Ciliberto, S.

Gollub, Jerry P.

English

Ciliberto, Sergio

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Haverford College Haverford Scholarship Faculty Publications Physics 1984 Pattern Competition Leads to Chaos S. Ciliberto Jerry P. Gollub Haverford College Follow this and additional works at: https://scholarship.haverford.edu/physics_facpubs Repository Citation "Pattern Competition Leads to Chaos," S. Ciliberto and J.P. Gollub, Phys. Rev. Lett. 52, 922 (1984). This Journal Article is brought to you for free and open access by the Physics at Haverford Scholarship. It has been accepted for inclusion in Faculty Publications by an authorized administrator of Haverford Scholarship. For more information, please contact nmedeiro@haverford.edu. VOLUME 52, NUMBER 11 PHYSICAL REVIEW LETTERSPattern Competition Leads to Chaos12 MARCH 1984S. Cilibertot' and J. P. GollubDepartment ofPhysics, Haverford College, Haverford, Pennsylvania 19041, and Department ofPhysics,University ofPennsylvania, Philadelphia, Pennsylvania 19104(Received 23 December 1983)Pattern competition has been studied experimentally in the free surface modes of a circularfluid layer that is forced vertically. Interaction between nearly degenerate modes of differentsymmetry gives rise to periodic and chaotic fluctuations. The time-dependent mode ampli-tudes have been measured with digital imaging methods. Many of the observed phenomenacan be explained by a low-dimensional coupled oscillator model that is related to the Navier-Stokes equations.PACS numbers: 47.20.+m, 03.40.6c, 47.35.+iSome hydrodynamic systems, such as Rayleigh-Benard convection in a layer whose lateral dimen-sions are sufficiently small, or Couette-Taylor flowin a system whose vertical height is not too great,are known to have chaotic states that can bedescribed by few-dimensional strange attractors. 'Unfortunately, the physical origin of the chaoticbehavior in these systems is still unclear, andpredictive models are not generally available. Inthis paper we report experiments in which chaoticbehavior clearly arises from a simple mechanism,competition between two different spatial modes orpatterns. We suggest a phenomenological modelthat is related to the wavier-Stokes equations andexplains many of the observations. We alsodemonstrate that the time-dependent mode ampli-tudes can be obtained by digital imaging methods,even when several modes are present.The system of interest is a cylindrical fluid layerin a container that is subjected to a small verticaloscillation. This system was first studied experi-mentally by Faraday, 4 and the primary instabilitywas explained in an inviscid linearized approxima-tion by Benjamin and Ursell. Although there havebeen several recent experimental studies of thedynamics of this system, the interaction betweendifferent spatial patterns has not received quantita-tive study. It is well known that if the driving am-plitude exceeds a critical value A, (v) which is afunction of frequency, the free surface develops apattern of standing waves. The surface deformationS(r, 0, t) can then be written as a superposition ofnormal modesS(„&t) = X, a, ~(r) Jl(k, ~r) cos(&&)where JI are Bessel functions of order l and the al-lowed wave numbers kI are determined bythe boundary condition that the derivativeJI'(k& R) = 0, where R is the radius of the cylinder.The modes may be labeled by the indices i (givingthe number of angular maxima) and m (related tothe number of nodal circles). The mode amplitudeal (r) develops an instability when the correspond-ing eigenfrequency (given by the dispersion law forcapillary-gravity waves) is approximately in reso-nance with half the driving frequency v, and Aexceeds A, . This parametric instability leads tostanding waves in which the mode amplitude oscil-1lates at —,v.In our experiments, the container (R = 6.35 cm)is mounted on the cone of a loudspeaker oscillatingaccurately in the vertical direction, and the fluid iswater of depth 1 cm. Surface wave patterns are stu-died by refraction: An expanded parallel laser beamtraverses the cell vertically and impinges on atranslucent screen located 9 cm above the fluid sur-face. The intensity field on the screen is convertedto an analog signal by a vidicon camera and then di-gitized in —„s(with a resolution of 320 by 240points, and 8-bit precision) with use of a fastanalog-to-digital converter and frame store residingon a computer bus. We synchronize the digitizationwith the forcing to insure a constant phase relativeto the fundamental oscillation at —,v (typically 7—10Hz). This allows us to study the slowly varying partsof the mode amplitudes, which we denote by al (r).We omit the second subscript because in practiceonly a single value of m is significant for a givenvalue of jt.Examples of digitized optical intensity fieldsformed by stable patterns involving only a singlemode (and possibly harmonics) are shown in Fig. 1for the (7,2) and (4,3) modes. The index l is obvi-ous from the symmetry, awhile m was determined bymatching the frequency to the known dispersionlaw. (The accuracy of the calculated and observed1984 The American Physical SocietyVOLUME 52, NUMBER 11 PHYSICAL REVIEW LETTERS 12 MARCH 1984120-10080E60401fjy'l2015.7 15.8 15.9 16.0 16.1 16.2 16.3 16.4v (Hz)FIG. 2. Portion of the phase diagram as a function ofdriving amplitude A and frequency v. Stable patterns oc-cur in the regions labeled by (l, m). Slow periodic andchaotic oscillations occur in the cross-hatched regions.The solid curves are the predictions of a phenomenologi-cal mode-coupling model (see text). The crosses are ex-perimental measurements of the phase boundaries.FIG. 1, Digitized optical intensity patterns for the(4,3) and (7,2) modes, respectively. The first indexgives the number of angular maxima.frequencies is sufficient to eliminate any ambiguityin m. ) The white areas correspond to surfacedepressions (typically about 0.5 mm), and the blackones to surface elevations. The driving amplitudeA was about 1.1A, and the frequency was at theminimum of the stability curve in each case. Wehave observed at least 30 distinct modes with t andm between 1 and 7.A small part of the phase diagram of the systemas a function of A and v is shown in Fig. 2. Thereare three nearly degenerate modes separated byonly about 0.2 Hz. The parabolic curves show thecritical amplitude A, (v) for the labeled modes.Below these curves, the surface is nearly flat (ex-cept for a small response at the driving frequency).Above but near the minimum of each curve, themotion is dominated by a single mode, and the pat-tern is stationary. Pattern competition takes placein regions close to the intersections of two stabilitycurves. In these regions, we find that both of theneighboring amplitudes are nonzero and that theyoscillate periodically or chaotically at a mean fre-quency that is more than two orders of magnitudesmaller than v. We have made a detailed study ofthe slow variations by digital imaging and spatialFourier analysis.We proceed by first integrating the digitized lightintensity field over radial segments to reduce noise.We then obtain the relative contributions of dif-ferent values of l by calculating the magnitudesquared of the complex Fourier series for theresulting angular intensity function I(0). Theheight of this "angular power spectrum" P(l) atgiven l is approximately proportional to the squareof the mode amplitude aIO(t). (The constants ofproportionality depend on radial integrals of the JI.)Although it is possible to do a two-dimensionalnormal-mode analysis in both the angular and radialvariables, this dramatically increases computationaltime without providing additional insight.An example of the angular intensity functionI(H) and corresponding angular spectrum P(l) forthe competition between l = 7 and l = 4 modes isshown in Fig. 3. There are seven angular maxima,and the dominant spectral peaks are at l=7 andI = 14. However, there is also a small peak at I = 4corresponding to the admixture of a small amountof this mode into the surface displacement. In theregime of pattern competition, we find that theheights of the peaks are time dependent. By doingangular Fourier analysis at many times and plottingthe square roots of the spectral heights P(7) andP(4), we obtain (Fig. 4) the time dependence ofthe mode amplitudes a7 and a4. The slow oscilla-tion resulting from mode competition is periodic inthis case, and a70 leads a40 by about 90'. This phaserelationship is significant; it implies that (7,2)923NUM&ER 1yo+UME 52~ Ey)EW LETTERH YSICAL 12 MARcH 19840C]41048 60 72012 240I {s)modee of the slo yandl varyinge dePendence& 99 ~mF]G 4. Timnd af ( I)f „nd at ]oweramplitudhaotic osc1lla) ( I a .are puii13 Hdriving frequenc8I=.„,f the i=4 '"~ the interactdescribes weset2mpdes,+p74a4i2 and t]'7e4=&4+ "",and P74h""coefficients P47 "may a»oe coup]ing pschemesposite s g"b] results. ) Fin Yt to expresspduceles are wide Y,l ]y varyingsensi el separa eatl tudes illthe timterms ofode am, (,).unc3216—5011(g) and c rattern I .t 7ar intens1 y "a state in w»cFIG 3. Ang "m p(i)for '99 p, m «dspectrumt (g = 'pnding angboth presethe angularspo=4 modes areeak heights 1nand I=,The relat1ve p16113Hz .spectrum osctions +I( I)+ g, (r) sin(~vr) 'g ( I) cos(~v Iai(r) =t fr0( I) resu pm sam']la-d amp»tud ' 'the fast p' 'The measurewith respect of C, andfixed phaseinatlPnpllng at aare hence„d ( are a]inear corndjusted totion, andfftctents yl ancoefficientsNpw the coethe nonlinear coli-;];ty curves ant saturation ampfit the stabi Ybtain the correc'le mode is(Iarechoion where 0 yhe resultingsen to oa singea]] in aintegratetudes,umerically»l variablesresenl sy~t~m fprfour-dimensi'"d find that reg'"'fact pro-p&, C4 a;odic and chapd „stthe twptionsteractlon regan oscillato yduced». ff ients toh exPeri-the int robtainode-couP ' g .to that fou. a« the]in coe icnd in t emmIlar in sizeboundariesdomain sredicted phased of the bifurca-ents. The P. A detailed stu Y»]] bes~lld /ines in ' f th, s non]inc;nsomestructure9 ]t resemb 'fourtipnelsewher~endulum, apresentedd spherical psed recent-ts the fPrcebeen discusresPectstern that has]y byshow in Figf frequency as oFina]]y wa function Pslow o 1 t'pns asling tp obtain3pcps2~v I Ia =(Ial 'a( +~ a(+ ~(als damping'enfrequency, 7l thresho ld;s the elgthe critlcawhere O'Id terminesb term thatt that e.the cu lccpefficien;s the coee and ~tII 'fficient othe gaing (v), ~ih f the moded to describee 'rowth onprderimits the gf rcing term.e that the' ' ntpfthe oe assumcoeffictenb tween modesample, tothe lnteractire couPled.lpns eFQr e&fficients again coe i'as pne pr'llatiPn varyth„mps( . f the slow '2. Moso erties on fFtgtThe Pr P-hatched regifrequencYthe cross-ln driving .f'traverses11 reductionion of Figa smashaded regio;mportan yd f the largelearly have(to the lefpscillationsbserved at third o .that cd to chaoticWe have o2)d frequency sput cannot yelea sectra.t say lbroadbanb'furcation, b"d doublings.period oulete sequencet; n structure ph study ptability i"A thorpug;rebetter s .d becausewill requlrr achieve 'this syst™have so fahas a widthproPertlesf mode comPthan weetltiontjre regime o;„vthat cted a simPh fact that ine have'motivated yNavier-constru.b t eose fPrm is'tipn to t emodel who' 'd approxima'de fp lows ainvlSCle amplitulineanze,each moded ce d~mpingStokes quatipn. W) and mpde uations,5 e intro ued Mathieu egdivergence,force(tp preventonlinearityf r each mpdecou924VOLUME 52, NUMBER 11 PHYSICAL REVIEW LETTERS 12 MARCH 19g450'UCL30"cdXX5This work was supported by Grants No. MEA-7912150 and No. MEA-8310933 from the NationalScience Foundation. One of us (S.C.) ac-knowledges the support of a NATO-Consiglio Na-zionale delle Ricerche fellowship. We thank P.Hohenberg, T. Lubensky, and L. Narducci for help-ful discussions.1016.08 16.13v (1-iz)16.18FIG. 5. Mean period of the regenerative oscillation asa function of driving frequency for 2 =99 p, m. Solidpoints are experimental, and crosses are obtained fromthe model.crosses the 0.1-Hz wide interaction region. Thesolid points. are experimental data, while the crossesare the results of the numerical model. We do notshow numerical results below 16.10 Hz because thefrequency spectrum is extremely broad in that re-gion. The numerical results for the period and itsdependence on frequency are roughly correct andcould be improved further by tuning the model.Summarizing, we have shown that the interactionof two modes with different symmetries gives riseto slow oscillations and chaos, and have described acoupled oscillator model whose form is suggested inpart by the hydrodynamic equations. We believethat pattern competition might well be a fairly com-mon source of chaos.(~)Permanent address: Istituto Nazionale di Ottica,Florence, Italy.&B. Malraison, P. Atten, P. Berge, and M. Dubois, J.Phys. (Paris), Lett. 44, L-897 (1983).2A. Brandstater, J. Swift, H. L. Swinney, A. Wolf, J. D.Farmer, E. Jen, and P. J. Crutchfield, Phys. Rev. Lett.51, 1442 (1983).3J. Guckenheimer and G. Buzyna, Phys. Rev. Lett.51, 1438 (1983).4M. Faraday, Philos. Trans. Roy. Soc. London 299, 319(1831).5T. B. Benjamin and F. Ursell, Proc. Roy. Soc. London,Ser. A 225, 505 (1954).6J. P. Gollub and C. W. Meyer, Physica (Utrecht) 6D,337 (1983).7R. Keolian, L. A. Turkevich, S. J. Putterman, and I.Rudnick, Phys. Rev. Lett. 47, 1133 (1981).8This assumes that the optical intensity field is linear inthe surface wave height, and that only a single value of mis significant for each l. In fact, optical nonlinearitycreates extra spatial harmonic content.9Nonlinear models have been used to describe thebehavior of a single mode by F. T. Dodge, D. D. Kana,and H. N. Abramson, AIAA J. 3, 685 (1965).~OJ. Miles, to be published.925

Pattern Competition Leads to Chaos

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Pattern Competition Leads to Chaos

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