFz...QNPF~vh)<_ l + - ~ f l ) /)~N-'}- Nfl ~'-N "var(h)+)~-ldNl[hll

Now choose fl so small that 2(1 + 3~)2N/)~N + Nfl/~. N < (3/)~U) l/u < 1 and observe that )~N > 4. This

proves the proposition for k = N with C1 = 1, ~ = (3)l/U and some C2 = C2(N, C, f ) . Iterated application of

this inequality extends it to integer multiples k of N with a new C2 = [1/(1 - ~)]C2,ola. The extension to general

k follows from the elementary inequality

2 4 var(/Slh) < -~var(hlxd\y) + (C + Const)llh[I < ~var(h) + (C + Const)llhll

with a constant Const depending only on F.

Remark 3.4. To obtain an arbitrary small value of/3 above it is enough to choose fine enough subpartitions for our

maps fi .

This finishes the proof of Theorem 1.2. It may seem that the same idea could be applied for a general multidimensional PE map. However, there are two

obstacles. The first one is ptire topological existence of periodic turning points, which we shall discuss in Section 4.

The second obstacle is due to the fact that in the general case elements of the partition for F (and especially for F n) are not rectangles. As we mentioned earlier this may lead to large (and even infinite) coefficients in the corresponding

estimates for the Perron-Frobenius operator PF.

4. Operator approach for CMLs. Localization for the case with periodic turning points

Consider now the case when Turn[fi] ~ 0. To prove Theorems 1.3 and 1.4 it is enough to construct examples

of CMLs with periodic turning points, satisfying the assumptions of these theorems. We shall show that this can be

done even in the simplest two-dimensional case (d = 2) with identical maps f l = f2 = f , having only two fixed turning points cl = c, c2 = l - c and the following symmetrical coupling:

c = f ( c ) ~ l - - c = f ( 1 - c ) , Yll = Y22 = O, ),'12=),'21 = 1 .

M. Blank/Physica D 103 (1997) 34-50 45

\ J/ I

c c + c /b 1 - c

(a)

i••// I \ \ / '~k

~ \ i / \ l \

~. / \ 1 I

/ l ~ \ l I / I " ~ I

c 1/2 1 - c (b)

Fig. 3. Local maps: (a) f~l:, (b) ¢12) • Jb,c"

1 - b

II

I I I

I I

I,I

I I I C_ C+

(~)

I I I

SL ~ - -

IT--- I I II ,

l - - e _

~ m

I

K

e_ c.

~"~ ~b) I~!~! Fig. 4. Space localization: (a) Jb,c'

I I I

I I I

I

L l - - C _

(b)

Consider the following two families of maps (see Figs. 3 and 4):

f~l c) (x) =

1 x / 2 i f 0 < x < c - c / b ,

2 c - c / b '

b ( x - c ) + c , i f c - c / b < x < c ,

- b ( x - c) + c, i f c < x < c + c / b ,

x + c + c / b 1 - - 2 " 1 - 2 c - 2 c / b ' i f c + c / b < x <

1 - f ( 1 - x ) , i f l < x < l .

46 M. Blank/Physica D 103 (1997) 34-50

b - c - x + b , i f 0 < x < c ,

c

f t Z ~ ( x ) = c I - - x + 0.5, i f c < x < ~, 0.5 - c

1 - f ( 1 - x ) , i f I < x < 1.

{c c I )ce(t) : = b > 1, ,~f(2) : = min , - - < 1, ~.(f~2))2 : = > 1.

Jb,c b,c 112-- C C b,o 1/2 - c

.(1) Let us start the investigation from the first family Jb,c" Fix some 1 < b < 2, 0 < c < ½, e > 0. Then the

expanding constant Xf = b > 1. Consider the first two points of the trajectory of the point (c, 1 - c) of the CML

FE constructed by means of this map:

1 - c 1 - c - e(1 - 2 c ) / 1 - c - e b ( 1 - 2 c ) /

q,~ ( c - e ( b ( 1 - 2c) + 2 c - 1 ) + 2e2b(l - 2c) ) ),

1 - c + e(b(1 - 2c) + 2 c - 1) - 2eZb(l - 2c)

Denote now

c_ : = c - e ( b - 1)(1 - 2c), c+ : = c + e(1 - 2c),

K1 : = [ c _ , c + ] x [1 - c + , 1 - c _ ] , K2 : = [1 - c + , 1 - c _ ] × [ c _ , c + ] .

Lemma 4.1. Fe Ki C Ki for every i = 1, 2.

Proof The restriction of the map Fe to the rectangle K1 is a continuous map, such that the boundary of the rectangle

Kl is mapped by F~ into Kl and at least one inner point of K1 (the point (c, 1 - c)) is also mapped by F~ into

Kl . To show this, it is enough to prove that every comer of the rectangle Kl is mapped into Kl . Let us prove it

for the left lower comer (c_, 1 - c+). Denote t : = e(1 - 2c), /7 := b - I < 1, y : = 1 - 2c + tb + tb/7. Then

c_ = c - t/7, c+ = c + t and

c- c - t/7 c - bt/7 ~ c - bt/7 + ey ( 1 - c + ) = ( 1 - c - t ) f ' ( 1 - c + b t l " ~ ' ( 1 - c + t b - e y l "

= ( c - t b + t b - t b / 7 + e y ~ ( c - t b ) ( c- = , 1 - c - t + t b ( 1 - e ( l + / 7 ) ) ] \ l - c - t l - c +

where by ~ > r / w e mean that every coordinate of the vector ~ is greater than the corresponding coordinate of the

vector O.

On the other hand,

( c - b t / 7 + e y ) ( c + t - b e ( ( 1 - 2 c ) , - t - t / 7 ) ) ( c + t ) ( c__+ ) 1 - c + tb - ell 1 - c + t - e((2 - b)(1 - 2c) + tb + tb/7) 1 - c + t 1 _

In the same way one can show that three other comer points are mapped into the rectangle Kl . The proof for the

rectangle K2 is analogous. Therefore these rectangles are mapped into themselves under the action of Fe. []

Lemma 4.2. The restriction of the map Fe to a rectangle Kl (K2) has a smooth invariant measure (SBR measure)

#1 (#2).

M. B l a n k / P h y s i c a D 103 ( 1 9 9 7 ) 3 4 - 5 0 47

Proof Denoting cl = c, c2 = 1 - c we consider for e > 0 the following change of variables:

x i - - c i Yi . - - - - , (4.1)

E

which gives new local maps j~ from the neighborhoods of the points ci (whose direct product gives the rectangle

K i) into some new intervals with lengths of order 1:

( eyl+c )__~ ( (1-e ) ( -eb ly l l+C)+e(eb ly21+l-c ) ) . -eye + (1 - c) \e(-eblyll + c) + (1 - e)(ebly2[ + 1 c)

Thus denoting by a := (1 - 2c)/(1 - 2e) > 0 we obtain

( ( 1 - e ) ( - b l Y l [ W a ) + e ( b , y 2 1 - : I ) "

( ; i ) ~ \e(-bly,] +a)+ ( 1 - e ) ( b l y 2 ,

This new system is also a CML, constructed by the maps:

f l(Yl) := --blyl[ +a, f2(Y2) := bly21 - a .

The shape of the new local map f l in a neighborhood of a fixed turning point of the original map is shown by a

broken line in Fig. 5(a). The maps f l are PE maps with the expanding constant b > 1 and without periodic turning

points. Note that the constant a depends on e and is close to 1 - 2c for e small enough. Therefore for a small positive

fixed e we can apply here the results of Theorem 1.2 to prove the existence of the smooth invariant measure for any

e > 0 small enough. To finish the proof it is enough to do the inverse change of variables. []

Lemma 4.3. There are only two SBR measures #i , i = 1, 2, of the map F~ and their supports supp(#i) C K i .

Proof. Let us construct a new map F from X 2 into itself, which will differ from the map F only on the rectangles

K i "

I /

C ~ S C

S- - ,S

Fig. 5. (a) Local action of coupling near fixed turning points, (b) bifurcation diagram for the map ¢(2) 0 < b < 1, 0 < c < ½. d b , c , . . . .

48 M. Blank/Physica D 103 (1997) 34-50

F ( x ) - 2 e ( 1 , - 1 ) i f x E KI,

~'(x) = F(x ) + 2e(1, - l ) i fx E K2,

F (x ) otherwise.

Then for e > 0 small enough the map Pe := ~E o/~ is a PE map. The difference between this map and the map

Fe is that it has no "traps" around turning points c and 1 - c. Now using the same argument as in the proof of

Theorem 1.2 one can prove the existence and the uniqueness of a smooth invariant measure #,~ of this map, which

will have a positive density on the whole rectangle X 2. Therefore a.a. trajectory of the map/~E is dense on X 2.

Thus a.a. trajectory of the original map FE hits eventually into one of the sets Ki, because it coincides with some

trajectory of the map FE up to the moment, when it hits into one of the sets Ki. []

This finishes the proof of Theorem 1.3. f(2) We shall use the same notation as for the In a close way one can also investigate the second family of maps Jb,c"

first family. Fix some 1 < b < 1, 0 < c < ½, e > 0. Then the expanding constant ~.f2 = (b - c ) / ( ) - c) > 1,

however, ~,f • min{(b - c)/c , c/(½ - c)} may be less than 1. Consider the first two points of the trajectory of the

point (c, 1 - c) of the CMLs Fe constructed by means of this map:

1 - c l - c - e ( 1 - 2 c ) 1 - c - 2 ~ c 1 - c + e ( 4 c - 1 ) - 4 e 2 c ] "

Denote now

c_ := c - e ( 4 c - 1), c+ := c + e(1 - 2c),

KI := [c_, c+] × [1 - c+, 1 - c_], K2 :----- [1 - c+, 1 - c_] × [c_, c+].

Lemma 4.4. Let 4c < 1, then FeK i C Ki, i = 1, 2.

The proof of this statement is analogous to the proof of Lemma 4.1.

Lemma 4.5. Let 4c < 1, then restriction of the map Fe to a rectangle K~ (K2) has a globally stable fixed point

Pl (P2)-

Proof The main difference between the considered situations and the result of Lemma 4.2 is that the restriction of

this CML to one of the rectangles Ki is not a PE map, because for any of its iterations the expanding constant is

strictly less than 1. Indeed, the derivative of the local map is equal to c / ( 1 - c) < 1 for 4c < 1, and the coupling

can only decrease the expansion. []

Theorem 1.4 is proved. Using the same argument as in the proof of Lemmas 4.1 and 4.2, one can show that the situation will be quite

different when 0 < b < ½. The point is: in this region of parameters the uncoupled system is not expanding and

has two locally stable fixed points.

Lemma 4.6. Let 4c > 1 and 0 < b < / , then FeKi C Ki, i = 1, 2 and CML Fe has two singular SBR measures with the supports on cycles of period 2.

The bifurcation picture for the second family of maps is shown in Fig. 5(b). The uncoupled system is chaotic when b > ½ and stable otherwise. We denoted these two possibilities by letters "C" and "S". Consider the coupled

M, Blank/Physica D 103 (1997) 34-50 49

system for a small enough e > 0. Lemmas 4.4 and 4.5 show that when 4c < 1 there is a transition from the chaotic

behavior to the stable one (C ~ S). Lemma 4.6 describes another transition from stable fixed points to stable

period-2 cycles (S --+ S). When 4c < 1, 0 < b < ½ or 4c > 1, ½ < b < 1 there are no "traps" around the fixed

turning points and therefore there are no "phase" transitions. Generalization of our results to the general case of periodic turning points (instead of fixed points) and arbitrary

dimension d is straightforward and it is easy to formulate some simple sufficient conditions of the localization

phenomenon. However, any CMLs, whose local map has only one fixed turning point, shows the absence of the

localization. Therefore the investigation of necessary conditions is a more complex problem and will be published

elsewhere.

5. Localization in CMLs, constructed by smooth maps

Actually the appearance of the localization in CMLs is quite unexpected and may seem to be a consequence of

the discontinuity of PE maps. To show that it is not so and that this phenomenon is not artificial and is generic,

we shall prove in this section its existence for smooth local maps. The simplest way to do it is to smooth the maps

f/i~ near periodic turning points. However, we shall consider a more general situation - a well-known family of

quadratic maps f , ( x ) := ax(1 - x), 3 < a < 4. This situation differs from the results of Theorems 1.3 and 1.4

in the sense that the localization takes place only for nonzero coupling strength. Let the value of the parameter a

be close enough to 4 and the map f , have a smooth SBR measure. It is well known that the set of such values of

the parameter a is of positive Lebesgue measure. We shall consider a CML, constructed by means of two identical

square maps with the parameter a.

Lemma 5.1. There exists a constant 3 < a0 < 4, and an interval of values of the coupling strength 0.14 < el <

e < e2 < 0.2 such that the CML Fe has a stable periodic trajectory with period 2:

F,.(pl, pz) = (p2, pl); F~.(pz, p l ) = (pi , p2),

~here Pl < ½ < P2 < 1 and fat(Pl) x Ifa1(p2)l > 1, and therefore an SBR measure, localized on this trajectory.

Proof We can follow the same construction as in the case of PE maps to find two "traps" for CMLs, but in the case of

square maps this way is too complex. Therefore we shall use the fact that the map f,, is smooth and smoothly depends

on the parameter a. Therefore it is enough to prove the existence of a stable periodic trajectory for a given pair of

values (a, e). Let us set a = 4, e = 0.17. A simple calculation shows that the pair of points pl = 0 .484989. . . and

P2 = 0 .893799. . . defines the periodic trajectory with period 2 for this pair of values (a, e). To prove its stability

~.e calculate Jacobian matrices of the CMLs at these points and show that eigenvalues of their product are positive

and less than 1. The Jacobian matrices and their product are

0.0997 0 . 5 3 5 6 " ] × ( 2 . 6 1 4 8 0.0204"~ : (0 .5476 0 . 0 6 4 6 ) ,

0.0204 2.6148} \ 0 . 5356 0.0997} \1 .4538 0.2904

The eigenvalues of the product matrix are 0.7512 and 0.0867. []

References

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50 M. Blank/Physica D 103 (1997) 34-50

[3] M.L. Blank, Singular phenomena in chaotic dynamical systems, Dokl. Akad. Nauk (Russia) 328 (1) (1993) 7-11. [4] M.L. Blank and G. Keller, Stochastic stability versus localization in chaotic dynamical systems, preprint Schwerpunktprogramm der

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[10] G. Keller and M. Kunzle, Some phase transitions in coupled map lattices, Physica D 59 (1992) 39-51. [11 ] M. Kunzle, Invariante Masse for Gekoppelte Abbildungsgitter, Ph.D. Thesis, Erlangen-Nurnberg University (1993). [ 12] A. Lasota and J.A. Yorke, On the existence of invariant measures for piecewise monotone transformations, Trans. Amer. Math. Soc.

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