The geometry of chaos synchronizationcomplex.gmu.edu/people/barreto/publications/chaos/chaos.pdf ·...

CHAOS VOLUME 13, NUMBER 1 MARCH 2003

The geometry of chaos synchronizationErnest Barretoa)

Department of Physics and Astronomy and the Krasnow Institute for Advanced Study,George Mason University, Fairfax, Virginia 22030

Kresimir JosicDepartment of Mathematics and Statistics and the Center for BioDynamics, Boston University,Boston, Massachusetts 02215

Carlos J. MoralesDepartment of Mathematics and Statistics, Boston University, Boston, Massachusetts 02215

Evelyn SanderDepartment of Mathematical Sciences, George Mason University, Fairfax, Virginia 22030

Paul SoDepartment of Physics and Astronomy and the Krasnow Institute for Advanced Study,George Mason University, Fairfax, Virginia 22030

~Received 16 April 2002; accepted 16 August 2002; published 21 February 2003!

Chaos synchronization in coupled systems is often characterized by a mapf between the states ofthe components. In noninvertible systems, or in systems without inherent symmetries, thesynchronization set—by which we mean graph~f!—can be extremely complicated. We identify,describe, and give examples of several different complications that can arise, and we link each toinherent properties of the underlying dynamics. In brief, synchronization sets can in general becomenondifferentiable, and in the more severe case of noninvertible dynamics, they might even bemultivalued. We suggest two different ways to quantify these features, and we discuss possiblefailures in detecting chaos synchrony using standard continuity-based methods when these featuresare present. ©2003 American Institute of Physics.@DOI: 10.1063/1.1512927#

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Since the surprising discovery that chaotic systems cansynchronize,1,2 many different kinds of nonlinear syn-chrony have been considered in the literature.3 We focushere on the geometry of synchronization structures thatarise in such systems and point out several complicatingfactors that are not generally appreciated. These morecomplicated states may arise in noninvertible systems, insystems with critical points, and in systems of signifi-cantly dissimilar nonlinear elements. An important caseis that of neuronal systems, which consist of many verydifferent interacting classes of neurons, each of which exhibit great variation in their morphology. Nontrivial syn-chronous relationships in such systems may have physological significance, and may even correspond toperceptual events.4 We identify several different geomet-ric features that can arise in these more complicated situ-ations, giving several examples and providing an intuitiveexplanation for each case. A good understanding of thesinherent mathematical features is important for theproper interpretation of experimental data, especially insituations in which the detection and classification of suchsynchronous states is of interest. In particular, we findthat most existing methods for detecting synchrony5–7

will be hampered by the inherent geometric features thatwe identify.8

a!Web site: http://complex.gmu.edu

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I. INTRODUCTION

In this work we consider a general drive/response sysF:X3Y→X3Y, where (xn11 ,yn11)5F(xn ,yn) and is ofthe form,

xn115f~xn!,

yn115g~xn ,yn ,c!. ~1!

The drivexPX and the responseyPY are state vectors,XandY are compact finite dimensional spaces, and bothf andg are smooth or piecewise smooth maps. The parametccharacterizes the coupling or interaction strength. We expthat many of our observations hold in the case of bidirectially coupled systems.9,10 Unidirectionally coupled flows maybe reduced to this form via a Poincare´ or time-T map.

Generalized synchrony2,7 is a useful concept in theanalysis of coupled nonlinear systems, and is usually defiby the existence of a smooth, continuous map betweenphase spacesX andY of the two component systems. For opurposes, we will be concerned with the mapf:X→Y whichassociates astateof the first system with astateof the sec-ond such that graph~f! is invariant and attracting under thevolution of the coupled system.11 Throughout this work, wewill refer to the synchronization set, by which we meangraph~f!.12 If the underlying dynamical equations have iherent symmetry, graph~f! typically has a simple structureFor example, in the frequently studied case of couplediden-tical oscillators, thex5y symmetry plane is invariant. In this

© 2003 American Institute of Physics

P license or copyright, see http://ojps.aip.org/chaos/chocr.jsp

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152 Chaos, Vol. 13, No. 1, 2003 Barreto et al.

case, when the components are synchronized, trajectorieattracted to the plane of symmetry, and the synchronizaset is trivial.

The continuity and smoothness off are important con-siderations, especially with regard to the practical detecof nonlinear synchronous states in experimentally obtaidata. The existence of such states has been demonstraboth physical13 and biological systems.6 However, we illus-trate below that many systems of current interest exhibit schronization sets with geometric complications that can ha detrimental effect on the detection of nonlinear synchrofrom measured data. In particular, noninvertible coupled stems and systems lacking intrinsic symmetries can exhsynchronization sets with very complicated structures. Simismatches among coupled components are unavoidabindeed heterogeneous coupled systems are of more geinterest, especially when considering biological systemsgood understanding of the geometry of the synchronizasets of these more general systems is needed.

A closely related concept isasymptotic stability.14 A uni-directionally coupled system is asymptotically stable if tw~or more! identical copies of the response synchronwhen subjected to the same driving signal. More precisfor any two initial conditions (x0 ,y08),(x0 ,y09)PX3Y whichshare the same initial drive statex0 , we havelimn→`iyn(x0 ,y08 ;c)2yn(x0 ,y09 ;c)i50, whereyn(x0 ,y0 ;c)is the y coordinate of thenth iterate of (x0 ,y0) under thedynamics of the full system, andc is the coupling betweenthe driver and response systems. This is similar to the ide‘‘reliable response’’ in the generation of neuronal signals15

Asymptotic stability and generalized synchrony are equilent if the driving system isinvertible and has a compacattractor.16 We have observed, however, that it is possiblehave asymptotic stability in systems which are noninvertibfor which graph~f! is not even single-valued.17

In this paper we propose a categorization of the strtures which arise in such synchronization states, andthese structures to specific features of the underlying dynics. In Sec. II we describe and illustrate these categoriesing a general driver/response dynamical system. In Sec.two methods of quantifying these structures is presented,we review the implications of these structures for the pracal detection of synchronization from measured data. Ccluding remarks appear in Sec. IV.

II. CATEGORIZATION

It is convenient to use a specific form of Eq.~1! to illus-trate the various cases of interest. In the following, we tathe driving system to be a modified baker’s map and ussimple filter of one of the driver variables for the responSpecifically,

un115H lv~un ,s!, vn,a,

l1~12l!un, vn>a,

vn115H vn /a, vn,a,

~vn2a!/~12a!, vn>a,~2!

yn115cyn1cos~2pun11!,


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where 0,l,1 and 0,a,1. The driver, by which we meanthe (u,v) subsystem, maps the unit square into itself aslustrated in Fig. 1~a!. The functionv(u,s) is a cubic poly-nomial in u defined by the following conditions:v(0,s)50; v(1,s)51; v(1/2,s)51/2, and ]v/]u(1/2,s)5s,where 21<s<1. This function, graphed in Fig. 1~b! forvarious values of the parameters, determines how the arewithin the lower rectangle of Fig. 1~a! is redistributed hori-zontally on each iterate. The responsey is a filter of thedrive’s variableu with c controlling the transverse contraction rate. The parameterc, although technically not a coupling parameter in Eq.~2!, is meant to model the more general case in which the dominant dynamical effect of couplis to alter the contraction rate transverse to the synchrontion set. One should also note that the limit atc50 is some-times singular. In particular, nontrivial synchronous strutures such as cusps and multivalued sets described innext section will persist for all values ofc except atc50. Atc50, the synchronization set collapses to the curvey5cos(2pu). Below, we varys and c to obtain various syn-chronization sets with nontrivial geometric structures. In pticular, we will be concerned with the mapy5f(x) whichassociates astate of the drive system with astate of theresponse such that graph~f! is invariant and attracting undethe evolution of the coupled system.

A. Wrinkling

The first type of nontrivial structure is best illustratewith s51, so thatv(u,s)51 and the drive reduces to thstandard baker’s map. Referring to Fig. 1~a!, this corre-sponds to uniform horizontal contraction and vertical streting of the shaded rectangle. This case has been studieRefs. 18 and 19, and we include it here for completenes

If, in Eq. ~2!, ucu,1, the response is asymptoticalstable for allx. As pointed out in Refs. 18 and 19, the sy

FIG. 1. ~a! The modified generalized baker’s map used as the driver in~2!. The functionv(u,s) determines how the area of the lower rectangle~a! is stretched and contracted horizontally, as schematically depicted bshading.~b! Plot showingv for three values of the parameters.


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153Chaos, Vol. 13, No. 1, 2003 The geometry of chaos synchronization

chronization set is typically not differentiable if the averacontraction within the synchronization set~as determined bythe drive! is larger than the contraction transverse to it~asdetermined by the response viac). In particular, lethd be themost negative past-history Lyapunov exponent20 of the driveand lethr be the Lyapunov exponent corresponding to

FIG. 2. Plots of the synchronization set of Eq.~2!. The insets show thefunction v. We sets51, l50.8, anda50.7, for whichhd520.64. Notethat hr5 ln c. ~a! Smooth case;c50.3 anduhdu,uhr u; ~b! wrinkled case;c50.8 anduhdu.uhr u. The curve in~b! is Holder continuous with exponenuhr /hdu;0.35.


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transverse contracting direction. Ifuhr u,uhdu, thenf is gen-erally not differentiable, but is only Ho¨lder continuous withHolder exponent21 equal touhr /hdu,1 at typical points.

Since the attractor of the generalized baker’s map is uform in v, the synchronization set can be accurately visuized in theuy plane. Graphs demonstrating both the diffeentiable and nondifferentiable cases are given in Figs. 2~a!and 2~b!, respectively. We call the development of nondiffeentiability in this fashion ‘‘wrinkling.’’

One can gain intuition about the wrinkling processconsidering the following iterative geometric constructionthe synchronization set. This construction is the essencthe graph transform method used first by Hadamard and lby a number of mathematicians to prove the existenceinvariant manifolds.22,23Begin with the graph of cos(2pu) inthe unit interval.~The choice of the initial curve is not important; any smooth curve with appropriate boundary contions may be used.! Next, join two copies of the initial curveat u5l, each scaled vertically byc and horizontally byland 12l, respectively. Note that since the initial curveperiodic in the interval and its first derivative is zero at boends, the combined curve is continuous in its value andfirst derivative at the connection point. Finally, adgraph(cos(2pu)) to the result to obtain the image of the intial curve under one complete iteration. Figure 3 illustrathe procedure. This entire process is then repeated to obfurther iterates of graph(cos(2pu)).

The above process produces a sequence of curveslimits to the synchronization set observed in Fig. 2. Theretwo competing factors responsible for the wrinkling. Onethe vertical scaling factorc, which for ucu,1 tends to reducethe slope of thenth-stage curve. The other is the horizontcompression, which tends to increase this slope. Whethenot the slope at a given point in the limiting set is bounddepends on the competition between these two opposingtors. If the horizontal compression is stronger than the ve

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FIG. 3. Iterative geometric construction of the synchronization set of Eq~2!. ~a! Begin with graph(cos(2pu)).~b! Two copies of the curve, eachscaled vertically byc and horizontallyby l and 12l respectively, are joinedat u5l. c50.3 andl50.8 are usedfor this example. ~c! The resultingcurve is then added tograph(cos(2pu)), giving the resultshown ~note the vertical scale!. Theprocess is then repeated.


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FIG. 4. Sequence of curves resultinfrom the iterative graph transform procedure forl50.8 andc50.3 ~smoothcase!. ~a! Initial curve; ~b! result afterone iteration; ~c! two iterations; ~d!eleven iterations.

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cal compression, the slope will tend to grow, and the curvethe limiting set will not be differentiable. Figures 4 andillustrate the smooth and the wrinkled cases as sequenccurves generated by the graph transform procedure. Insequences, the initial curve and the resultant curves aftefirst, second, and eleventh iterations are shown.~Moviesshowing the development of the synchronization set usthe graph transform procedure are available on the web24!As one can see from these two sequences, the iterated capproach the actual synchronization set after only a fewerations. Note that the curve that is obtained after any finumber of iterations is differentiable (C1), but the limitingcurve in the wrinkled case is only Ho¨lder continuous.

The wrinkling of the synchronization manifold is alocalfeature, and the smoothness in the vicinity of a single odepends on the ratio of the exponentshr and hd along this


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orbit. Thus there typically exist invariant sets embeddedthe synchronization set on whichf has differing degrees oregularity. As we will demonstrate below, there are situatioin which nondifferentiability on these smaller sets may bcome important. A quantification of this multifractality igiven in Sec. III.

The degree of wrinkling is also closely connected to tconcept of ‘‘reliable response’’ in the generation of neuronsignals. A system is said to respond reliably to an input ifresponse is identical each time the input is presented.~Ex-perimental evidence for such behavior in neuronal tissuebeen reported in Ref. 15.! Consider two trajectories beginning at nearby initial conditions in the drive system. Torbits of the drive will remain close to each other for a timt'1/h, where h is the largest Lyapunov exponent of thdriver. If the synchronization manifold is approximate

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FIG. 5. Sequence of curves resultinfrom the iterative graph transform procedure for l50.8 and c50.8~wrinkled case!. ~a! Initial curve; ~b!result after one iteration;~c! two itera-tions; ~d! eleven iterations.


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smooth, then the difference between the two orbits ofdrive will not affect the response system in significantly dferent ways during this time. However, if the synchronizatiset is severely wrinkled, then any small difference betwethe two states of the drive will be amplified in the responeven for times shorter than 1/h. Thus, a reliable responscannot be expected.

B. Cusps

The second type of structure that can develop withinsynchronization set results from the presence of critpoints in the drive. At such a point the Jacobian matrixsingular, and we expect to find orbits inX near the criticalpoints along which the contraction is arbitrarily large.

We illustrate this situation using the map in Eq.~2! withs50. In this case, the contraction rate in theu direction isnot uniform. Instead,v, which remains invertible, has a critcal ~inflection! point atu51/2, and thus the contraction raalong the lineu51/2 is infinite. As in the previous example

FIG. 6. Cusped case withs50 andc50.2. The inset shows the shape ofv.We setl50.2 anda50.3, for which hd520.90 anduhdu,uhr u. Cuspsoccur at the forward iterates of the critical point atu51/2 and decrease insize due to the cumulative effect of the transverse contraction.


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the synchronization set can be visualized as a graph in theuyplane. The result for the current case is shown in Fig. 6.choosec sufficiently close to zero such that numerically, wfind uhr /hdu.1 for a typical orbit. This suggests thagraph~f! is smooth almost everywhere. However, graph~f!is not completely smooth, since ‘‘cusps’’ are formed at poincorresponding to the orbits ofu51/2 that are affected byv,i.e., that visit the shaded rectangles of Fig. 1~a!. Thus, themain cusp occurs atu5l/2. The next largest cusp occursu5lv(l/2,0). More cusps appear at subsequent iterabut these get progressively smaller in the figure becausthe cumulative effect of the transverse contraction. Note tthe Holder exponent at each cusp is zeroregardlessof c. Thesignificance of critical sets in general noninvertible mapsone and higher dimensions has been considered by Miraco-workers~see Ref. 25, and references therein!.

In addition, we have found through numerical measeveral periodic orbits sufficiently near cusps such tuhr /hdu calculated along the orbit is less than one. We aexpect that there are aperiodic orbits with the same propeFor these orbits, however, the ratiouhr /hdu depends on thevalue ofc, and the size of the set of such points decreasethe rate of transverse contraction increases.

The iterative geometric construction discussed inprevious section applies to the current case with only ochange: the horizontal rescaling is no longer linear, buinstead specified bylv for the left ‘‘copy’’ of the previousstate. A sequence of pictures showing the first few stagethe construction of the synchronization set using the grtransform method is shown in Fig. 7. Beginning again wgraph(cos(2pu)) as the initial condition, the resulting curvis seen to begin to resemble the actual synchronizationafter only two iterations~compare Fig. 6!.

Although graph~f! is not smooth in either the cusped othe wrinkled case, its global structure in the two casesdifferent. The occurrence of wrinkling depends on tstrength of the contraction rate in the direction transverse

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FIG. 7. Iterative geometric construction of the cusped synchronization seshown in Fig. 6.~a! The initial curve isgraph(cos(2pu)); ~b! result after oneiteration; ~c! two iterations;~d! eleveniterations.


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the synchronization set. In particular, graph~f! is everywheredifferentiable forc,min(l,12l) and is nonsmooth otherwise. In our second example, the functionv has a criticalpoint atu51/2 regardless of the value ofc. Hence the driv-ing system has a critical line, and the infinite contractionthe vicinity of this line ~and its forward iterates! leads tocusps in the synchronization set where the Ho¨lder exponentvanishes. We emphasize that this occurs forall values ofc,in contrast to the first example. In addition, we expect incusped case that there is typically an additional small sepoints along which the synchronization set is only Ho¨lder,and that the size of this set decreases with increasing trverse contraction.

In Sec. III, we introduce statistics which describe tdifferences between these two cases in a more quantitafashion.

C. Multivalued synchronization sets

In this section we discuss the development of multivued synchronization sets. Such structures are associatedthe presence of noninvertibility in the underlying equatioNoninvertible mathematical models are very importantnonlinear dynamics, despite the usual assumption that phcal processes are fundamentally described by inherentlyvertible ordinary differential equations. The use of the logtic map in the study of population dynamics in biology26 is awell-known example. More generally, dynamics recostructed from measured time series of systems with strdissipation are frequently best and most usefully appromated by noninvertible maps.27 Models with time-delays,important for describing neuronal and more general biolocal processes, are even more complicated since a propescription requires delay differential equations; temporalvertibility in these systems cannot be taken for granted.28

The synchronization set can be characterized as theof points that have preimages within the boxX3Y N stepsinto the past for every positive integerN. That is, a point(x,y) is in the synchronization set ifF2N(x,y) stays withinthe box X3Y for every positive integerN.29,30 From thispoint of view, the connection between the multivalued natof a synchronization set and noninvertibility can be undstood as follows. A noninvertible drive implies that there adrive states that have more than one inverse image undf.Consider such a drive statex. Associate the set of all possibly values with each of the preimages ofx. Typically, theseiterate forward underF to a disjoint union of sets ofy-valuesassociated withx such that each component of this uniocorresponds to one preimage ofx. This is depicted schematically in Fig. 8~a!. The above argument may be repeated usthe set of preimages underf of drive statex j steps into thepast, for j 52,3,. . . . This yields a disjoint union of an increasing number of smaller sets ofy-values that is associatewith the drive statex, as shown in Fig. 8~b! for j 52. Thus,fis multivalued. Intuitively, each particularj th preimage ofxgives rise to a different orbit that lands onx after j iterates.These different trajectories~also called histories! provide dif-ferent driving signals to the response system, and thereonce the drive lands onx, the response can be in any


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several possible states. Multivalued synchronization shave been described in Refs. 8 and 17 and observed exmentally in Ref. 31. A different and less severe form of mtivalued synchronization, in which the drive and responserelated by a 1:m ratio, has been recently reported.32,33

We now give several examples of this phenomenFirst, we note that the driver in Eq.~2! is noninvertible whenv has a negative slope atu51/2, i.e., fors,0. The synchro-nization set fors521/2 is illustrated in Fig. 9. Note that as is progressively decreased from 1 to 0 to21/2, the syn-chronization set goes from being smooth to having cusps;cusps then ‘‘push through’’ to form loops. Thus the synchnization set becomes multivalued. An animation of this pcess is available on the web.24

The development of the multivalued synchronizationin this case is particularly clear in terms of the iterative gemetric construction described above. Beginning again wthe graph of cos(2pu), the first step in the construction is trescale this curve vertically byc and horizontally bylv. Thelatter step can be thought of as occurring in three sepapieces as indicated in Figs. 10~a! and 10~b!; the result is theformation of the looped curve shown in Fig. 10~c!. This isthen joined to another copy of the original curve which

FIG. 8. The connection between multivalued synchronization sets andinvertibility. ~a! The statex of the drive has two preimages underf asshown. Each preimage is then associated with all possible response valuy;this situation is then iterated forward under the full dynamicsF. The resultis two typically disjoint sets ofy values, both associated with the drive stax. ~b! The same argument considering two steps into the past.

FIG. 9. Multivalued ‘‘looped’’ case withs520.5 andc50.3. The upperinset shows the shape ofv. We set l50.2 and a50.3, for which hd

520.7 anduhdu,uhr u. The lower inset magnifies the largest loop, in whicadditional loops, going both inward and outward, are evident. The appagaps in the curve are due to finite iteration.


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FIG. 10. Iterative geometric construction of the multivalued synchronization set shown in Fig. 9.~a! The functionv(u,21/2), which is noninvertible, has twocritical points atuc

1 and uc2 as shown. The images of these critical points under the cubic functionu85v(u,21/2) arevc

1 and vc2 . ~b! The initial curve

cos(2pu) in the iterative process. It is useful to consider the horizontal stretching and contraction by the factorlv as occurring in three pieces as showSpecifically, the curve in region I maps to the curve starting fromu850 to u85vc

1 ; the curve in region II maps to the U shaped curve in reverse order withe domainu8P@vc

2 ,vc1#; and the curve in region III maps to the curve starting fromu85vc

2 to u851. The result is shown in~c!. To complete one cycle ofthe iterative process, the curve resulting from part~c! is added to graph(cos(2pu)), giving the result shown in~d!. The process is then repeated.

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scaled uniformly both vertically and horizontally, and thresult is added to graph(cos(2pu)). The curve after one fullstep of the iterative process is shown in Fig. 10~d!. Note thatin the course of iteration, it is possible to obtain loops withloops. This feature is evident in Fig. 9; see the lower inswhich enlarges the main loop.

Another way to introduce noninvertibility into Eq.~2! isto allow the rectangles depicted in Fig. 11 to overlap. Rethat for s51, the driver in Eq.~2! reduces to the standarbaker’s map. Rewriting this system to explicitly account fthe overlap, we consider the following system:

un115H ~l1r~12l!!un, vn,a,

~12r!l1~12~12r!l!un, vn>a,~3!

vn115H vn

a, vn,a,

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yn115cyn1cos~2pun11!,

where rP@0,1# determines the degree of overlap. Figur11~a! and 11~b! show the synchronization sets that resultr50.3 andr50.8 with c50.3.


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We now give an example for which the synchronizatiset is multivalued for all drive states, and the structure canunderstood exactly. Consider the following two-dimensionpiecewise linear system:

xn115 f ~xn!5H 2xn, xn,0.5,

2~xn20.5!, xn>0.5,

yn115g~xn ,yn ,c!5cyn1xn11 , ~4!

where f is noninvertible with two preimages for eachxn11 .For ucu,1, the system is asymptotically stable. Figureshows the synchronization set, which consists of a selines. The topology in this case is unusual because the dis not continuous; we expect that for the more typical casea continuous noninvertible drive-response system, the schronization set will be connected. However, we believe tthe one-to-many structure illustrated by this example is tycal of many cases.

The structure of this synchronization set can be undstood using a linear transformation of the full (x,y) system.In particular, let (x y)T5T(x y)T, where

T~c!5S 1 0

22~12c!/c ~22c!~12c!/cD . ~5!


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In the new coordinates, Eq.~4! becomes the ‘‘skinny’’ bak-er’s map given by

xn115H 2xn, xn,0.5,

2~ xn20.5!, xn>0.5,

yn115H cyn, xn,0.5,

cyn1~12c!, xn>0.5.~6!

Under one iteration, the two halves of the unit squaremapped into two horizontal rectangles as shown in Fig.For c,1/2, this map contracts area at a rate given by 2c.After n iterations, the original unit square is mapped inton

horizontal strips of heightcn, and the limiting set of thisprocess is a Cantor set of lines. The attracting set oforiginal map~Fig. 12! is the image of this Cantor set of lineunder the transformationT21(c).

FIG. 11. The multivalued synchronization sets resulting from Eq.~3! for ~a!r50.3 and~b! r50.8. In both cases,c50.3.

FIG. 12. The synchronization set of Eq.~4! with c50.35.


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This example also demonstrates how a multivalued schronization structure is directly related to the history of tdrive. In terms of the thin baker’s map, one particular histoof a drive statexn can be uniquely described by a sequenof the symbolsR and L, constructed by listing anL eachtime that a preimage lands to the left ofx51/2, and anReach time that a preimage lands to the right ofx51/2. Anillustration is given in Fig. 14 in which we have drawn thimages of two halves of the unit square as we iterate Eq.~6!forward twice. At the end of the second iteration, we cassociate different symbol sequences with the po( xn( i ),y( i )) ( i 51,...,4), where each such point is locatwithin a different horizontal strip and all correspond to tsamex value at timen. A finer resolution of the striatedstrips corresponds to additional steps backwards in tiwhich in turn corresponds to more symbols in the symsequence. Each point has a distinct infinite symbol sequeUsing a metric on the space of symbol sequences, onealso determine the difference between the orbits of anystriations. Two striations are close together if the most recsymbols in the sequence are identical.

The case of a noninvertible drive discussed in this stion may be reduced to the case with an invertible driverthe expense of replacing the driving system with one thamore complex. LetV5(X,f) be the space of all infinite sequences (x0 ,x1 ,...) such thatxi5f(xi 11) ~the inverse limit

FIG. 13. The skinny baker’s map.

FIG. 14. Dynamics under the skinny baker’s map of Eq.~6!. Top left: Theoriginal domain is divided into regionsR andL. Top right: The image ofRand L after one iterate. Bottom: After two iterates, the original regionRmaps to the regions labeledRL andRR, and the original regionL maps tothe regions labeledLL andLR.


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space!. We define aninvertible map f on V by f(x0 ,x1 ,...)5(f(x0),x0 ,x1 ,...). It iseasy to check that the response stem in Eq.~1! and the response system in

x85 f~ x!,

y85g~p0~ x!,y!, ~7!

receive the same input, wherep0( x)5x0 is the projection ofthe sequencex onto its first coordinate. Therefore, the dnamics of these response systems is the same in both cThe driving system in Eq.~7! is invertible, and thus thegraph transform method converges to a single-valued mf:V→Y. The multivalued functionf whose graph is thesynchronization set in Eq.~1! can be obtained by projectingraph(f) onto the first coordinate.

The complexity of the synchronization set becomesparent when the topology of this inverse limit spaceV isanalyzed. This space has an extremely complicated struceven in the relatively simple case of unimodal maps ofinterval.34 To get a better description of the synchronizatiset, we briefly outline an extension of the idea of symbsequences introduced in the case of the ‘‘skinny’’ bakemap. The construction of systems with invertible driversinverse limit spaces will be addressed in more detail ewhere.

We will describe this method in the case of the tent mf of the intervalI 5@0,1# with critical point 1/2, and suchthat f (1/2)51. The critical point separates the interval insubintervalsI 1

15@0,1/2# and I 215@1/2,1#. The preimage of

each of these intervals consists of the intervalsI 12 , I 2

2 andI 32 ,

I 42 such that f (I 1

2)5 f (I 22)5I 1

1 and f (I 32)5 f (I 4

2)5I 21. This

process can be continued to create a tree of intervalsthat a pair of intervals on the (n11)-st level of the tree mapto an interval in thenth level of the tree.

If the mapf defines the driving map of the system in E~1!, the graph transform method can be performed on infinpaths starting at any nodeI m

n in this tree of intervals. Themethod converges to a functionfm

n :I nm→Y over the first

interval in such a path. The graphs of these functions areinvariant themselves, however, under the dynamics of~1!, graph(fm

n ) maps into graph(fm8n21) when f (I m

n )5I m8n21 ,

and thus the family$graph(fnm)%n,m is an invariant family of

graphs. Moreover the graphs in this family are joined othe points in the history of the critical point 1/2, i.e., over tpointsk/2n.

This idea can be extended to describe the invariantover more complicated driving systems. In the case of~2!, we again obtain a family of graphs which are joined ovthe orbits of the critical points. If the map of the drivinsystem is smooth, we again encounter the problemscussed in Sec. II B, since the contraction becomes very laclose to the orbits of the critical points. Thus, smoothnessthe graphs in the invariant family cannot be expected atorbit that is close to the orbit of the critical point over sufciently many iterates.

We conclude this section by stressing that the multivued structure of the synchronization set for all these casean intrinsic feature of the underlying dynamics resulti


-

ses.

p

-

ree

lsn-

p

ch

e

otq.

r

ts.

r

is-gefy

l-is

from the presence of noninvertibility. Thus, we expect ththe multivalued nature off(x) persists for almost all valueof c.

D. Combinations

Finally, we briefly note that the various cases describabove can coexist. For example, Fig. 15 shows a case thboth multivalued and wrinkled; this is obtained using Eq.~2!and settingl50.2, a50.3, s520.8, andc50.7. Further-more, since a smooth noninvertible map must have critpoints, the situations described in Sec. II B and Sec. Ishould be expected to occur together.

Animations showing the evolution of the synchroniztion set as boths andc are varied are available on the web.24

III. QUANTIFICATION

Here we present two methods for quantifying the fetures of the synchronization set that we have descriabove.

A. emaxÀd test

Most practical methods of detecting nonlinear synchroin data rely strongly on the continuity off, and in generalalso require a certain degree of smoothness off.5–7 Thesemethods generally proceed by checking if clusters of poin X correspond to similarly small clusters of points inY. Itis important to note that the presence of the intrinsic geomric features that we have discussed above can significahinder the experimental detection of more complicated~andperhaps more interesting! synchronous relationships.

Consider the following numerical test based on the dnition of continuity. Iterate the system of [email protected]., Eq.~1!# for a sufficiently long time to allow transients to die ouPick a point~x,y! on the attractor and a small numberd, anditerate the full system until thex-component of the trajectorylands in the ballBx(d) a large number of times. Keep tracof these points, and letemax denote the largest distance btween the correspondingy-components. Iff is differen-tiable, then typicallyemax→0 linearly asd→0. When uhdu.uhr u, the functionf is typically only Holder continuous,andemax→0 sublinearly asd decreases. The Ho¨lder exponentof f at x can be estimated from the slope of the graph

FIG. 15. A multivalued and wrinkled synchronization set obtained from E~2! with l50.2, a50.3, s520.8, andc50.7.


ltol


FIG. 16. Graphs of ln(emax) versusln(d) for Eq. ~2!. In each case, severacurves are shown, correspondingseveral randomly chosen fiduciapoints in the driver. The thick line hasslope 1. The graphs correspond to~a!the smooth case in Fig. 2~a!; ~b! thewrinkled case in Fig. 2~b!; ~c! thecusped case in Fig. 6; and~d! the mul-tivalued ~looped! case in Fig. 9.

ynan

ssiin

s

r

leer

gs

lleho

d

toceotioela

-hip

canf adsther,

ll asof

thetion

ce-ierentsInto a

m

let

ntletsvan-sistare

ln(emax)-ln d, and to probe the overall smoothness of the schronization set, an ensemble of scaling curves with rdomly chosen fiducial pointsx can be studied.

Figures 16~a!–16~d! show the results of applying thiprocess to the various cases considered above. First, conthe smooth/wrinkled transition discussed in Sec. II A,which s51 in Eq. ~2!. Whenuhdu,uhr u, f is smooth almosteverywhere, and the ln(emax)–lnd curves have slope 1, ashown in Fig. 16~a!. In contrast, whenuhdu.uhr u, the syn-chronization set is only Ho¨lder continuous and thusf is notdifferentiable almost everywhere. The scaling curves thefore have smaller slopes, as shown in Fig. 16~b!. For thiscase, synchronization detection methods may not be abdetect synchronization at all, even in the absence of nois

In the cusped case, cusps of decreasing size occu~and near! forward iterates ofu51/2 for all c. However, thesynchronization set may be otherwise smooth, dependinc. Thus, the ln(emax)–lnd graphs show a variety of slopedepending on the location of the fiducial pointx; see Fig.16~c!. While most curves have slope 1, a few have smaslopes. In this case, standard synchrony detection metfail to detect the presence of the cusps.

Figure 16~d! demonstrates the effect of the multivaluestructure of Eq.~4! ~Fig. 12! on the ln(emax)–lnd graphs.These are seen to saturate at a scale that corresponds‘‘thickness’’ of the synchronization set. As a consequenthe ability to predict the state of the response system frthe state of the drive is severely affected, and this situacannot be improved by increasing the precision of the msurements. Although there is a dynamically coherent retionship between the driver and the response~in fact, thesystem is asymptotically stable!, most synchronization detection methods will fail to detect any synchronous relationsin such multivalued cases.


--

der

e-

to.at

on

rds

the,

mn

a--

B. Wavelets

As shown in Sec. II A, the Ho¨lder exponent off at apoint x depends on the ratio of two contraction rates, andtherefore vary from point to point. This is characteristic omultifractal set. In this section we use the wavelet methodescribed in Ref. 35 as a more accurate way to examineregularity of multifractal synchronization sets. In particulawe are interested in estimating the typical Ho¨lder exponents,i.e., those that occur on some set of full measure, as wethe distribution of other, nontypical exponents. The detailsthis method are briefly discussed in the Appendix, andinterested reader is referred to the literature on the estimaof the regularity of functions using wavelet techniques.35,36

Since the functionf(u,v) obtained from Eq.~2! is constantin the v direction, we will keepv constant in our analysisand treatf as a function from the real line to itself.

The wavelet transform may be thought of as a spalocalized counterpart of the Fourier transform. The Fourtransform decomposes a function into sinusoidal componof varying frequency, which are not localized in space.contrast, the wavelet transform decomposes a signal infamily of wavelets of varying location and sizeC((u2b)/a). All the wavelets in such a family are obtained froa single functionC(u), which is shifted byb and dilated bya factora. The ‘‘mother wavelet’’C(u) is typically chosento be localized in space, so that we can think of the wavetransform as a microscope magnifying an area aroundu5bby a factora.

A smooth function looks nearly constant after sufficiemagnification, and hence the coefficients of the waveneeded to describe such a function at small scales areishingly small. On the other hand, cusps or wrinkles perunder magnification, and hence wavelets of all sizes


te

-

d

in


FIG. 17. The wavelet transform of thewrinkled f discussed in Sec. II A. Themiddle figure represents the wavelecoefficients at different scales. Thlight regions correspond to largewavelet coefficients. Wavelet coefficients fora5212 are plotted in the bot-tom figure. Note that the self-similarity of the graph is reflected inthe wavelet coefficients. We have useMatlab routines from the packageWaveLab in the analysis presentedFigs. 17–20.

el

dt

toon,cal

17sedlet

needed to describe a function around such points. It canshown that, for appropriately chosen wavelets, the wavcoefficients scale as

WC~u0 ,a!;ah(u0)

at a pointu0 , wherea is the scaling factor of the wavelet anh(u0) is the Holder exponent of the function at tha


beetpoint.35,36Unlike the Fourier transform, which can be usedgive general information about the roughness of a functithis fact allows us to use wavelets to estimate the loroughness of a function.

These ideas are illustrated in Figs. 17 and 18. In Fig.we have chosen the wrinkled synchronization set discusin Sec. II A. The white regions show regions of high wave

FIG. 18. The wavelet transform of thecuspedf discussed in Sec. II B. Thegraphs are the same as in Fig. 17.


f

-dt

n

i-


FIG. 19. The typical smoothness othe synchronization manifold as afunction of the coupling strengthc.The solid line is obtained from the theoretical prediction, while the dasheline is obtained from the waveletransform. For any finite resolutionlevel, the wavelet method provides aupper bound forg t which is consistentwith the observed numerical overestmate.

ntleae

hp,

ing
coefficients. Note that the graph of the wavelet coefficiereveals the self-similar structure of the graph. The wavecoefficients computed for the cusped case of Sec. II Bshown in Fig. 18. In this case, the cusps are distributed vsparsely, but occur at many different scales.
Returning to the wrinkled case of Sec. II A, in whicsettings51 in Eq. ~2! reduces the drive to the baker’s ma


st

rery

the typical past-history Lyapunov exponent in the contractdirection is given by

hd5a ln l1~12a!ln~12l!,0, ~8!

and the response Lyapunov exponent ishr5 ln c.18 Therefore,the typical Holder exponentg t of the synchronization set is

ity

FIG. 20. The functionsDH(g) for thewrinkled case withc50.7, a50.7,and l50.8 ~solid line!, and for thecusped case withc50.2, a50.3, andl50.2 ~dashed line!. In the wrinkledcasegmin'0.2 is predicted correctly.In the cusped case due to the sparsof cusps, it is difficult to computeDH(g) numerically for small Ho¨lderexponents.


-y i

en

is-

-

ka

ionwff

idotiom

heticeeed

min

ivechtioposou

retezaigthmra

ndal

a-g

rd

the. Itng inn-

e

ty

d in

ribededto

od

s

es-

ffi-


g t5ln c

a ln l1~12a!ln~12l!. ~9!

In Fig. 19, g t computed numerically as a function ofc iscompared with the results obtained analytically from Eq.~9!.

The smallest~largest! Holder exponent on the synchronization set corresponds to points whose orbits lie entirelthe part of the square which is more contracting~less con-tracting!. Therefore, we obtaingmin5ln c/ln l and gmax

5ln c/ln(12l) if l.1/2. The Holder exponents off there-fore range fromgmin to gmax, and are equal tog t on a set offull measure.

We introduce the dimensionDH(g) of the set of allpoints in the domain off at whichf is Holder with expo-nent g. SinceDH(gmin)5DH(gmax)50 and DH(g t)51, thisfunction varies between 0 and 1. As explained in the Appdix, wavelets provide a natural way of computingDH(g) asa function ofg. The functionDH(g), frequently called thesingularity spectrumf (a), is expected to have a charactertic ù shape, hitting zero atgmin and gmax and attaining amaximum atg t . The g t in Fig. 19 were obtained by computing DH(g) numerically for eachc, and finding themaxima.

In Fig. 20 we show theDH(g) functions for a wrinkledand a cusped synchronization set. Note that the peaDH(g) in the cusped case is to the right of 1, showing ththe function is differentiable on a set of Hausdorff dimens1. The function corresponding to the wrinkled case shothat thef is only Holder continuous on a set of Hausdordimension 1.

The wavelet analysis of signals presented here prova computationally robust estimation procedure, and is msystematic than the method described in the previous secMoreover, the graphical display of the singularity spectruis an interpretable way of quantifying the irregularities of tsignal. Using these methods it is possible to make statisdistinctions between different signals. This idea has bsuccessfully used to distinguish between signals in biomcal applications.37

IV. CONCLUSION

In summary, we have shown that for coupled systewithout symmetries, systems can be coherent without haveasily-detectable synchronization properties. We have gexamples of invertible and noninvertible drivers for whithe system is asymptotically stable, yet the synchronizaset is nonsmooth or multivalued. We have given an examof a test for synchrony which incorporates features of mstandard tests. We illustrate the misleading nature of thecome of this test~and most standard tests! when it is appliedto synchronization sets containing the complicated featuof our examples. Furthermore, we give a sophisticamethod of detection designed for multifractal synchronition sets. These coherent yet complicated structures maffect the generation of ‘‘reliable response’’ in neuronal paways within actual biological systems. In conclusion, we ephasize that complicated synchronization sets and their p


n

-

oft

s

esren.

alni-

sgn

nlett-

sd-ht--c-

tical implications are not pathological: asymmetry anoninvertibility are typical in many biological and physicsystems.

ACKNOWLEDGMENTS

The authors gratefully acknowledge helpful converstions with S. J. Schiff, and support from the followinsources: NIH 1K25MH01963~E.B.!; a George Mason Uni-versity College of Arts and Sciences Junior Faculty Awa~E.S.!; NSF-IBN 9727739 and NIH 2R01MH50006~P.S.!.

APPENDIX: WAVELET FORMALISM

In recent years much research has been devoted tostudy of the multifractal properties of singular measureshas been observed that many fractal measures appearipractice scale differently at different points. Thus if we cosider a measurem on a spaceX and ballsBx(e) of radiusecentered atx, and define thesingularity strengthof the mea-sure atx by a(x) where

m~Bx~e!!;ea(x),

then it is frequently observed thata(x) is not constant.To characterize the size of the sets over whicha(x) is

constant one may cover the entire support ofm with balls ofradiuse and letNa(e) be the number of balls that scale likea for a given value ofa. The Hausdorff dimensionf (a) ofthe set on whicha(x)5a is then obtained by examininghow Na(e) scales ase→01, i.e.,

Na;e2 f (a). ~A1!

As noted in Ref. 35, instead of a measurem, we mayconsider a functionF and define the strength of a singulariof F at a pointx by

uF~Bx~e!!u;ea(x),

whereu•u is used to denote the size of a set. As discusseprevious sections,a(x) is exactly the Ho¨lder exponentg(x)of the functionF at the pointx and we may think off (a) asthe Hausdorff dimensionDH(g) of the set of points at whichF is Holder with exponent exactlyg. It follows that the ther-modynamic formalism that has been introduced to descthe statistical properties of singular measures, and extento the case of functions in Ref. 38, can be applied directlythe present problem.39

Arneodoet al. have introduced a wavelet based methto numerically estimateDH(g).35 It can be shown that thewavelet transform

WC@F#~b,a!51/aE2`

`

C~~x2b!/a!F~x!dx

for a waveletC that is orthogonal to linear functions scaleas

WC~x0 ,a!;ah(x0)

in the limit a→01.40 ~In the preceding, the overbar denotcomplex conjugation.! This means that ideally one could determine the Ho¨lder regularity of a function at a pointx0 bycomputing the exponential decay rate of its wavelet coecients.


etho

lyw

eiotho

ap

n-

eie

l,

di-geas

d

ono--nc

ib

gy,

.

it

nse

,

cal

tion

ifur-nda-

ork,ld

altheise istruc-

ni-esials


This approach does not circumvent the difficulty posby the fact that singularities accumulate on each other incase of fractal functions, and therefore an indirect methneeds to be used to approximate quantities likeDH(g). Toexamine the properties of a functionF, we cover its domainwith N(e) balls Bx(e) and define apartition function,

Z~q,e!5 (i 51

N(e)

uF~Bx~e!!uq.

The functionZ(q,e) scales aset(q) ase→01. A fundamen-tal result in the multifractal formalism states thatt(q) is theLegendre transform of the singularity spectrumDH(g) of F.Assuming thatt is differentiable, we can determineDH(g)from the following relations:

DH~g!5qg2t~q!,

g5d

dqt~q, j !.

We briefly mention that since theu coordinates of theiterates of the map defined by Eq.~2! are not distributeduniformly, standard wavelet transform algorithms which reon tools such as the FFT cannot be used directly. Insteadstarted with a large number of points, linearly interpolatthem, and subsampled the resulting partially linear functon a uniform grid. This procedure tends to smooth outfunction at small scales, so care needs to be taken to usewavelets on scales that are sufficiently large. A similarproach has been used in Ref. 41.

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9The choice of unidirectional coupling is for ease of analysis. With birectional coupling, crises may occur and additional dynamical chanmight obscure the discussion. However, in certain cases, unidirectioncoupled systems arelocally equivalent to bidirectionally coupled system~Ref. 10!. Most importantly, we expect the complications~such as wrin-kling, cusps, and fractal structures! in the synchronization set discussehere to persist in the bidirectional case.

10K. Josic, Phys. Rev. Lett.80, 3053~1998!.11The definition of generalized synchrony given in Ref. 7 requires a c

tinuous map betweentrajectories in the phase spaces of the two compnent systems. Most authors~e.g., Refs. 5, 16, 18! define generalized synchrony via a stronger but more practical condition requiring the existeof a continuous map betweenstatesof the component phase spaces.

12The synchronization set is a graph overX, but attractors for this coupledsystem may only be subsets of this graph. This is true for both invertand noninvertible systems.


ded

ednenly-

u-,

slly

-

e

le

13L. M. Pecora, T. L. Carroll, G. A. Johnson, D. J. Mar, and J. F. HeaChaos7, 520 ~1997!; N. F. Rulkov, ibid. 6, 262 ~1996!.

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20For a typical point~x, y! in the synchronization set, consider an orbsegment consisting of the pointsF2 j (x,y) for j 51,2,...,T. The time-TLyapunov exponents over the orbit segment emanating fromF2T(x,y) canbe calculated. If, forT→`, this spectrum of Lyapunov exponents remaithe same for almost all~x, y! with respect to the natural measure, then wcall these the past-history Lyapunov exponents of the system.

21The Holder exponent off at x is lim infd→0$ loguf(x1d)2f(x)u/logudu% ifthis quantity is less than 1, and equal to 1 otherwise.

22N. Fenichel, Indiana Univ. Math. J.21, 193 ~1971!.23M. W. Hirsch, C. C. Pugh, and M. Shub,Invariant Manifolds, Lecture

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28H. Steinlein and H.-O. Walther, J. Dyn. Diff. Eqs.2, 325 ~1990!.29This is similar to the classification of the invariant set within a dynami

horseshoe~see Ref. 30!.30K. T. Alligood, T. D. Sauer, and J. A. Yorke,Chaos: An Introduction to

Dynamical Systems~Springer-Verlag, New York, 1997!.31J. C. Chubb, E. Barreto, P. So, and B. J. Gluckman, Int. J. Bifurca

Chaos Appl. Sci Eng.11, 2705~2001!.32Reference 33 describes a 1:2 multivalued state that results from a b

cation of the response system as the coupling is decreased. This is fumentally different from the mechanism described in the present wwhich is based on the intrinsic noninvertibility of the drive. One shounote that the examples in Ref. 33 are notglobally asymptotically stable.This means that for a given history of the drive, two different initiresponse states do not necessarily have the same asymptotic state onmbranches (m52 in this case!. In contrast, the situation considered heremore coherent in the sense that for a given history of the drive, theronly one unique asymptotic response state despite the Cantor-type sture of the synchronization set.

33N. F. Rulkovet al., Phys. Rev. E64, 016217~2001!; N. F. Rulkov and C.T. Lewis, ibid. 63, 065204~2001!.

34M. Barge and W. T. Ingram, Top. Appl. Phys.72, 159 ~1996!.35A. Arneodo, E. Bacry, and J. F. Muzy,Wavelets in Physics~Cambridge

University Press, Cambridge, 1999!.36S. G. Mallat,A Wavelet Tour of Signal Processing, 2nd ed.~Academic,

New York, 1999!.37C. J. Morales and E. Kolaczyk, Anal. Biomed. Eng.30, 588 ~2002!.38S. Jaffard, SIAM~Soc. Ind. Appl. Math.! J. Math. Anal.28, 944 ~1997!;

28, 971 ~1997!.39See, for example, C. Beck and F. Schlo¨gl, Thermodynamics of Chaotic

Systems~Cambridge University Press, Cambridge, 1993!.40Since in this paper we are only interested in the Ho¨lder exponent we only

requireC to be orthogonal to first degree polynomials. If the synchrozation manifold is differentiable, the regularity of its higher derivativcan be studied using wavelets orthogonal to higher order polynom~Ref. 35!.

41R. de la Llave and N. Petrov, Exp. Math.~in press!.


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