We carry out a systematic study of a novel type of chaos at onset ("soft-mode
turbulence") based on numerical integration of the simplest one dimensional
model. The chaos is characterized by a smooth interplay of different spatial
scales, with defect generation being unimportant. The Lyapunov exponents are
calculated for several system sizes for fixed values of the control parameter
$\epsilon$. The Lyapunov dimension and the Kolmogorov-Sinai entropy are
calculated and both shown to exhibit extensive and microextensive scaling. The
distribution functional is shown to satisfy Gaussian statistics at small
wavenumbers and small frequency.Comment: 4 pages (including 5 figures) LaTeX file. Submitted to Phys. Rev.
  Let

A. Pikovsky

B. A. Malomed

B. I. Shraiman

C. C. Chow

D. A. Egolf

D. Ruelle

E. D. Siggia

F. H. Busse

H. Chate

H. Greenside

H. Richter

H. W. Xi

Hao-wen Xi

I. A. Beresnev

I. L. Kliakhandler

J. D. Gunton

M. C. Cross

M. I. Tribelskii

M. I. Tribelsky

Michael I. Tribelsky

P. C. Hohenberg

P. Manneville

R. Carretero-Gonzalez

R. Livi

R. W. Wittenberg

Raúl Toral

S. Kai

S. Tajima

T. S. Parker

V. N. Nikolaevskii

V. S. L’vov

V. Yakhot

English

arXiv

Different types of spatiotemporal chaos (STC) at onset (`soft-mode turbulence') based on numerical integration of the Nikolaevskii model were studied. The Lyapunov exponents were obtained by linearizing the equation along the trajectory. The Lyapunov exponents and Kolmogorov-Sinai entropy exhibit extensive and microextensive scaling. The power spectrum is described by Gaussian statistics for small wavenumbers and frequencies.This work was supported by NATO Grant No. CRG.CRG972822, by NSF Grant No. DMR9810409, Project Nos. PB94-1167 and PB97-0141-C02-01 ~Spain!, and the Grant-in-Aid for Scientiﬁc Research ~No. 11837006! from the Ministry of Education, Science, Sports and Culture ~JapanPeer Reviewe

Xi, Haowen

Toral, Raúl

Gunton, James D.

Tribelsky, M. I.

Digital.CSIC

RAPID COMMUNICATIONSPHYSICAL REVIEW E JULY 2000VOLUME 62, NUMBER 1Extensive chaos in the Nikolaevskii modelHao-wen Xi,1 Rau´l Toral,2 J. D. Gunton,2,3 and Michael I. Tribelsky41Department of Physics and Astronomy, Bowling Green State University, Bowling Green, Ohio 434032Instituto Mediterra´neo de Estudios Avanzados (IMEDEA), CSIC-UIB, E-07071 Palma de Mallorca, Spain3Department of Physics, Lehigh University, Bethlehem, Pennsylvania 18015*4Department of Applied Physics, Faculty of Engineering, Fukui University, Bunkyo 3-9-1, Fukui 910-8507, Japan~Received 6 October 1999!We carry out a systematic study of a different type of chaos at onset ~‘‘soft-mode turbulence’’! based onnumerical integration of the simplest one-dimensional model. The chaos is characterized by a smooth interplayof different spatial scales, with defect generation being unimportant. The Lyapunov exponents are calculatedfor several system sizes for fixed values of the control parameter e . The Lyapunov dimension and theKolmogorov-Sinai entropy are calculated and both shown to exhibit extensive and microextensive scaling. Thedistribution functional is shown to satisfy Gaussian statistics at small wave numbers and small frequency.PACS number~s!: 05.45.2a, 47.20.Ky, 47.27.Eq, 47.52.1jSpatiotemporal chaos ~STC! is a subject of considerableexperimental and theoretical importance and occurs in awide variety of driven, dissipative systems @1–3#. Such cha-otic behavior in spatially extended systems is extremely dif-ficult to characterize quantitatively, as the dynamics involvesa large number of degrees of freedom. The most commonand useful tool for the characterization of chaos is given bythe Lyapunov exponents $l i%. Knowledge of this Lyapunovspectrum permits one to estimate the number of effectivedegrees of freedom of the system ~i.e., the dimension of theattractor!, using for example the Kaplan-Yorke @4# formulafor the Lyapunov dimension D(L), where L is the linearsystem size. It also permits one to test the important conceptof extensivity of chaos, defined as the case in whichlimL→‘ D(L);Ld, where d is the spatial dimension of thesystem @1,5#. An interpretation of extensive chaos is that thewhole system can then be thought of in some sense as theunion of almost independent subsystems. This was originallyproposed by Ruelle @6#, who argued that widely separatedsubsystems of a turbulent system should be weakly corre-lated, so that the spectrum of Lyapunov exponents would bethe union of exponents associated with each of the sub-systems. The question is closely related to the fundamentalproblem of ergodicity of nonequilibrium systems. If thechaos is extensive and each subsystem evolves in time prac-tically independently of the others, then in a steady ~nontran-sient! chaotic state the time average is equivalent to the en-semble average and the system should be ergodic. Muchwork has focused on attempting to characterize spatiotempo-ral dynamics in these terms ~see, e.g., @7–11#!. However, inspite of the fundamental importance of the question practi-cally all the results are related just to a few discrete coupledmap lattices @12–14# and two continuous systems, namely,the complex Ginzburg-Landau ~CGL! and Kuramoto-Sivashinsky ~KS! equations ~see, e.g., Refs. @13,15,16#!. Thisis partly due to the computational complexity of the problembut primarily to the lack of simple models exhibiting the*Permanent address.PRE 621063-651X/2000/62~1!/17~4!/$15.00requisite chaotic behavior. It would therefore be of consider-able interest to characterize quantitatively other types ofSTC.Recently attention was drawn to the existence of a newwide class of systems displaying such a behavior @17–20#.Their properties are qualitatively different from those of theCGL and KS models. In contrast to both these models thechaos is associated with smooth, random long-wavelengthmodulations of a short-wavelength pattern, with defect gen-eration being unimportant. The short-wavelength patternarises due to a single supercritical bifurcation of the Turingtype such as occurs in Rayleigh-Be´nard convection. Thelong-wavelength modes belong to a Goldstone branch of thespectrum originated in a broken continuous symmetry. Thesymmetry makes the system degenerate to the extent thatinstead of a single, unique spatially uniform state, it has acontinuous family of equivalent spatially uniform states,which may be obtained from each other by the symmetrytransformation. This symmetry, which is additional to thetrivial groups of translations and rotations, can be one ofmany different types. For this reason the STC in question isquite a common phenomenon and occurs, for example, inelectroconvection in liquid crystals @17,18#, in convection ina fluid with stress-free boundary conditions @21–24#, etc.;see Ref. @20# for further discussion. The chaos observed insuch cases may be interpreted as a macroscopic dynamicalanalog of second-order phase transitions, where the orderparameter is related to the amplitudes of turbulent modes.Due to this analogy it has been called soft-mode turbulence~SMT! @18#. The simplest model exhibiting SMT was intro-duced by Nikolaevskii @25,26# to describe longitudinal seis-mic waves in viscoelastic media. In what follows we exploitthe simplicity of this model to shed light on general featuresof this different type of STC.We present a detailed systematic study of the Nikolae-vskii model, including calculation of the Lyapunov expo-nents, fractal dimension, etc. We show that the Lyapunovdimension and Kolmogorov-Sinai entropy are extensivequantities, which supports the validity of the Ruelle’s ideasfor SMT. We also show that the power spectrum can beR17 ©2000 The American Physical SocietyRAPID COMMUNICATIONSR18 PRE 62XI, TORAL, GUNTON, AND TRIBELSKYdescribed by Gaussian statistics for small wavenumbers andfrequencies, in agreement with a general argument of Hohen-berg and Shraiman @5#.The model is defined by the following partial differentialequation for the real scalar field v(x ,t) ~longitudinal mode ofthe displacement velocity in the original formulation!@25,26#:]v]t1]2]x2F e2S 11 ]2]x2D 2Gv1v ]v]x 50, ~1!with 0<x<L and periodic boundary conditions. This modelhas two control parameters, e ~the distance from onset! andL, in contrast to, e.g., the KS model, where the only non-trivial control parameter is the system size L. An essentialfeature of the model is that even at small e it cannot bereduced to any e-free form @20,27#, which makes the hierar-chy of characteristic scales much more complex than those inthe KS and CGL models @28,29#. Equation ~1! may be re-garded as a generalized Burgers equation and shares with itthe same group of symmetry, namely, trivial symmetries un-der shifts of the spatiotemporal coordinate system, and non-trivial invariance with respect to the Galilean transformationv(x ,t)→v(x2vot ,t)1vo , where vo is an arbitrary constant.The Galilean invariance plays the role of the above specifiedadditional symmetry, generating for Eq. ~1! the continuousfamily of solutions v5vo . The Nikolaevskii equation admitsa continuous set of spatially periodic, stationary solutions,whose instability has been proved analytically @28#. Com-puter simulations @19# of an equivalent version of this modelfor an order parameter u(x ,t)[*v(x ,t)dx showed that evenat extremely small e51024, the system exhibits STC. How-ever, this simulation did not provide any quantitative resultsabout the STC. The only result of this kind is in Ref. @30#, inwhich just a single quantitative characteristic, namely, thedependence of the mean amplitude of chaotic patterns on thecontrol parameter, was studied. In this Rapid Communica-tion we span the gap in our knowledge of this type of STC,providing a detailed quantitative description of its most im-portant properties based on numerical integration of Eq. ~1!.The simulations were carried out using the pseudospectralmethod combined with a fourth-order predictor-corrector in-tegrator, for several different values of L and two values of e~0.2 and 0.5, respectively! @31#. The Lyapunov exponentswere calculated by linearizing the equation along the trajec-tory, performing a reorthonormalization after a few integra-tion steps to prevent the largest Lyapunov exponent fromswamping all the others @32#. A typical pattern v(x ,t) as afunction of space and time in the steady chaotic regime isshown in Fig. 1. The time averaged power spectrum^uvˆ (k)u2& obtained over a time period of T5104 for systemsize L578 with e50.2 is shown in Fig. 2. As can be seen,the dominant modes occur in the vicinity of k561. Note,however, the smaller peak near zero wave number, whicharises from coupling between unstable short-wavelengthmodes centered about k561 and the slowly decayingmodes from the Goldstone branch of the spectrum centeredat k50 ~using terminology based upon the linear stabilityanalysis of the spatially uniform solution!. The power spec-trum for uˆ (k)5vˆ (k)/ik ~shown in the inset in Fig. 2! has thedominant peak near k50, as was originally demonstrated in@19#. One would also expect on general grounds @5# eachFourier transform variable vˆ (k ,v) to be governedby a Gaussian probability distribution functional,exp@2Duvˆ (k ,v)u2# , for small k and v . We have verifiedthis for several values of k and v by fitting the probabilitydistribution for the variable z[uvˆ (k)u2 to an exponentialform: f (z)5De2Dz. For instance, in the particular case L578, e50.5, we obtain a good fit for v50.6, k51.5 byusing D’47.As has already been emphasized, the key question for anysystem exhibiting STC is whether it can be represented as aunion of weakly correlated subsystems. If this is the case,then the spectrum of Lyapunov exponents for the entire sys-tem should be intensive in the sense that l i is a function onlyof the intensive index i/V , i.e., l i5 f (i/V), where V standsfor the volume of the system @3#. The question is not trivialfor the type of STC considered here, because of the impor-tance of the long-wavelength modes and the divergence ofthe two point correlation length j2 @33# as e→0 @20#. Toanswer this question, a detailed study of the Lyapunov spec-trum for the Nikolaevskii model was conducted. The resultsare shown in Fig. 3 where the number of Lyapunov expo-nents greater than a particular value l i , scaled by the systemsize L, is plotted versus l i for e50.5. A similar curve isFIG. 1. Typical space-time configuration for v(x ,t) in thesteady chaotic regime is shown here. The configuration shown hasevolved from random initial condition within a system size L578and e50.5.FIG. 2. Time average power spectra ^uvˆ (k)u2& and ^uuˆ (k)u2& ink-space are plotted. The system size is L578 and control parametere50.2. Note that the power spectrum has symmetry with respect tok→2k .RAPID COMMUNICATIONSPRE 62 R19EXTENSIVE CHAOS IN THE NIKOLAEVSKII MODELfound at e50.2, but the maximum positive eigenvalue isnow smaller ~one expects this eigenvalue to vanish as e→0.! The intensive nature of the Lyapunov density is evi-dent. In this case, one also expects the fractal dimensionD(L) of an attractor to be extensive for large enough L ~ex-tensive chaos!, as was first shown by Manneville @15# forchaotic solutions of the Kuramoto-Sivashinsky equation. Wehave checked this for the Nikolaevskii model using theKaplan-Yorke formula @4# for the Lyapunov dimension,D~L !5K11ulK11u(i51Kl i, ~2!where the integer K is the largest integer such that the sum ofthe first K Lyapunov exponents is non-negative. We havealso calculated the Kolmogorov-Sinai entropy H(L) fromthe definitionH~L !5(i51i1l i , ~3!where the sum is over the positive Lyapunov exponents. TheKolmogorov-Sinai entropy @32# is a measure of the meanrate of information production in a system, or the mean rateof growth of uncertainty in a system subjected to small per-turbations. We find that for large enough L both D(L) andH(L) are extensive. Our results for the Lyapunov dimensionD(L) are shown in Fig. 4 for e50.5. The same behavior isfound at e50.2, but with a different slope ~naturally theslope of the straight line D(L) at e50.2 is smaller than thatat e50.5). In addition, we find that the upper index i1 in Eq.~3!, which corresponds to the smallest positive Lyapunovexponent, is also proportional to L.The important characteristic of STC is the dimension cor-relation length jd @1,3#. This length is defined as jd[d21/d,where d[limL→‘ D(L)/Ld. It can be thought of as the ‘‘ra-dius’’ of a volume that contains one degree of freedom, or,as the linear size of the subsystem described above. Thevalue of this dimension correlation length for Eq. ~1! is jd53.0 for e50.5 and jd53.3 for e50.2. In contrast to jd theFIG. 3. Here N.i is the number of Lyapunov exponents greaterthan a particular value l i . We plot N.i /L ~scaled by the systemsize L) vs l i in the case e50.5.above-mentioned two point correlation length j2 governs thespatial decay of the correlation function. In general these twolengths are different @3#. We found that j2>4.9 for the Ni-kolaevskii model at e50.5 and j2>5.6 at e50.2.The use of periodic boundary conditions ensures that thedynamics observed is not influenced by the boundaries andhence, in some sense, it is intrinsic to the system. Also, for alarge enough system size, periodic boundary conditions ap-proximate the thermodynamic limit of infinite size and havebeen used previously in many other similar studies@34,16,15#. It would be interesting to analyze the finite sizeeffects introduced by the use of other types of boundary con-ditions, such as rigid-rigid. However, the fact that in all casesconsidered here the system size is much larger than any ofthe correlation lengths obtained, suggests to us that similarresults would be obtained with other boundary conditions.Finally, Tajima and Greenside @35# have recently foundfor the one dimensional Kuramoto-Sivashinsky model thatD(L) is also ‘‘microextensive.’’ Namely, they found that ifone increases L by a small amount dL , with dL!jd (dL50.8 in our simulation!, one finds that D(L) satisfies thesame linear relationship as that characterizing extensivechaos. We have examined this for two different domains of Lfor our model and found that microextensivity holds for bothD(L) and H(L). Our results for microextensivity for D(L)are shown in the inset in Fig. 4.In conclusion, our detailed study of this type of STCbased on the numerical integration of the Nikolaevskii modelshows that for sufficiently large system size the chaos is bothextensive and microextensive. We also found that the systemsatisfies Gaussian statistics at sufficiently small wave num-bers and frequencies. We believe these results are quite gen-eral and reflect intrinsic features of this type of STC, ratherthan specific peculiarities of the model.There are several interesting questions to investigate inthe limit e→0, including the dependence of quantities suchas the correlation lengths and Lyapunov exponents on e asFIG. 4. Lyapunov dimension D(L) vs system size L for e50.5. dL515.5 for extensive, and dL50.8 for microextensive.RAPID COMMUNICATIONSR20 PRE 62XI, TORAL, GUNTON, AND TRIBELSKYwell as the possible scaling of the power spectrum. Thisstudy is in progress and will be reported elsewhere.We would like to thank Henry Greenside for a helpfuldiscussion. This work was supported by NATO Grant No.CRG.CRG972822, by NSF Grant No. DMR9810409, ProjectNos. PB94-1167 and PB97-0141-C02-01 ~Spain!, and theGrant-in-Aid for Scientific Research ~No. 11837006! fromthe Ministry of Education, Science, Sports and Culture ~Ja-pan!. We also wish to acknowledge an allocation of time onthe Pittsburgh Supercomputer Center, where some of thiswork was carried out. J.D.G. acknowledges support from theFundacio´n Banco Bilbao Vizcaya in the Programa Ca´tedra.@1# M. C. Cross and P. C. Hohenberg, Rev. Mod. Phys. 65, 851~1993!.@2# P. Manneville, Dissipative Structures and Weak Turbulence~Academic Press, New York, 1990!.@3# H. Greenside, in Proceedings of the CRM Workshop, Semi-Analytic Methods for the Navier Stokes Equations, edited byK. Coughlin ~American Mathematical Society, Providence, RI,1999!, and references therein.@4# J. L. Kaplan and J. A. Yorke, Chaotic Behavior of Multidimen-sional Difference Equations, Lecture Notes in MathematicsVol. 730 ~Springer, New York, 1979!.@5# P. C. Hohenberg and B. I. Shraiman, Physica D 37, 109~1989!.@6# D. 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Lett. 76, 1631~1996!.@20# M. I. Tribelskii, Usp. Fiz. Nauk. 167, 167 ~1997! @Phys. Usp.40, 159 ~1997!#.@21# E. D. Siggia and A. Zippelius, Phys. Rev. Lett. 47, 835 ~1981!.@22# F. H. Busse and E. W. Bolton, J. Fluid Mech. 146, 115 ~1984!.@23# H. W. Xi, X. J. Li, and J. D. Gunton, Phys. Rev. Lett. 78, 1046~1997!.@24# It is relevant to mention that the consideration of convectionwith the stress-free boundary conditions @21,22# was the veryfirst analysis of pattern formation in the degenerate systems.However, at that time the dramatic destabilization of steadypatterns obtained in this case was regarded as a peculiarity ofthe particular problem, rather than a generic attribute of thedegeneracy.@25# V. N. Nikolaevskii, in Recent Advances in Engineering Sci-ence, edited by S. L. Koh and C. G. Speciale, Lecture Notes inEngineering Vol. 39 ~Springer-Verlag, New York, 1989!, p.210.@26# I. A. Beresnev and V. N. Nikolaevskii, Physica D 66, 1 ~1993!.@27# B. A. Malomed, Phys. Rev. A 45, 1009 ~1992!.@28# M. I. Tribelsky and M. G. Velarde, Phys. Rev. E 54, 4973~1996!.@29# M. I. Tribelsky, Int. J. Bifurcation Chaos Appl. Sci. Eng. 7,997 ~1997!.@30# I. L. Kliakhandler and B. A. Malomed, Phys. Lett. A 231, 191~1997!.@31# Although the original numerical study @19# was carried out interms of the order parameter u(x ,t), we have found that Eq.~1! is more convenient for numerical study as it exhibitsgreater numerical stability. The reason is that the conservationlaw dv¯ /dt50 holding for this equation ~where the bar denotesa spatial average! helps to stabilize the code. In our simula-tions the value v¯50 was imposed by the initial, random, con-ditions. For the space discretization we have found that Dx50.31 provides a good enough resolution.@32# T. S. Parker and L. O. Chua, Practical Numerical Algorithmsfor Chaotic Systems ~Springer-Verlag, New York, 1989!.@33# At small e in this model the two point correlation function(^v(x ,t)v(x1x8,t)&, where the ^ . . . & denotes the time aver-age! is generally oscillatory with a spatial period of the oscil-lations close to 2p and a slow decay of its amplitude over ascale j2. Thus, the model has two correlation length scales,with j2 being the larger of the two.@34# B. I. Shraiman, A. Pumir, W. van Saarlos, P. C. Hohenberg, H.Chate´, and M. Holen, Physica D 57, 241 ~1992!.@35# S. Tajima and H. S. Greenside, Bull. Am. Phys. Soc. 44~1!,327 ~1999!.

Extensive chaos in the Nikolaevskii model

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Extensive Chaos in the Nikolaevskii Model

Extensive Chaos in the Nikolaevskii Model

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