A method for reconstructing dimensions of subspaces for weakly coupled dynamical systems is offered. The
tool is able to extrapolate the subspace dimensions from the zero coupling limit, where the division of
dimensions as per the algorithm is exact. Implementation of the proposed technique to multivariate data
demonstrates its effectiveness in disentangling subspace dimensionalities also in the case of emergent synchronized
motions, for both numerical and experimental systems

Boccaletti, S. (S.)

Carroll, T. (T.)

Geffert, H.P. (H. P.)

Pecora, L.M. (Louis M.)

Valladares, D.L. (D.L.)

Dadun, University of Navarra

RAPID COMMUNICATIONSPHYSICAL REVIEW E, VOLUME 65, 035204~R!Reconstructing embedding spaces of coupled dynamical systems from multivariate dataS. Boccaletti,1,2 D.L. Valladares,2,3 Louis M. Pecora,4 Hite P. Geffert,4 and T. Carroll41Istituto Nazionale di Ottica Applicata, Largo E. Fermi, 6, I50125 Florence, Italy2Departamento de Fisica y Matematica Aplicada, Universidad de Navarra, Pamplona, Spain3Departamento de Fisica, Universidad Nacional de San Luis, San Luis, Argentina4Code 6343, Naval Research Laboratory, Washington, D.C. 20375~Received 4 October 2001; published 21 February 2002!A method for reconstructing dimensions of subspaces for weakly coupled dynamical systems is offered. Thetool is able to extrapolate the subspace dimensions from the zero coupling limit, where the division ofdimensions as per the algorithm is exact. Implementation of the proposed technique to multivariate datademonstrates its effectiveness in disentangling subspace dimensionalities also in the case of emergent synchro-nized motions, for both numerical and experimental systems.DOI: 10.1103/PhysRevE.65.035204 PACS number~s!: 05.45.Tp, 05.45.Xt, 45.10.2bThe characterization of chaotic and irregular measuredtime series with the help of nonlinear analysis tools is asubject of great scientific interest @1#. A preliminary funda-mental task in all analysis techniques is how to reconstructcorrectly the strange chaotic attractors from observed scalardata. To this purpose, it has been shown that an embeddingcan be obtained from time-delayed coordinates of the ob-served variables @2#, by means of which one can reconstructthe chaotic attractor of a given dynamics, and then use it forcharacterization @3#, prediction @4#, measurement, modeling,or control @5# purposes. Alternative procedures have beenproposed based on the reconstruction of the main features ofthe chaotic dynamics by means of the interspike intervaltechnique @6#, or by adaptive methods @7#. The embeddingtechnique of Ref. @2# depends on the suitable choice of twoparameters, namely the embedding time and the embeddingdimensions. While the former may be easily obtained as thefirst minimum of the mutual information function @8#, a cor-rect determination of the system dimensionality is an essen-tial problem to be solved in order to approach correctly anyfollowing steps of a nonlinear data analysis.A popular method that is used for measuring the minimalembedding dimension is the so-called false nearest-neighbor~FNN! method, originally introduced by Kennel et al. @9#,and later improved and reelaborated in order to face specificanalysis tasks @10#. The method consists in marking as falsenearest neighbors at dimension m those pairs of nearest-neighbor m-dimensional embedded vectors whose distance atdimension m11 exceeds a given number of times their dis-tance at dimension m, thus accounting for possible self-intersections of the flow due to insufficient dimensionality inthe embedded space. A vanishing fraction of FNN marks theminimum dimensionality needed to properly reconstruct thechaotic flow. This technique has been later improved @10#,also complementing this analysis with the one on the signalsurrogates @11#.In this paper, we discuss how to extend the dimensionreconstruction problem to the case of multivariate dataanalysis, that is, to the case in which an observer is presentedwith a system composed by n weakly coupled nonidenticaldynamical subsystems ~of dimensions l1 ,l2 , . . . ,ln , respec-1063-651X/2002/65~3!/035204~4!/$20.00 65 0352tively!, and extracts separately scalar quantities xi(t) out ofeach subsystem i (i51, . . . ,n). If the observer is interestedin probing global properties of the system under study ~and ifthe subsystem variables are all to all coupled!, then the usualreconstruction methods work regardless of the particularvariable xi(t) on which embedding is performed. However,this gives information on the full phase-space dimensionality.There are relevant cases, such as detection of synchroniza-tion processes @12,13#, or detection of interdependence ofdynamics @14#, for which the determination of the dimen-sionality of each subsystem is needed. Synchronization fea-tures have been largely studied for both coupled chaotic@12,13# and space-time chaotic @15# systems, as well as ana-lyzed and studied in experiments and natural phenomena@16#. Recently, various attempts at providing a unifying for-malism encompassing the major synchronization featureshave been made @17#, whose applications to real data rely ona proper determination of the subspace dimensions ontowhich the global dynamics should be projected to look forfunctional relationships.The problem can be stated as follows. Consider having nweakly coupled nonidentical dynamical systems, and sup-pose that an observer is able to measure separately scalarquantities xi(t) out of each subsystem i (i51, . . . ,n). Inorder to properly reconstruct the dimensions l1 ,l2 , . . . ,ln ,let us consider the vectorz¯[x1~ t !,x1~ t2t1!,x1~ t22t1!, . . . ,x1@ t2~m121 !t1# ,x2~ t !,x2~ t2t2!, . . . ,x2@ t2~m221 !t2# , . . . ,xn~ t !,xn~ t2tn!, . . . ,xn@ t2~mn21 !tn#, ~1!where t i (i51, . . . ,n) are n different embedding times. z¯PRm (m5( i51,nmi) is a vector whose first~second, third, . . . ,nth) m1 (m2 ,m3 , . . . ,mn) componentsresult from the embedding of the x1 (x2 ,x3 , . . . ,xn) scalarvariable with embedding time t1 (t2 ,t3 , . . . ,tn). The em-bedding times t i can be different from each other, since dif-ferent observed variables xi(t) may show different mutualinformation properties @8#.©2002 The American Physical Society04-1RAPID COMMUNICATIONSS. BOCCALETTI et al. PHYSICAL REVIEW E 65 035204~R!Suppose now we start at dimension m realized with achoice of an initial set of subspace dimensions $mi% ~usuallyone begins with mi51, i51, . . . ,n), and to consider allm-dimensional vectors z¯ j , j51, . . . ,N , N being the totalnumber of available measurements. We associate to eachvector z¯ j its nearest-neighbor z¯NN, j at dimension m. Further-more, we introduce n counters Ni(m) (i51, . . . ,n), and agiven threshold s .For each pair of nearest neighbors z¯ j ,z¯NN, j we calculatethe distance d(z¯ j ,z¯NN, j ,m)5A( l51, . . . ,m(z¯ j ,l2z¯NN, j ,l)2. Wethen pass from dimension m to dimension m11.This latter operation can be performed in n different ways.Precisely from m[(m1 , . . . ,mi , . . . ,mn) we can pass toany space m11[(m1 , . . . ,mi11, . . . ,mn) (i51, . . . ,n).In those spaces, we calculate the new distancesdi(z¯ j ,z¯NN, j ,m11). The condition di(z¯ j ,z¯NN, j ,m11)>sd(z¯ j ,z¯NN, j ,m) is taken as a signature of the falseness ofnearest neighbors with respect to increasing by one the di-mension of the ith subspace. Consequently, the counterNi(m) is increased by one.After having probed all pairs of nearest neighbors at di-mension m, the set of counters Ni(m) (i51, . . . ,n) retainsinformation on how many nearest neighbors are false withrespect to increasing by one the dimension of the corre-sponding ith subspace. For any Ni(m) above a preassignedthreshold d we increase by one the dimension of the corre-sponding subspace, and we perform the whole process againat dimension m1p , p being the number of Ni(m) countersthat overcome d . The process is stopped when all Ni(m) arebelow d at once, thus gathering simultaneous information onboth the dimension of the full reconstructed phase space andthe dimensions mi of each subsystem. We emphasize thatthis procedure should work well for weak coupling and infact be exact for zero coupling, thus we think of it as anextension of the zero coupling case.The above algorithm makes use of the two thresholds sand d . s discriminates the falseness of a pair of nearestneighbors, and therefore must exceed unity. On the otherhand, a too large s value would lead to the consequence thatthe condition di(z¯ j ,z¯NN, j ,m11)>sd(z¯ j ,z¯NN, j ,m) is neversatisfied. The criteria for a proper choice of s are largelydiscussed in Refs. @1,9#. In what follows, we always take s510, which in our case represents a good compromise be-tween the two discussed extrema. Furthermore, we havechecked that the dimension estimate was insensitive to swithin some reasonable range around that value.As for d , it indicates the percentage of FNN at which thereconstruction process is stopped. In fact, for ideal data, oneshould put d50 and wait for a vanishing fraction of FNN.However, in all practical cases, data are corrupted by noise,or by a finite measurement resolution. A direct consequenceof the above is that the FNN counters Ni(m) saturate to anonzero value, thus one has to estimate a minimal acceptablefraction of FNN to stop the dimension reconstruction processby estimating the level of noise corrupting the available data.In order to demonstrate the effectiveness of the proposedmethod, we provide here three applications for n52. Thefirst application is the reconstruction of the subspaces for asystem described by03520x˙ 1,252v1,2y1,22z1,21«~x2,12x1,2!,y˙ 1,25v1,2x1,210.15y1,2 , ~2!z˙ 1,250.21z1,2~x1,2210!,where v1,250.9760.02. System ~2! is constituted by twocoupled nonidentical chaotic Ro¨ssler systems @18#.By increasing the coupling strength « , several differentkinds of synchronized motions emerge. In particular, Ref.@13# identified phase, intermittent lag, lag, and eventuallyalmost complete synchronization @a situation where the dif-ferences ux1(t)2x2(t)u, uy1(t)2y2(t)u, and uz1(t)2z2(t)uare all bounded in time by a quantity much smaller than theattractor diameter#. By numerically integrating system ~2!,we take the scalar signals x1,2(t) as two distinct measure-ments for the reconstruction task.The second application deals with structurally nonequiva-lent subsystems described byx˙ 152vy12z11«~x22x1!,y˙ 15vx110.15y1 ,z˙ 150.21z1~x1210!,x˙ 25w210.25x21z21«~x12x2!, ~3!y˙ 2531y2w2 ,z˙ 2520.5y210.05z2 ,w˙ 252x22y2 .For v50.925, system ~3! is constituted by a chaoticRo¨ssler oscillator @18# coupled with a hyperchaotic Ro¨ssleroscillator @19#. As a function of « , Ref. @13# discriminatesbetween nonsynchronized and phase-synchronized dynam-ics. As above, we take for reconstruction the scalar signalsx1,2(t) as they come from numerical integration of system~2! with initial conditions x150.1,y150.2,z150.3,x25y250,z2515,w25220.The third application uses scalar signal x1,2(t) as mea-sured over an experiment, whose setup is depicted in Fig. 1.The two circuits that were built were similar to the Ro¨sslersystem, but they each contained only one piecewise linearelement @20#. One of these circuits, which were similar butnot identical, is shown in Fig. 1. For the first ~the second!circuit, the resistor R1 was 221 kV (178 kV).A coupling circuit took the difference between x signalsfrom the two circuits and multiplied this difference by a cou-pling constant c. The resulting signal, D , was fed back intoeach circuit at the place indicated, so that the two circuitslooked like a pair of similar Ro¨ssler oscillators coupled bythe difference of x variables.4-2RAPID COMMUNICATIONSRECONSTRUCTING EMBEDDING SPACES OF COUPLED . . . PHYSICAL REVIEW E 65 035204~R!The x signals from the two circuits were digitized at a rateof 20 000 samples per second, which was about 20 points percycle of the Ro¨ssler oscillation. Time series of length100 000–500 000 points were acquired.FIG. 1. The experimental setup. Ro¨ssler-like circuit used to gen-erate data. For the first ~the second! circuit, the resistor R1 was 221kV (R15178 kV).TABLE I. Detection of subspace dimensionality for system ~2!.First column indicates the coupling value, second ~third! columnreports the calculation of false nearest-neighbor dimension m @9#over the signal x1 (x2). Fourth column indicates the dimensions ofsubspaces m1 and m2 as calculated by the proposed method. Theconsidered coupling values determine the following dynamicalstates: «50.02 unsynchronized evolution, «50.05 phase synchro-nization, «50.13 intermittent lag synchronization, «50.16 lag syn-chronization ~LS!, «50.50 almost complete synchronization ~ACS!@13#. Notice that, for both LS and ACS, the subspaces dimension-ality reconstruction is affected by the emergence of the correspond-ing synchronization manifolds, due to large couplings between thetwo subsystems.Coupling value m(x1) m(x2) m1 ,m2«50.02 6 6 3,3«50.05 6 6 3,3«50.13 6;7 6;7 ;3,3«50.16 6 6 3,2«50.50 6 6 3,203520In all cases we have taken t1 (t2) from the first minimaof the mutual information function calculated on x1 (x2) @8#.Calculations of nearest-neighbor dimensions have begunwith m15m251. In Table I ~II,III! we show the process ofdetection of subspace dimensionality for the system ~2! @thesystem ~3!, the experimental outputs of the system describedin Fig. 1#. The first column indicates the coupling value atwhich variables x1,2 were measured. The second and thirdcolumn report the dimensionality of the full phase space, ascalculated via the application of the usual false nearest-neighbor technique @9# over the signal x1 (x2). Finally, in thefifth column, we show the results of the application of theabove-described techniques for the calculation of the sub-space dimensions m1 and m2. For the experimental data~Table III!, m , m1, and m2 have been taken as the dimensionat which the fraction of false nearest neighbors was smallerthan 0.5% of the total number of data points, since this wasa satisfactory estimation of the level of noise corrupting thedata.At weak-coupling values, the proposed reconstructingmethod is successful in disentangling the dimensions of thecoupled subspaces, thus giving either information of the di-mension of the reconstructed full attractor, and on the dimen-sion of the two coupled subspaces onto which the dynamicsshould be projected to inspect for synchronization or othercollective phenomena. For larger coupling strengths ~fourthand fifth row of Table I and fifth row of Table III!, the de-TABLE II. Same as in Table I, but for system ~3!. First columnindicates the coupling value, second ~third! column reports the cal-culation of false nearest-neighbor dimension m @9# over the signalx1 (x2). Fourth column indicates the dimensions of subspaces m1and m2 as calculated by the proposed method. The considered cou-pling values determine the following dynamical states: «50.008unsynchronized evolution, «50.0012 phase synchronization @13#.Coupling value m(x1) m(x2) m1 ,m2«50.08 7 7 3,4«50.012 7 7 3,4TABLE III. Detection of subspace dimensionality for the experi-mental output of the system described in Fig. 1. First column indi-cates the coupling value. Second ~third! column reports the calcu-lation of false nearest-neighbor dimension m @9# over the signal x1(x2). Fourth column indicates the dimensions of subspaces m1 andm2 as calculated by the proposed method. As the coupling strengthincreases, the systems experience a smooth transition from no syn-chronized evolution to generalized synchronization.Coupling strength~units of 1024) m(x1) m(x2) m1 ,m25 6 6 3,35.5 7 6 3,36.25 7 6 3,37.14 7 6 3,38.33 7 6 2,44-3RAPID COMMUNICATIONSS. BOCCALETTI et al. PHYSICAL REVIEW E 65 035204~R!tection of subspaces dimensionality is affected by the emerg-ing synchronization manifolds.In conclusion, we have introduced a multivariate dataanalysis tool that is able to extrapolate the dimensions ofweakly coupled subspaces from the zero coupling limit,where the division of dimensions as per the algorithm isexact. Implementation of the proposed technique demon-03520strates its effectiveness in disentangling subspace dimension-alities also in the case of emergent synchronized motions.The authors acknowledge F. T. Arecchi, J. Kurths, H. L.Mancini, and A. Pelaez for many fruitful discussions. Thiswork was partly supported by the European Contract No.HPRN-CT-2000-00158.@1# See H.D.I. Abarbanel, Analysis of Observed Chaotic Data~Springer-Verlag, New York, 1996!; H. Kantz and T. Schreiber,Nonlinear Time Series Analysis ~Cambridge University Press,Cambridge, UK, 1997!, and references therein.@2# F. Takens, in Detecting Strange Attractors in Turbulence, ed-ited by D. A. Rand and L. S. Young, Lecture Notes in Math-ematics Vol. 898 ~Springer-Verlag, New York, 1981!; T. Sauer,M. Casdagli, and J.A. Yorke, J. Stat. Phys. 65, 579 ~1991!.@3# P. Grassberger and I. Procaccia, Phys. Rev. Lett. 50, 346~1983!; R. Wayland, D. Bromley, D. Pickett, and A. Passa-mente, ibid. 70, 580 ~1993!; Dov Sherman and Ilan Be’ery,ibid. 80, 540 ~1998!.@4# A.K. Alparslan, M. Sayar, and A.R. Atilgan, Phys. Rev. E 58,2640 ~1998!; Christian G. Schroer, Tim Sauer, Edward Ott, andJames A. Yorke, Phys. Rev. Lett. 80, 1410 ~1998!; GongXiaofeng and C.H. Lai, Phys. Rev. E 60, 5463 ~1999!.@5# See S. Boccaletti, C. Grebogi, Y.C. Lai, H. Mancini, and D.Maza, Phys. Rep. 329, 103 ~2000!, and references therein.@6# T. Sauer, Phys. Rev. Lett. 72, 3811 ~1994!.@7# F.T. Arecchi, G. Basti, S. Boccaletti, and A.L. Perrone, Euro-phys. Lett. 26, 327 ~1994!; S. Boccaletti, A. Giaquinta, andF.T. Arecchi, Phys. Rev. E 55, 5393 ~1997!.@8# A.M. Fraser and H.L. Swinney, Phys. Rev. A 33, 1134 ~1986!;T. Buzug and G. Pfister, Physica D 58, 127 ~1992!.@9# M.B. Kennel, R. Brown, and H.D.I. Abarbanel, Phys. Rev. A45, 3403 ~1992!.@10# L. Cao, A. Mees, K. Judd, and G. Froyland, Int. J. BifurcationChaos Appl. Sci. Eng. 8, 1491 ~1998!; T. Aittokallio, M. Gyl-lenberg, J. Hietarinta, T. Kuusela, and T. Multamki, Phys. Rev.E 60, 416 ~1999!; Rainer Hegger and Holger Kantz, ibid. 60,4970 ~1999!; David M. Walker and Nicholas B. Tufillaro, ibid.60, 4008 ~1999!.@11# J. Theiler, S. Eubank, A. Longtin, B. Galdrikian, and J.D.Farmer, Physica D 58, 77 ~1992!; T. Schreiber and A. Schmitz,Phys. Rev. Lett. 77, 635 ~1996!.@12# H. Fujisaka and T. Yamada, Prog. Theor. Phys. 69, 32 ~1983!;L.M. Pecora and T.L. Carroll, Phys. Rev. Lett. 64, 821 ~1990!;N.F. Rulkov, M.M. Sushchik, L.S. Tsimring, and H.D.I. Abar-banel, Phys. Rev. E 51, 980 ~1995!; L. Kocarev and U. Parlitz,Phys. Rev. Lett. 76, 1816 ~1996!; S. Boccaletti and D.L. Val-ladares, Phys. Rev. E 62, 7497 ~2000!.@13# M.G. Rosenblum, A.S. Pikovsky, and J. Kurths, Phys. Rev.Lett. 76, 1804 ~1996!; 78, 4193 ~1997!.@14# S.J. Schiff, P. So, Taeun-Chang, R.E. Burke, and T. Sauer,Phys. Rev. E 54, 6708 ~1996!.@15# D.H. Zanette, Phys. Rev. E 55, 5315 ~1997!; P. Parmananda,ibid. 56, 1595 ~1997!; A. Amengual, E. Herna´ndez-Garcı´a, R.Montagne, and M. San Miguel, Phys. Rev. Lett. 78, 4379~1997!; S. Boccaletti, J. Bragard, F.T. Arecchi, and H.L. Man-cini, ibid. 83, 536 ~1999!; H. Chate´, A. Pikovsky, and O. Rudz-ick, Physica D 131, 17 ~1999!; S. Boccaletti, D.L. Valladares,J. Kurths, D. Maza, and H. Mancini, Phys. Rev. E 61, 3712~2000!.@16# C. Schafer, M.G. Rosenblum, J. Kurths, and H.H. Abel, Nature~London! 392, 239 ~1998!; P. Tass, M.G. Rosenblum, M.G.Weule, J. Kurths, A. Pikovsky, J. Volkmann, A. Schnitzler, andH.J. Freund, Phys. Rev. Lett. 81, 3291 ~1998!; G.D. Van Wig-geren and R. Roy, Science 279, 1198 ~1998!; A.Neiman, X. Pei, D. Russell, W. Wojtenek, L. Wilkens, F. Moss,H.A. Braun, M.T. Huber, and K. Voigt, Phys. Rev. Lett. 82,660 ~1999!; G.M. Hall, S. Bahar, and D.J. Gauthier, ibid. 82,2995 ~1999!; A.R. Yehia, D. Jeandupeux, F. Alonso, and M.R.Guevara, Chaos 9, 916 ~1999!; C.M. Ticos, E. Rosa, Jr., W.B.Pardo, J.A. Walkenstein, and M. Monti, Phys. Rev. Lett. 85,2929 ~2000!.@17# R. Brown and L. Kocarev, Chaos 10, 344 ~2000!; S. Boccaletti,Louis M. Pecora, and A. Pelaez, Phys. Rev. E 63, 066219~2001!.@18# O.E. Ro¨ssler, Phys. Lett. 57A, 397 ~1976!.@19# O.E. Ro¨ssler, Phys. Lett. 71A, 155 ~1979!.@20# T.L. Carroll, Am. J. Phys. 63, 377 ~1995!.4-4

Reconstruction Embedding Spaces of Coupled Dynamical Systems from Multivariate Data

Deposito Adademico Digital Universidad De Navarra

English

Universidad de Navarra

https://dadun.unav.edu/bitstream/10171/1607/1/2002.PRE65.035204.pdf

Reconstruction Embedding Spaces of Coupled Dynamical Systems from Multivariate Data

Abstract

Similar works

Full text

Available Versions

Dadun, University of Navarra

Deposito Adademico Digital Universidad De Navarra

Universidad de Navarra