Large populations of globally-coupled identical maps subjected to independent
additive noise are shown to undergo qualitative changes as the features of the
stochastic process are varied. We show that for strong coupling, the collective
dynamics can be described in terms of a few effective macroscopic degrees of
freedom, whose deterministic equations of motion are systematically derived
through an order parameter expansion.Comment: Phys. Rev. Lett., accepte

A. Lemaître

A. S. Pikovsky

Erik Mosekilde

Francesco d’Ovidio

Hugues Chaté

I. Z. Kiss

K. Kaneko

M. Chabanol

O. Popovytch

S. Danø

S. De Monte

Silvia De Monte

T. Shimada

English

arXiv

Crossref

Noise-Induced Macroscopic Bifurcations in Globally Coupled Chaotic Units

4 pages.-- PACS nrs.: 05.45.Ra, 05.40.Ca, 05.45.Jn, 87.10.+e.-- Pre-print version available at ArXiv: http://arxiv.org/abs/cond-mat/0405251.S. D.M. acknowledges support from the ESF
Programme REACTOR. F. d’O. acknowledges financial
support from MCyT (Spain) and FEDER under Project
No. BFM2000-1108 (CONOCE).Peer reviewe

Monte, Silvia de

D'Ovidio, Francesco

Mosekilde, Erik

Chaté, Hugues

Digital.CSIC

P H Y S I C A L R E V I E W L E T T E R S week ending25 JUNE 2004VOLUME 92, NUMBER 25Noise-Induced Macroscopic Bifurcations in Globally Coupled Chaotic UnitsSilvia De Monte,1,2 Francesco d’Ovidio,3 Hugues Chate´,2 and Erik Mosekilde11Chaos Group, The Technical University of Denmark, DK 2800 Lyngby, Denmark2CEA—Service de Physique de l’E´ tat Condense´, CEN Saclay, 91191 Gif-sur-Yvette, France3IMEDEA, CSIC University of Balearic Islands, E 07071 Palma de Mallorca, Spain(Received 5 June 2003; published 24 June 2004)254101-1Large populations of globally coupled identical maps subjected to independent additive noise areshown to undergo qualitative changes as the features of the stochastic process are varied. We show that,for strong coupling, the collective dynamics can be described in terms of a few effective macroscopicdegrees of freedom, whose deterministic equations of motion are systematically derived through anorder parameter expansion.DOI: 10.1103/PhysRevLett.92.254101 PACS numbers: 05.45.Ra, 05.40.Ca, 05.45.Jn, 87.10.+exj  1 Kfxj  Khfxi  jt; (1) cascade (Fig. 2). The onset of collective oscillations isDynamical units coupled all to all constitute relevantmodels of a wide range of physical and biological systemssuch as Josephson junction arrays [1], electrochemicaloscillators [2], interacting neurons [3], and yeast cells ina continuous-flow, stirred tank reactor [4]. Globallycoupled dynamical systems are also widely used as pro-totypical models for addressing the collective behavior ofsystems with many degrees of freedom [5]. In the limit ofstrong coupling of identical units, full synchronizationusually occurs; i.e., the collective dynamics is an exactcopy of the individual one. However, in the generic casewhere some diversity and/or noise is present at the micro-scopic level, macroscopic measurements do not directlyreflect individual dynamical features. This simple obser-vation is of large experimental importance. In the yeastcell experiment, for instance, single-cell measurementsare not available so far. The idea is thus to extract in-formation about metabolic processes at the individuallevel from the observed collective oscillations in cellsuspensions [4].In this Letter, we consider the influence of microscopicadditive noise on the fully synchronous regime of globallycoupled systems. This allows us to isolate the effect ofnoise from the possibly already complex collective dy-namics that systems of globally coupled identical andnoiseless units may themselves exhibit [5]. Restricting,for simplicity, our discussion to identical scalar maps, weshow that noise plays a nontrivial role, ‘‘unfolding’’ thenoiseless collective dynamics while progressively sup-pressing the direct connection between average and localevolution. We build a systematic expansion that accountsquantitatively for the collective dynamics observed, re-vealing that its essential features can be described interms of few effective degrees of freedom and that itshierarchical structure is captured in progressively greaterdetail as the truncation order is increased.Our system of noisy and globally coupled identicalone-dimensional maps is defined as follows:0031-9007=04=92(25)=254101(4)$22.50 where xj 2 R denotes the state of the jth populationelement, j  1; . . .N, f:R ! R is a smooth function de-fining the dynamics of the uncoupled element, andhfxi : 1NPNj1 fxj is the average over the population.Every map is subjected to a white, but not necessarilyGaussian, noise of zero average and variance 	2.Without noise and for strong enough coupling, thetrajectories of all the population elements asymptoticallycoincide (full synchronization). The mean field X  hxithen evolves on a one-dimensional manifold of the high-dimensional phase space of the full system.Although noise smears any individual trajectory, forlarge population sizes these fluctuations average outand the mean field X evolves deterministically up tofinite-size effects. Somewhat counterintuitively, themacroscopic dynamics is qualitatively different fromthat of the uncoupled map and remains apparently low-dimensional for any noise intensity. For chaotic logisticmaps, increasing noise drives the mean field X through acascade of period-doubling bifurcations which take placeat smaller and smaller	 values as the coupling strengthKis lowered [Figs. 1(d)–1(f)]. The difference betweensingle-element noisy evolution and regular averagebehavior can be easily seen comparing the sharplypeaked probability distribution functions (PDF) of Xwith the broad, smooth PDFs of individual elements[Figs. 1(a)–1(c)].The effect of noise is not always the ‘‘simplification’’ ofthe collective dynamics; rather, noise unravels the under-lying complexity of the phase space. This is particularlystriking in the case of excitable units, such asfx  x x3ex2 : (2)(For other examples of excitable maps, see [6].) While theorigin is the unique global attractor, map (2) possessesan unstable fixed point, excitation across which triggersa chaotic transient. Increasing the noise intensity overa threshold 	, the mean field displays intermittentbursts, due to the crisis of a chaotic attractor whichdisappears, for stronger noise, via a period-doubling2004 The American Physical Society 254101-1FIG. 2. Population of N  222 excitable maps [Eq. (2) with  0:4,   1,   8, and K  0:9] subjected to Gaussiannoise of variance 	2. (a) Bifurcation diagram for X vs 	.(b) Critical value 	 of the noise intensity above which themean field displays large amplitude oscillations for the popu-lation (K  1, ; K  0:8, ; K  0:6, ) and for the orderparameter reduction (solid line).FIG. 1. System of N  220 globally coupled logistic maps inthe chaotic regime [fx  1 ax2, a  1:57] subjected touniformly distributed noise of variance 	2. (a)–(c) K  0:7:PDFs for the mean field (solid line) and one single element(dashed-dotted line; dotted line at even times, dashed line atodd times) for different noise intensities: (a) 	2  0:0025,(b) 	2  0:0324, and (c) 	2  0:09. (d) Bifurcation diagramfor X as a function of 	2 for K  1. (e) First period-doublingbifurcations for the mean field () and for the second-orderreduced system (solid line, see text) in the 	;K plane.(f) Same as (d) but for the reduced system Eqs. (9) with K 0:7 (the diagram of the full system almost coincides). Thedotted lines in (d) and (f) indicate the location of the bifurca-tions reported in (e).P H Y S I C A L R E V I E W L E T T E R S week ending25 JUNE 2004VOLUME 92, NUMBER 25nothing but an explicit mechanism for ‘‘noise-induced’’and ‘‘coherence’’ resonance [7]: the single-element powerspectrum displays an enhanced peak in the periodicregimes occurring at intermediate 	 values [8].Let us now investigate analytically the apparent low-dimensional nature of the collective motion by trying todecouple the macroscopic effect of noise from the dy-namical ‘‘skeleton’’ provided by the microscopic map. Wewrite the position of each population element in terms ofthe mean field and of its displacement from it: xj  Xj. The uncoupled element equation can now be expandedas a power series around the mean field:fxj  fX X1q11q!DqfXqj ; (3)254101-2where Dqf indicates the qth derivative of the function f.Substituting Eq. (3) into Eq. (1), the macroscopic firstiterate is obtained by averaging:X  hfxi  fX X1q11q!DqfXhqi: (4)The mean field is then coupled to a full set of new orderparameters:q : hqi; q 2 N: (5)It is worth stressing that, at this point, we have taken theinfinite-size limit, discarding the term hi which scaleslike 1=Npand plays the role of a macroscopic noiseacting on the mean field. This accounts for the mean-fieldfluctuations obeying the law of large numbers. The evo-lution of the order parameters q can be computed usingj  1 KX1p11p!DpfXpj p  jand the fact that, in the limit N ! 1, the positions andthe noise are uncorrelated variables and, hence,hhX; qi  hhX; ihqi. Equation (5) then yieldsq mq Xqi1qi1 KimqiX1p11p!DpfXp pi	; (6)where mq  hqi is the qth moment of the noise distribu-tion. The macroscopic dynamical system defined byEqs. (4) and (6) is still infinite dimensional, like thefull system, but it can be truncated to finite order whenassuming that 1 K is sufficiently small. This leads to ahierarchy of finite-dimensional reduced systems whichcapture, as we show below, the properties of the collectivedynamics in increasing detail.254101-2FIG. 3 (color online). Population of Fig. 1. (a) First returnmap of X for the full system (dark dots), its zeroth-orderapproximation (solid line), and its second-order approximation(light dots) (K  0:4 and 	2  0:001). (b) 4 versus 2 for thesame parameters (dark dots, full system; zeroth order, triangle;second order, solid line; fourth-order, light dots). (c) and(d) Distances dx and d defined in text (circles for the fullsystem, dashed lines for the second-order approximation, solidline for fourth-order system). [(c) 	2  0:03; (d) K  0:6].P H Y S I C A L R E V I E W L E T T E R S week ending25 JUNE 2004VOLUME 92, NUMBER 25A first approximation of Eq. (6) consists in keepingonly terms of zeroth order in 1 K, so that any q mq. The scalar equation for the mean field,X  fX X1q11q!DqfXmq; (7)describes exactly the bifurcation diagram for K  1[Fig. 1(d)]. Here the macroscopic effect of noise is aone-dimensional unfolding of the uncoupled element dy-namics, whose nonlinearities become progressively morerelevant as the noise intensity increases. In particular, ifthe map is polynomial, only a finite number of momentsinfluence the mean-field dynamics. This suggests a crite-rion for dividing the noise distributions into classes ac-cording to the collective behavior they generate.For logistic maps, Eq. (7) takes the simple form:X  1 a	2  aX2: (8)This map is easily rescaled to a logistic map in which 	2plays the role of the usual nonlinearity parameter.Although Eq. (8) provides a good approximation to theregimes of very strong coupling, it is independent of K.Hence, it cannot account for the changes with K of theperiod-doubling bifurcation lines [Fig. 1(e)]. TruncatingEqs. (4) and (6) to the second order in 1 K (the firstorder term vanishes for zero-average noise), the couplingstrength appears as a parameter for the macroscopicequations of motion. In this case, the higher-order mo-ments obey a recurrence relation leading, when f ispolynomial, to a two-dimensional map. For our exampleof logistic maps, this reads:X  1 aX2  a2;2  	2  1 K2a2m4  6	4 4X2 2  6	22: (9)Figure 1(e) shows that even for intermediate couplingstrengths this equation accounts quantitatively for themain features of the mean-field dynamics.Looking at the bifurcation cascade of Fig. 1(f), onecould think that Eqs. (9) might be further reduced to ascalar map. This is not possible, however, as the mean-field first return map is folded (Fig. 3) and hence themacroscopic system must have more than one variable.Remarkably, our approximation at order 1 K2 [Eqs. (9)for logistic maps] reproduces this folded structure. Forboth the full system and the second-order approximationEqs. (9), the distance dX between the first return map of Xand the parabola identified by the zeroth-order truncationscales, as expected, like 1 K2 or 	2 [Figs. 3(c) and3(d)] and vanishes on K  1, where the zeroth-ordertruncation Eq. (8) is exact. The ‘‘thickness’’ of the trans-verse folded structure shown in Fig. 3(a) scales similarly[8], indicating that, apart from finite-size effects thatmay prevent us from observing it, the dimension of the254101-3macroscopic attractor becomes larger than that of theuncoupled element as soon as K < 1 and 	 > 0.Going one step further, it is easily seen that there arestill features of the mean-field dynamics that cannot becaptured by Eqs. (9). The macroscopic variables q forthe population satisfy only approximately the relationsused for reducing the dimensionality of the truncatedsystem. For instance, 4 displays a complex dependenceon 2 [Fig. 3(b)], while it is constant at zeroth order andlinear at second order. In turn, this complex structure isretrieved if Eqs. (4) and (6) are truncated to fourth orderin 1 K. In the 2;4 plane, the distances d sep-arating the second-order truncation from the full systemor the fourth-order truncation are comparable andthey scale like the expected 1 K4 and 	4 [Figs. 3(c)and 3(d)].Extrapolating from the above results, we conclude thatthe presence of noise unfolds the fully synchronizedregime into an infinitely complex but hierarchicallyorganized structure whose main features are capturedby the lowest orders of our approximation scheme.Nevertheless, in the parameter region considered here,we expect the collective chaos to remain embedded in afinite-dimensional space. This is perhaps best seen whencalculating the Lyapunov exponents of the finite-dimen-sional maps approximating the full system. The mapspossess only one positive exponent in the chaotic region,whereas increasing the order of the approximationadds ‘‘new,’’ more negative exponents to those existingat previous order. These exponent values agree wellwith those numerically computed by simulating the254101-3P H Y S I C A L R E V I E W L E T T E R S week ending25 JUNE 2004VOLUME 92, NUMBER 25Perron-Frobenius dynamics of the population PDF [8], sothat we are confident that they do represent the collectivedynamics of the full system.In general, when the local map f is not polynomial, theorder parameter expansion of Eq. (2) contains an infinitenumber of terms even for K  1, and it is then necessaryto introduce the further closure assumption 	 1 toobtain closed form approximations. We can illustratethis for the excitable map example mentioned above. Atsecond order in both 1 K and 	, the approximationaccounts well for the establishment of collective oscilla-tions. The linear stability analysis of the origin for thereduced system provides an analytical condition for theonset of the intermittent behavior, which is in good agree-ment with the simulations of the population [Fig. 2(b)].To summarize, the collective dynamics of infinitelylarge populations of globally coupled identical maps sub-jected to independent additive noise is deterministic andundergoes qualitative changes as the features of the sto-chastic process are varied. For strong coupling, the col-lective dynamics can be described in terms of a feweffective macroscopic degrees of freedom, whose equa-tions of motion have been systematically derived throughan order parameter expansion valid for any smooth mapand noise distribution. Such a reduced system allows us tounderstand how the nonlinearities of the map interactwith the specific features of the microscopic noise. Atthe experimental level, our work indicates that the ob-servation of unusual PDFs of local variables such as thoseshown in Fig. 1 may be traced back to the presence of anunderlying chaotic local dynamics.Our approach reveals that in a large region of parame-ter space the collective motion, in spite of being endowedwith a single positive Lyapunov exponent, takes place on ahierarchically organized complex attractor. We believethat this complexity is but another facet of the ‘‘anoma-lous scaling’’ properties of the cloud of points represent-ing the population in the local phase space which wereuncovered recently by Teramae and Kuramoto [9], bothphenomena being rooted in the structure of the globalphase space around the fully synchronized solution.The method proposed here, together with Ref. [10],allows one to compare the macroscopic bifurcations in-duced by different sources of microscopic disorder, suchas noise and parameter mismatch (quenched noise). Inthis last case, an order parameter expansion with otherclosure assumptions provides a different low-dimensionalunfolding of the local dynamics, accounting for the ob-served diversity of the bifurcation scenarios [8].Future work should consider how all these aspectsevolve as one lowers the coupling strength towards theweak-coupling regimes considered by Shibata et al. [11],254101-4where collective chaos is high dimensional. Our work canbe extended to the case of continuous-time systems,where we expect a similar effect of microscopic noise[12]. Finally, it would be most interesting to make theconnection with lattices or networks of locally coupledmaps and the so-called nontrivial collective behavior theyexhibit. There, as suggested in [13], the equivalent of thenoise considered here lies in the finite-amplitude fluctua-tions introduced by the neighbors and the finite correla-tion length in the system.S. D. M. acknowledges support from the ESFProgramme REACTOR. F. d’O. acknowledges financialsupport from MCyT (Spain) and FEDER under ProjectNo. BFM2000-1108 (CONOCE).[1] P. Hadley, M. R. Beasley, and K. Wiesenfeld, Phys. Rev. B38, 8712 (1988); S. Nichols and K. Wiesenfeld, Phys. Rev.E 49, 1865 (1994).[2] I. Z. Kiss, Y. Zhai, and J. L. Hudson, Science 296, 1679(2002).[3] H. Sompolinsky, D. Golomb, and D. Kleinfeld, Phys.Rev. A 43, 6990 (1991).[4] S. Danø, P. G. Sørensen, and F. Hynne, Nature (London)402, 320 (1999); S. Danø, F. Hynne, S. De Monte,F. D’Ovidio, H. Westerhoff, and P. G. Sørensen, FaradayDiscuss. 120, 261 (2002).[5] K. Kaneko, Phys. Rev. Lett. 63, 219 (1989); Physica(Amsterdam) 41D, 137 (1990); A. S. Pikovsky andJ. Kurths, ibid. 72D, 1644 (1994); T. Shimada andS. Tsukada, Physica (Amsterdam) 168–169D, 126(2002); O. Popovytch, Y. Maistrenko, andE. Mosekilde, Phys. Lett. A 302, 171 (2002).[6] R. Toral, C. R. Mirasso, E. Herna´ndez-Garcı´a, andO. Piro, Chaos 11, 665 (2001); Y. Hayakawa andY. Sawada, Phys. Rev. E 61, 5091 (2000).[7] W. Rappel and A. Karma, Phys. Rev. Lett. 77, 3256(1996); A. Pikovsky and J. Kurths, ibid. 78, 775 (1997).[8] S. De Monte, F. d’Ovidio, H. Chate´, and E. Mosekilde (tobe published).[9] J. N. Teramae and Y. Kuramoto, Phys. Rev. E 63, 036210(2001).[10] S. De Monte and F. d’Ovidio, Europhys. Lett. 58, 21(2002); S. De Monte, F. d’Ovidio, and E. Mosekilde,Phys. Rev. Lett. 90, 054102 (2003).[11] T. Shibata, T. Chawanya, and K. Kaneko, Phys. Rev. Lett.82, 4424 (1999).[12] C. Kurrer and K. Schulten, Phys. Rev. E 51, 6213 (1995);M. Chabanol, V. Hakim, and W. Rappel, Physica(Amsterdam) 103D, 273 (1997); H. Hong and M.Y.Choi, Phys. Rev. E 62, 6462 (2000).[13] A. Lemaıˆtre, H. Chate´, and P. Manneville, Phys. Rev.Lett. 77, 486 (1996); Europhys. Lett. 39, 377 (1997);A. Lemaıˆtre and H. Chate´, ibid. 46, 565 (1999).254101-4

Physical Review Letters

Noise-induced macroscopic bifurcations in globally-coupled chaotic units

Large populations of globally coupled identical maps subjected to independent additive noise are shown to undergo qualitative changes as the features of the stochastic process are varied. We show that, for strong coupling, the collective dynamics can be described in terms of a few effective macroscopic degrees of freedom, whose deterministic equations of motion are systematically derived through an order parameter expansion

Noise-induced macroscopic bifurcations in globally-coupled chaotic units

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