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Chaotic Traveling Waves in a Coupled Map Lattice

arXiv:chao-dyn/9303009v1 8 Mar 1993Chaotic Traveling Waves in a Coupled Map LatticeKunihiko KANEKO ∗Department of Pure and Applied Sciences,College of Arts and SciencesUniversity of Tokyo, Komaba, Meguro-ku, Tokyo 153, JAPANAbstractTraveling waves triggered by a phase slip in coupled map lattices are studied.A local phase slip affects globally the system, which is in strong contrast with kinkpropagation. Attractors with different velocities coexist, and form quantized bandsdetermined by the number of phase slips.

The mechanism and statistical and dy-namical characters are studied with the use of spatial asymmetry, basin volumeratio, Lyapunov spectra, and mutual information. If the system size is not far froman integer multiple of the selected wavelength, attractors are tori, while weak chaosremains otherwise, which induces chaotic modulation of waves or a chaotic itiner-ancy of traveling states.

In the itinerancy, the residence time distribution obeys thepower law distribution, implying the existence of a long-ranged correlation. Super-transients before the formation of traveling waves are noted in the high nonlinearityregime.In the weaker nonlinearity regime corresponding to the frozen randompattern, we have found fluctuation of domain sizes and Brownian-like motion ofdomains.

Propagation of chaotic domains by phase slips is also found. Relevanceof our discoveries to B´enard convection experiments and possible applications toinformation processing are briefly discussed.1IntroductionSpatiotemporal chaos (STC) is high-dimensional chaos which involves spatial patterndynamics.

It covers turbulent phenomena in general, including B´enard convection, electricconvection in liquid crystals, boiling, combustion, magnetohydrodynamic turbulence inplasmas, solid-state physics (Josephson junction arrays, spin wave turbulence, chargedensity waves and so on), optics, chemical reactions with spatial structures, etc. It isalso important in biological information processing with nonlinear elements like neuraldynamics.The coupled map lattice (CML) is a dynamical system with discrete time (“map”),discrete space (“lattice”), and a continuous state.

It usually consists of dynamical elementson a lattice interacting (“coupled”) among suitably chosen sets of other elements [1-15].The CML was originally proposed as a simple model for spatiotemporal chaos.Modelling through a CML is carried out as follows: Choose essential procedures whichare essential for the spatially extended dynamics, and then replace each procedure by aparallel dynamics on a lattice. The CML dynamics is obtained by successive application ofeach procedure.

As an example, assume that you have a phenomenon, created by a localchaotic process and diffusion. Examples can be seen in convection, chemical turbulence,and so on.

In CML approach, we reduce the phenomena into local chaos and diffusion∗E-mail address : chaos@tansei.cc.u-tokyo.ac.jp

processes. If we choose a logistic map x′n(i) = f(xn(i)) (f(y) = 1 −ay2) to representchaos, and a discrete Laplacian operator for the diffusion, our CML is given byxn+1(i) = (1 −ǫ)f(xn(i)) + ǫ/2(f(xn(i + 1)) + f(xn(i −1)))(1)One of the merits of the CML approach lies in its predictative power of novel qualitativeuniversality classes, without being bothered by the details of phenomenology.

Classesdiscovered thus far include spatial bifurcation, frozen random chaos, pattern selection withsuppression of chaos, spatiotemporal intermittency, soliton turbulence, quasistationarysupertransients, and so on [2-7].In the present paper we report a novel universality class in CML, which is related torecent experiments in fluid convection and liquid crystals: (chaotic) traveling waves. Westudy the qualitative and quantitative nature of the chaotic traveling wave, with the helpof the Lyapunov analysis and co-moving mutual information flow.In our model (1), observed domain structures are temporally frozen when the couplingǫ is small (< .45), as has been studied in detail in [5].

For larger couplings, domain struc-tures are no longer fixed in space, but can move with some velocity. For weak nonlinearity(a < aps ≈1.55), the motion of a domain is rather irregular and Brownian-like, while pat-tern selection yielding regular waves is found for larger values of the nonlinearity.

Thesetwo regions correspond to the frozen random phase and (frozen) pattern selection in theweaker coupling regime [5], respectively. Our novel discovery here is that the pattern isno longer frozen but can slowly move.The organization of the paper is as follows.

In section 2, the coexistence of traveling-wave attractors with different velocities is shown. The quantization of selected velocitiesis noted, and the basin volume for each attractor is investigated.The mechanism oftraveling is attributed to the existence of phase slips, phase-advancing units, as will bestudied in detail in section 3.

The traveling wave suppresses chaos almost completely, as isconfirmed by the Lyapunov analysis in section 4. When the size is not close to an integermultiple of the selected wavelength, weak chaos remains, which induce the modulation oftraveling wave or a chaotic itinerancy over different traveling wave states due to chaoticfrustration in the pattern.

The long-term correlation of the itinerancy is studied in section5. The flow of information in the traveling wave is characterized by co-moving mutualinformation flow in section 6.

Quasistationary supertransients before falling on a travelingwave attractor are studied in section 7. Switching among attractors by a local input isstudied in section 8, where it is shown that a single input can induce a transition of anattractor’s velocity and thus affect the entire lattice.

In a weak nonlinearity regime, themotion of a domain is no longer regular. The Brownian-like motion of domains is studiedin section 9.

If the local dynamics is not chaotic, but periodic with the period 2n, wecan have traveling kinks in the strong coupling regime. These kinks are localized in spaceand do not have a global influence, in contrast with the phase slips, as will be shown insection 10.

Discussions and a summary are given in section 11 [1].2Selection of discrete velocitiesIn the CML (1), only a few patterns with some wavenumbers are selected for large non-linearity (a > 1.55 for ǫ = .5). Examples of attractors are given in Fig.1.

Besides thenon-traveling pattern, there are moving patterns which form a traveling wave. We notethat such traveling attractors are not observed in the weak coupling regime (ǫ < .4).

Theselected velocities of the attractors in the examples are rather low, in the order of 10−3.As can be seen in Fig.1, attractors with different velocities of waves coexist. In thesimulation, the admissible velocities vp for the attractors lie in narrow bands located at±v1, ±v2, · · ·, ±vk ( e.g., .8vk < vp < 1.2vk ).

For example, v1 = .95×10−4, v2 = 1.9×10−3,

and v3 = 2.9 × 10−3, v4 = 3.9 × 10−3, for a = 1.72, ǫ = .5, and N = 100. No attractorsexist with vp ≈0 but vp ̸= 0.

There is a clear gap between the velocities of the attractorsin each velocity band; No attractors are found with different velocity from these bandsaround vk. For all parameters, vk is approximately proportional to k.—Fig.1 —One might argue that this discreteness of the velocity bands may be an artifact of ourmodel, which is discrete in both space and time.

Since the speed is very slow ( i.e., theorder of 10−3 site per step), it is not easy to imagine a mechanism to which our originaldiscreteness (the order of 1 site per step ) is relevant. To examine possible effects of thespatial discreteness, we have also simulated a CML with a much longer coupling range,following the method of §7.5 in [5]; i.e., a repetition of the diffusion procedure in the CMLof ID times per local nonlinear mapping procedure.

With the increase of ID, the range ofthe diffusion is increased, making our attractor spatially much smoother, approaching acontinuous space limit. Our traveling attractors have then, much longer wavelengths, andhave higher quantized speeds.

For example the speed band at v1 is amplified roughly 4times by choosing ID = 8 (for a = 1.72, ǫ = .5, N = 200). Thus the discreteness in spaceis not relevant to the discrete selection of velocities.The wavelength of a pattern is almost independent of the velocity of an attractor.

Thevelocity is governed not by the (spatial) frequency but by the form of the wave. Since ourmodel has mirror symmetry, a traveling wave attractor must break the spatial symmetry.The wave form is spatially asymmetric.Here this spatial (a)symmetry is not a localproperty.

Indeed, the waveform differs by domains of unit wavelength. The asymmetryis defined only through the average over the total lattice.

We have measured the spatialasymmetry bys ≡< 1NNXj=1(xn(j + 1) −xn(j))3 >(2)with the long time average < ... >. This third power is chosen just because it isthe simplest moment of an odd power, since the first powerPNj=1(xn(j + 1) −xn(j))vanishes due to the periodic boundary conditions.

In the present paper the velocity ofan attractor is estimated by virtue of the following algorithm: Find the minimum k suchthat PNj=1(xn+2m(j) −xn(j −k))2 is a minimum. Up to some value of 2m, the minimumis found for the lattice displacement k = 0.

If the attractor is moving, at a certain delay2m, the minimum is not obtained for k = 0, but for k = ±1. With the help of this delaythe speed of the pattern is estimated as ±1/(2m).

In Fig.2, we have measured the above2m over time 160000 steps, after discarding 10000 initial transients, to obtain an accuratefor the average velocity.The relationship between s and vp is shown in Fig.2, for several values of the parametera. If a≥≈1.74 ≈atr 1, the relationship is rather simple.

The velocity of an attractor turnsout to be proportional to its asymmetry s, as is plotted in Fig.2c)d). Here we note thatthere is a gap of velocity between frozen attractors (vp = 0) and traveling attractors.For an attractor with velocity v = 0, s is zero within numerical accuracy.

Thus spatialsymmetry is attained through the attraction to the non-traveling attractor, starting froman initial condition with spatial asymmetry. Again, this spatial symmetry is not a local buta global property.

Indeed, for each domain over a single wavelength, its waveform is notgenerally mirror symmetric. The asymmetry in each wave form is cancelled through thesummation over the entire lattice.

This attainment of self-organized symmetry is possibleunder the existence of traveling attractors. Indeed, for a weaker coupling regime withouta traveling attractor, all attractors have a fixed structure [5], but they are not generally

spatially symmetric. Spatial asymmetry in the initial conditions is not eliminated in thiscase.For a < 1.74, the relationship between s and vp is more complicated.Attractorswith vp = 0 can have a small non-vanishing asymmetry.

The self-organized symmetry isnot complete. The velocity gap between frozen attractors and moving ones is not seen.Furthermore the linear relationship between vp and s does not hold, although we can seea band structure of velocities.

One of the reasons of this complication is coexistence ofattractors with different periods ( or frequencies), as will be studied in the next section.—Fig. 2 a)b)c)d)—For random initial conditions, the probability to hit an attractor with the velocity 0or ±v1 is rather high.

In Fig.3, we have measured the basin volume ratio for attractorsof different velocities. A band structure of admissible velocities is found.

In each bandthere are discrete sets of admissible velocities. We have confirmed that there are manyattractors with different velocities within each band by running a long-time simulation.—Fig.

3 a)b)—–As the velocity of an attractor increases, its basin volume shrinks rather drastically(see Table I). The basin volume for v1 is often rather large.

As is shown in Fig.3, basinvolumes decrease (approximately) in a Gaussian form with the velocity of the band (exp(−K2 × const.) for attractors in the band vp = Kv1).This Gaussian decrease isgenerally observed for any parameter value, although the basin ratios for the fixed and v1attractors may vary.Table I: Velocity, asymmetry, and basin volume of fixed and traveling attractors.a = 1.73, and ǫ = .5.

500 attractors from random initial conditions are chosen to estimatethe basin volume ratio.velocity0±v1 = .95 ±v2 = 1.95±v3 = 2.9±v4 = 3.8 ×10−3asymmetry s02.55.38.312×10−5basin volume ratio34.6%23.7%7.2%1.6%0.2%Dependences of the velocity vp on the parameter a and size N are given in Figs.4-5.In these figures, we have measured the velocity by taking the average over 160000 timesteps, after discarding 800000 transients starting from several initial conditions. Theseaveraged velocities are plotted for 1.6 < a < 1.85 for N = 50 ( in Fig.4), while they areplotted over 10 < N < 250 for a = 1.73, in Fig.5.

We can see the selection of discretevelocities rather well. Velocities lie in a narrow band around vk .—-Fig.

4 —-—-Fig. 5 —-As is given in Fig.5, selected velocities slowly decrease with the system size.Oursystem has a selected wavelength R, and the fractional part of NR is rather essential forthe nature of traveling wave 2.

Except for this additional dependence, the velocity decreaseis roughly fitted by 1/√N up to N = 200. We also note that higher bands successively1See section 7 for atr, where possible mechanism for the change of s-v relationship at atr is discussed.

appear (vk with larger k), with the increase of the system size, although the basin volumefor such higher bands is rather small due to the Gaussian decay as shown in Fig.3.As has been reported, no traveling state has been observed for ǫ < ǫc ≈.402. We notethat the velocity does not go to zero as ǫ approaches ǫc from above.

For .402 < ǫ < .45the velocity lies between 1.0 × 10−3 and 1.8 × 10−3 without displaying any symptoms ofa decrease. Rather, the basin volume for the traveling attractor vanishes with ǫ →ǫc,which is the reason why only non-traveling attractors are observed for ǫ < ǫc.

The basinvolume for traveling states ( i.e., all attractors with non-zero velocities) is shown in Fig.6.—- Fig. 6 —-3Phase Slips: Local units for global traveling waveTo understand the mechanism of this velocity selection, we note that xn(i) oscillates intime.

For ǫ = .5, the oscillation is almost periodic and the period is very close to 4.Then one can assign a phase of oscillation to a lattice site i relative to (xn(i), xn(i + 1)).It is possible to assign a phase change m′π/2 (m′ = ±1) between the lattice site i andthe lattice site i + j in a neighboring domain, according to the order of the period-4 likemotion.—-Fig.7 —-When there is a phase gap of 2π between sites i and i + ℓ, it is numerically found thatthis interval unit [i, i+ℓ] maintains the traveling wave. For example, in the attractor withvelocity v1 in Fig.1, the oscillation is close to period 4, with slow quasiperiodic modulation.For periodic boundary conditions, the total phase change should be 2kπ.

The velocity iszero for an attractor with k = 0. If k = 1, there must be a sequence of 5 domains withphases 0, π/2, π, 3π/2, 2π for corresponding lattice sites i (Fig.7).

This unit is a phaseslip of 2π. A phase slip with a negative sign is defined by the mirror-symmetric patternof a positive one.

An attractor with the velocity of the band vk has exactly k (positive)phase slips, in other words, 2kπ phase change over the total lattice. (k equals the numberof positive phase slips subtracted by negative ones).

Among attractors with the samenumber of phase slips, there can be various configurations of domains.The velocityvariance among attractors within the same band depends on this configuration. Since aphase slip is localized in space, one might think that the movement is a local phenomenonlike soliton propagation.

This is not the case. In the present case, this phase slip mustpull all the other regions to make them travel, changing the phases of oscillations of alllattice points.

Thus a local slip influences globally all lattice points. Our dynamics givesa connection between local and global dynamics.

One clear manifestation of the globalaspect is the additivity of velocity. In our system, the velocity of the wave is proportionalto the number of phase slips.

This proportionality gives a clear distinction between ourdynamics and soliton-type dynamics, where, of course, the velocity of a soliton does notincrease with the number of solitons present.The phase of oscillation can clearly be seen with the use of spatial return maps, 2-dimensional plots of (xn(i), xn(i + 1)). When there is a phase slip, the spatial return mapshows a curve as in Fig.8.

A point (xn(i), xn(i + 1)) rotates clockwise with time whenthere is a phase slip, while the point does not rotate for a non-traveling attractor. Whenthere are two phase slips, the rotation speed is twice in addition to a slight change of the2See section 4 for a novel dynamical state which appears when there is a mismatch between the sizeand the wavelength.

curve. We note that the motion is smooth without any remarkable change of rotationspeed even when the lattice site lies at the phase slip region.

In Fig.8., we have plottedspatial return maps for attractors with 1,2,3, and 4 phase slips. In the figure the systemsize is 64 lattice sites while each phase slip requires 16 lattice sites.

Thus the attractorwith 4 slips (see fig. 8d) consists only of a sequence of 4 repeated phase slip patterns.

Forattractors with less than 4 slips, there can be variable configurations of domains otherthan the phase slips. Depending on the configuration, the spatiotemporal return mapsare different.When the return map shows a closed curve, the attractor is on a projection of a 2-dimensional torus.

This is the case in the state consisting only of phase slips (see fig.8d).In general, the curve is not closed and the return map forms surface rather than a curve(see fig. 8.a-c), suggesting a higher-dimensional attractor.

Indeed, in Fig 8c), for example,we can see clearly another frequency modulation. As will be confirmed in §4, the attractoris on a higher-dimensional torus.

Frequencies of quasiperiodic modulation depend on thenumber of phase slips and the configuration of the domains.—-Fig. 8 —-Some of the numerical results in the previous section are explained by the phase slipmechanism in this section.The proportionality between the asymmetry and the velocity can partially be explainedby the fact that each (positive) phase slip gives rise to a certain contribution to theasymmetry s.In Fig.2c)d), however, the proportionality holds even in a level withineach band where the number of slips is identical.

The asymmetry can depend on theconfigurations of domains, besides the number of slips. So far it is not clear why theproportionality holds even for such small changes of the asymmetry by the configurations,when the nonlinearity is large.The basin volume vs. the velocity: Let us assume that by a random initial condition,a phase change between two domains (±π/2) are randomly assigned.

( We have to imposethe constraint that its sum should be a multiple of integers of 2π, but this is not importantfor the following rough estimate). Then the probability for the sum of the phase changeobeys the binomial distribution.

For large N, the probability to have K phase slips isestimated as exp(−(K/σ)2) with σ ∝√N. Thus the probability to have K phase slips isexpected to decay with a Gaussian form with K. Thus the Gaussian form of the basinvolume ( see Fig.3b) and the 1/√N dependence of the velocity (in Fig.

4) are explained.4Chaos and Quasiperiodicity in the traveling waveTo examine the dynamics of our attractors, Lyapunov spectra are measured numericallythrough the product of Jacobi matrices [4], and are plotted in Fig. 9.

For most parameters(1.65 < a < 2.0) and sizes, the maximal exponent is zero, irrespective of the velocity ofattractors. Thus chaos is completely eliminated by pattern selection, and the attractoris a torus.

As is expected from the spatial return maps, the attractor can be a higher-dimensional torus with more than one null exponent. Between attractors with v = 0 andv = vp, there are only slight differences in Lyapunov spectra ( see Fig.

9).—-Fig. 9 —-In our model a (traveling) pattern is selected such that it eliminates chaos (almost)completely.

If chaos were not sufficiently eliminated, it would be impossible to sustaina spatially periodic pattern during the course of propagation. Such elimination of chaos

is not possible for every wave pattern, since our dynamics has topological chaos. In oursystem a wavelength R is selected.

When the size N is not close to a multiple of theselected wavelength R, there can remain some frustration in any pattern configuration,and weak chaos can be observed.In narrow parameter regimes, we have seen a chaotic traveling waves for some sizes.For example, very weak chaos is observed around a ≈1.70, if N is large, as is shown in Fig.10. The corresponding spatial return map (see Fig.11) consists of curves ( correspondingto regular traveling) and scattered points (corresponding to chaotic modulation).

It shouldbe noted that the chaotic modulation propagates in the opposite direction as the travelingwave.Lyapunov spectra are given in Fig.12, where few positive exponents exist in the trav-eling wave attractor. The number of positive exponents is small (1 ∼3) compared withthe system size N. Chaos, localized in a domain, propagates as a modulation of the wave,as is shown in Fig.10.

We also note that the spectrum is almost flat near λ ≈0. Thepropagating wave leads to a Goldstone mode giving rise to a null exponent.—-Fig.10 —-—-Fig.11 —-—-Fig.12 —-5Chaotic Itinerancy of Traveling WavesAs is shown in the previous section, there remains some frustration when forming a wavepattern if the ratio N/R is far from an integer.

When the frustration due to this mismatchbetween the size and wavelength is large, it leads to spontaneous switching among patterns(see Fig. 13).

This spontaneous switching arises from chaotic motion of each pattern, andmay be regarded as a novel class of chaotic itinerancy [13]. Global interaction is believedto be necessary to obtain a chaotic itinerancy [13].

Although the interaction of our modelis local, the phase slip globally influences all the lattice points, and thus satisfies thecondition for chaotic itinerancy.Only few remnants of curves (corresponding to the traveling structure) can be seen inthe spatial return map (see Fig.14),while scattered parts are more dominant than in thechaotic traveling in the previous section. The direction of rotation also changes with time,through the scattered points.

Both amplitudes and phases of oscillations are modulatedstrongly here.—-Fig.13 —-—-Fig.14 —-For the spontaneous switching, we need some kind of modulation of the wave. Indeed,each waveform starts to be rather irregular in space and time in advance to the switching.The wavelength, on the other hand, is not affected by the course of this switching process.In general, there can be three types of modulation of the wave; frequency, phase, andamplitude modulations.

In our example, frequency is hardly modulated (as is seen in theinvariance of wavelength through the switching), while the phase modulation (followingthe amplitude one) is essential to the spontaneous switch of traveling states.The switching occurs through the creation or destruction of a phase slip. Frustrationin a pattern leads to the distortion of a phase slip, inducing chaotic motion.

This chaotic

motion breaks the phase slip. On the other hand, there can be the creation of a slip bychaotic modulation of the phase of oscillation.

This creation or destruction of a phaseslip is a local process, but influences globally the velocity of the traveling wave.In chaotic itinerancy, long time residence at a quasi-stable state is often noted. Wehave measured the residence time distribution of a state with a given velocity.As isshown in Fig.15, all the residence time distributions Pk(t) of a k-phase-slip state (fork = 0, ±1, ±2; i.e., vp = 0, vp = ±v1, vp = ±2v1) obey the power law Pk(t) ≈t−α withα ≈1.This power-law dependence clearly indicates the long time residence at eachtraveling state.

Similar power-law dependence of a quasistable state has already beenfound for spatiotemporal intermittency in a CML [5], although the power itself is clearlydistinct.Lyapunov spectra for this frustration-induced chaos are shown in Fig.16. The numberof positive exponents is again very few (3 in the figure), whose magnitudes are very small.The chaos by the frustration is very weak and low-dimensional.The spectra have aplateau at the null exponent, implying the existence of a Goldstone mode by travelingwave.

As seen in the previous section the accumulation at null exponent is characteristicof a (chaotic) system with a traveling wave.For larger system sizes, chaotic itinerancy of waves is hardly observed. The systemsettles down to a frozen or traveling pattern after transients.

Since the number of chaoticmodes is few (O(1)), the frustration per degree of freedom is thought to decrease with N.The distortion due to the mismatch of phases is still there, but it is distributed over alarge size and is too weak to switch the pattern. The remnant frustration in a travelingwave leads to chaotic modulation of wave as is studied in §4.—-Fig.

15 —-—-Fig. 16 —-6Co-Moving Mutual Information FlowCo-moving mutual information is often useful for measuring correlations in space andtime [4].

From the joint probability P(xn(i), xn+m(i + j)), we have calculated the mutualinformationI(m, j) =R dxn(i)dxn+m(i + j)P(xn(i), xn+m(i + j))log P(xn(i), xn+m(i + j))P(xn(i))P(xn+m(i + j)). (3)In a traveling wave, we have peaks in I(m, j) at I(t, vpt) for an attractor traveling withvp.

For a quasiperiodic attractor the peak height does not decay with the time delay t,while it slowly decays for a chaotic attractor. The transmission of correlations can clearlybe seen.In a chaotic attractor, however, there is also propagation of small modulations on thetraveling wave.

As can be seen in Fig.10, this propagation is in the opposite direction tothe wave. From the above mutual information, this reverse propagation could not easilybe measured so far.

The propagation of chaotic modulation implies the flow of informationcreated by chaos [16]. One way to measure this information creation may be the use ofthree point mutual information with the use of P(xn(i), xn+m(i + j), xn+m+ℓ(i + j))[17],while another possible way of charactering a chaotic traveling wave is the use of the co-moving Lyapunov exponent [4].

A slight increase of the exponent at the traveling velocityis observed. In our case, however, chaos is too weak to give a quantitative distinction.

—-Fig. 17 —-The mutual information in the chaotic itinerancy decays with time and space, withoutany peaks at some velocity.

By the switching process, all local traveling structures aresmeared out, leading to the destruction of peaks in the mutual information at somevelocities (see Fig.18).—-Fig. 18 —7Chaotic Transients before the formation of Trav-eling WavesTo fall on a traveling ( or fixed) attractor, the velocities of all local domains of unitwavelength must coincide.

Thus it is expected that the transient time before falling on anattractor may increase with the system size. As for the transient behavior, our transientwave phase splits into the following two regimes.

(i) For medium nonlinearity regime (a < atr ≈1.74), the transient length increases atmost with the power of N. Indeed, local traveling wave patterns are formed within a fewtime steps. Before hitting the final attractor, these local waves are slightly modulated toform a global consistency.

The formation of a global wave structure occurs for time stepssmaller than O(N). We need time steps in the order of O(N) for the slight modulationto adjust the phases of all domains.

( see Fig. 19a)b) for spacetime diagram).

(ii)For larger nonlinearity (a > atr), there are long-lived chaotic transients before oursystem falls on a traveling-wave attractor. The transient length increases with the systemsize rather rapidly: the increase is roughly estimated by exp(const.

× N) [14], althoughsome (number-theoretically) irregular variation remains.In the transient process, thedynamics is strongly chaotic, and is attributed to ”fully developed spatiotemporal chaos”in [5] 3. Lyapunov spectra during the transients are shown in Fig.20, in contrast withthe spectra of an attractor.

This transient process is quasistationary (see Fig. 19c forspacetime diagram); No gradual decay is observed for dynamical quantifiers such as theshort-time Lyapunov exponent [11] or Kolmogorov-Sinai entropy.

Such dynamical quan-tifiers fluctuate around some positive value, till a sudden decrease occurs at the attractionto the regular attractor. These observations are consistent with the type-II supertran-sients often observed in spatially extended systems [10].

In a strong coupling regime, wehave found traveling wave states up to the maximal nonlinearity a = 2. Thus the fullydeveloped spatiotemporal chaos in this regime may belong to supertransients [14].We note that the linear relationship between the asymmetry s and the velocity v (insection 2) is seen only for a > atr.

This relationship may be partially explained fromthe results in the present section, although further studies are necessary for a completeexplanation: For a < atr, the pattern selection can occur locally, and some local distortionin the wave pattern may not be removed. Then spatial asymmetry can remain even fora non-traveling attractor, and the s-v relationship can be very complicated.

For a > atr,on the other hand, slight distortion in wave pattern leads to global chaotic transients.Only patterns without distortion are admissible as attractors. The s-v relationship maybe expected to be monotonic and simpler.—-Fig.

19 —-—–Fig. 20 —-3If a is not so large ( near a ≈atr), we have often observed some local traveling wave patterns duringthe transients.

This dynamics can be attributed to the spatiotemporal intermittency of type-II [9].

8Switching among attractors with different veloci-tiesBy a suitable input at a site at one time step, we can make an external switch from oneattractor to another (with a different velocity). By a local input, the structure of anattractor is changed over the whole lattice.

Local information by an input is transformedinto a global wave pattern (Fig.21). In the medium nonlinearity regime(a < atr), theswitching process occurs within a short time, without any global chaotic transients.As is expected this switching is easily attained by applying an input at site(s) in aphase slip.

For example, assume that the phases at neighboring 5 domains are given by[0, +π/2, +π, +3π/2, 2π]. By applying an input at site(s) of the third domain, the phasescan be switched to [0, +π/2, 0, −π/2(= 3π/2), 0].

Thus a phase slip is removed, leadingto a switch to an attractor with v →vnext = v −v1. We can control a switch by choosingan input site and value so that the number of phase slips is in(/de)creased.For larger nonlinearity (a > atr), the chaotic transient lasts for many time steps duringthe course of switching.

In this case the control of switching is almost impossible; it ishard to predict the length of the switching process or the attractor after the switch. Thistype of chaotic transients in the search for an attractor can be seen in some models withchaotic itinerancy [13] and in the neural activity in an olfactory bulb [15].—–Fig.

21 —–9Fluctuating Domain by ChaosIn the weak coupling case, a frozen random pattern is observed [5] for weak nonlinearity(a < 1.55). In our strong coupling case, which phase corresponds to it?

In this couplingregime, domains with variable sizes are again formed. These domains, however, are notfixed in space.

The boundary of domains here fluctuates in time. Over some time stepssome region moves in one direction locally, but then it changes the direction of traveling (see Fig.22a).

In spatial return maps, the motion of (xn(i), xn(i+1)) along a curve changesits direction with time. The boundary motion is diffusive and Brownian-like (see Fig.22).Furthermore, the size of domains can also vary (chaotically).

Domain distribution is ratherrandom. We have plotted the spatial power spectrum S(k) =< | Pj xn(j)exp(ikj)|2 >with the temporal average < ... >.In contrast with the peaks corresponding to theselected wavelength in the regular traveling wave regime, there are no clear peaks in thespectra (see Fig 24).

The decay of the power spectra with the wavenumber is consistentwith the diffusive motion of domains, while the decay in the frozen random phase in theweak coupling regime is much slower, due to the absence of such diffusive motion.In this phase, some attractors have phase slips. Again a phase slip is defined by a unitwith a sequence of domains of 2π phase advance.

In an attractor with phase slips, thepattern moves (in average) in one direction with some fluctuation. Generally the patternhas some average velocity depending on the number of phase slips, although a fluctuatingboundary of domains brings about the fluctuation of the velocity.Here chaos is noteliminated in large domains.

In this case chaos is transported along with the travelingwave. Chaos localized in large domains moves together with the wave and in the samedirection.

An example of a pattern is given in Fig. 23 with the corresponding spatialreturn maps .If a is smaller, domains of various sizes coexist, while the appearance of larger domainsis less frequent as a approaches ≈1.55, where pattern selection sets in.Chaos in the internal dynamics in a large domain is confirmed by the Lyapunov spectragiven in Fig.

25. The slope of the spectra is small at λ ≈0.

These exponents near 0 are

thought to come from the diffusive motion of the domains. Co-moving mutual informationdecays exponentially in space and time (Fig.26), implying that there are no remainingpatterns in space and time.—-Fig.

22 —-—-Fig. 23 —-—-Fig.

24 —-—-Fig.25 —-—-Fig. 26 —-10Propagating Kinks in Period-doubling MediaTo clarify the difference between the phase slips and conventional solitons, we have studiedour CML in the period-doubling regime with a strong coupling.

As is known our CMLexhibits the period-doubling of kink patterns [2]. In the lower coupling regime (ǫ < .4),these kinks are pinned at their positions.

In the strong coupling regime, some kinks canmove when they form a phase gradient in one direction ( see for example fig.27). Thisphase in-(de-)crease is possible if the period of the kinks is larger than 2 [18].

If theperiod is 4, for example, there can be a series of domains with the increase of the phase0, π/2, 3π/2, 2π, separated by 3 kinks. These kink patterns form a phase gradient, whichdrives them to move with a constant speed.As for this phase advance, these moving kinks are similar to our phase slips.

However,the kinks here are completely local. When there are two kinks at a distant position,they move almost independently with their original speeds, (until they collide).SeeFig.27, where elimination of one kink by external input does not cause any change tothe propagation of the other kink.

Furthermore, there is no discrete selection of speeds.The speed of a kink gradually varies with the phase gradient within the kink pattern. InFig.

27, change of a tail length at a kink leads to a slight change of speed. Kinks herebelong to the same class as those studied in some partial differential equations like a φ4system.

In an oscillatory medium (without chaos), we can expect the existence of kinkswith period-doubling as in the present example.—-Fig. 27 —-11Summary and discussionsIn the present paper we have reported a traveling wave triggered by phase slips.

Thevelocity of traveling attractors forms quantized bands determined by the number of phaseslips. Frozen attractors (without any phase slips) and traveling attractors with differentvelocities coexist.

The velocity of each band increases linearly with the number of slips.When the nonlinearity is large, the proportionality between the asymmetry of a patternand the velocity holds even within each band. In this case, (approximate) symmetry isself-organized for frozen attractors.Through pattern selection of domains with some wavenumbers, chaos is completelyeliminated leading to quasiperiodicity.

When there is a mismatch between the size and thewavelength, remaining frustration leads to a chaotic motion of the wave. If the frustrationis large, chaotic itinerancy over many traveling (and frozen) states is observed.

Our systemitinerates over states with different velocities of traveling. The residence time in each stateobeys a power law distribution.

If the frustration is not so large, a chaotic traveling waveis observed, where the chaotic modulation is transmitted in the opposite direction totraveling wave

It should be noted that a local phase slip affects globally the motion of the totalsystem. This is in strong contrast with the kink type propagation (also observed in oursystem in a non-chaotic region), where it propagates as a local quantity.

The additivity ofvelocity of the wave with the number of phase slips is a clear manifestation of the globalnature.By local external inputs, one can create or destroy a phase slip, and to switch to anattractor with a different velocity. By the traveling wave, information is transmitted tothe whole space within the steps in the order of O(N).

Thus the transformation fromlocal to global information is possible through this switching, which may be useful forinformation processing and control.We have noted two types of transient processes in the course of attraction to trav-eling states. When the parameter for the nonlinearity is large, a supertransient (withquasistationary measure) is observed whose transient length increases exponentially withthe system size, while such rapid increase is not seen in the medium nonlinearity regime(a < 1.75).In the weaker nonlinearity regime (a < 1.55) corresponding to the frozen randompattern, we have found fluctuation of domain sizes and Brownian-like motion of domains.Coexistence of fluctuating domains and phase slips is also noted.

In this random pattern,chaos is localized in large domains, and propagates along the traveling wave.We have analyzed the dynamics of the traveling wave with the use of spatial returnmaps, Lyapunov spectra, and co-moving mutual information flow. When chaos is sup-pressed in the pattern selection regime, the maximal Lyapunov exponent is zero implyingthat the traveling wave attractor is on a torus whose dimension depends on the number ofphase slips.

This null Lyapunov exponent remains even in a chaotic wave or chaotic itin-erancy. This exponent is due to the Goldstone-type mode corresponding to the travelingstructure.

Through the mutual information flow, we note the creation and transmission ofinformation by a chaotic traveling wave. The chaotic modulation added on the travelingwave leads to the possibility of information transmission, created by chaos [16].There is no apriori reason to deny the possibility of the traveling wave in partialdifferential equation systems.

The existence of admissible velocity bands is thought tobe due to the suppression of chaos. For convenience of an illustration, consider a partialdifferential equation with two components∂⃗φ(r, t)/∂t = ⃗F(⃗φ(r, t)) + D∇2⃗φ(r, t).

(4)If there exists a traveling wave solution ⃗φ(r, t) = ⃗f(r −vt), it must satisfy−v⃗f ′ = ⃗F(⃗f) + D ⃗f ′′. (5)If this coupled second order differential equation has a periodic solution for a range ofvelocities v, then the traveling wave ⃗f(r −vt) can be a solution of eq.

(4).Generallyspeaking, nonlinear eq. (5) has a chaotic solution for some range of v, and has windows oflimit cycles in the parameter space (v) for chaos.

This scenario implies the existence ofadmissible velocity bands for stable traveling wave solutions, as in our CML example.Traveling waves have often been studied in various experiments [19]. Quite recently,Croquette’s group has observed traveling waves in B´enard convection with periodic bound-ary condition.

Indeed this traveling wave is triggered by a unit of a 2π phase advance[21]. They have also observed chaotic itinerancy over different traveling states, when themotion in a B´enard cell is strongly chaotic [22].

It is expected that this discovery belongsto the same class as our traveling wave. It is interesting to check if attractors with differ-ent velocities coexist by applying perturbations to such experimental systems.

A searchfor our velocity bands in experiments will also be of interest. Detailed comparison withd lditill b itt if t

In the weak nonlinearity regime, the suppression of chaos is not possible, where do-mains show chaotic Brownian motion without a traveling velocity. This type of floatingdomains has some correspondence with the dispersive chaos found in B´enard convectionby Kolodner’s group [20].acknowledgementsI would like to thank K. Nemoto, Frederick Willeboordse, K.Ikeda, M. Sano, S. Adachi,T.

Konishi, J. Suzuki, for illuminating discussions. This work is partially supported byGrant-in-Aids (No.

04231102) for Scientific Research from the Ministry of Education,Science and Culture of Japan.A preliminary version of the paper was completed inFebruary of 1992, although I have had many fruitful discussions since then. I would like tothank again Frederick Willeboordse for discussions, critical comments on the manuscript,and for sending me his Doctoral thesis in advance.

My stay at Paris in June 1992 gaveme an exciting chance to encounter a beautiful experiment by Croquette’s group. I amgrateful to Hugues Chat´e for his hospitality during my stay and discussions.

This travelwas supported by Grant-in-Aids (No. 04044020) for Scientific Research from the Ministryof Education, Science and Culture of Japan.References[1] Rapid communication of the present studies is given in K. Kaneko, Phys.Rev.

Lett.69 (1992) 905[2] K. Kaneko, Prog. Theor.

Phys. 72 (1984) 480, 74 (1985) 1033[3] K. Kaneko, Ph.

D. Thesis Collapse of Tori and Genesis of Chaos in DissipativeSystems, 1983 (enlarged version is published by World Sci. Pub., 1986)[4] K. Kaneko, Physica 23D (1986) 436[5] K. Kaneko, Physica 34D (1989) 1; 36D (1989) 60[6] J. P. Crutchfield and K. Kaneko,“Phenomenology of Spatiotemporal Chaos”, inDirections in Chaos (World Scientific, 1987)[7] K. Kaneko “Simulating Physics with Coupled Map Lattices —— Pattern Dynamics,Information Flow, and Thermodynamics of Spatiotemporal Chaos” in Formation,Dynamics, and Statistics of Patterns (ed.

K. Kawasaki, A. Onuki, and M. Suzuki,World. Sci.

1990)[8] I. Waller and R. Kapral, Phys. Rev.

30A (1984) 2047; R. Kapral, Phys. Rev.

31A(1985) 3868[9] for type-I spatiotemporal intermittency, see [1],[6], and H. Chat´e and P. Manneville,Physica 32D (1988) 409.for type-II intermittency, see J. D. Keeler and J. D. Farmer, Physica 23D (1986)413; [5].See also P. Grassberger and T. Schreiber, Physica 50 D (1991) 177[10] J.P. Crutchfield and K. Kaneko, Phys. Rev.

Lett. 60 (1988) 2715 and in preparation:K. Kaneko, Phys.

Lett. 149A (1990) 105[11] K. Kaneko, Prog.

Theor. Phys.

Suppl. 99 (1989) 263[12] R J D i ld K KkPhL tt 119A (1987) 397

[13] K. Kaneko, Phys. Rev.

Lett. 63 (1989) 219; Physica 41D (1990) 38; K. Ikeda, K.Matsumoto, and K. Ohtsuka, Prog.

Theor. Phys.

Suppl. 99 (1989) 295; I. Tsuda, inNeurocomputers and Attention, (eds.

A.V. Holden and V. I. Kryukov, ManchesterUniv.

Press, 1990)[14] F. Willeboordse, Phys. Rev.

E (Brief Reports), in press (1992); F. Willeboordse,PhD thesis, Tsukuba Univ., to be submitted[15] W. Freeman and C. A. Skarda, Brain Res. Rev.

10 (1985) 147; W. Freeman, BrainRes. Rev.

11 (1986) 259[16] R. Shaw, Z. Naturforshung, 36a (1981) 80; The Dripping Faucet as a Model ChaoticSystem, (Aerial Press, Santa Cruz, 1984)[17] K. Ikeda and K. Matsumoto, Phys Rev Lett. 62 (1989) 2265[18] This type of moving kink domains is first discussed in C. Bennett, G. Grinstein, Y.He, C. Jayaprakash and D. Mukamel, Phys.

Rev. A 41 (1990) 1932.

In the case ofperiod-2 attractor, possible attributed phases are π or −π. Thus it is impossible todistinguish between the phase increase or decrease (for example the phase changeswith the sequence of π, −π, π can have neither phase advance or retardation.

)[19] D.R. Ohlsen et al., Phys.

Rev. Lett.

65 (1990) 1431; P. Kolodner, Phys. Rev.

A 42(1990) 2475 ; V. Croquette and H Williams Pjysica 37 D (1989) 295; D. Bensimonet al., J. Fluid Mech. 217 (1990) 441[20] P. Kolodner, J.A.

Glazier, and H Williams, Phys. Rev.

Lett. 65 (1990) 1579[21] V. Croquette, J.M Flesselles and S. Jucquois, private communication[22] Yanagita has recently observed the traveling wave in B´enard convection in the CMLmodel for convection given by T. Yanagita and K. Kaneko, ”CML model for con-vection”, submitted to Phys.

Lett. A.

Fig.1Amplitude-space plot of xn(i) with a shift of time steps. 200 sequential patterns xn(i)are displayed with time (per 64 time steps), after discarding 25600 initial transients,starting from a random initial condition.

a = 1.71, ǫ = .5, and N = 64. Four examples ofattractors.

(a) vp = 0 (b)vp = −v1 (c)vp = v1 (d)vp = v2Fig.2 Asymmetry s versus velocity vp for attractors started from randomly chosen 300initial conditions. The asymmetry and velocity are computed from the average of 160000steps after discarding 100000 initial transients.

N = 64 and ǫ = .5. (a)a = 1.64(b)a =1.69(c)a = 1.75(d)a = 1.8Fig.3 The basin volume for each attractor with the velocity vp, calculated from 2000random samples, for a = 1.72, ǫ = .5, and N = 100.

Obtained from 200000 steps afterdiscarding 200000 steps. The number of initial conditions fallen on the attractor with thevelocity vp is plotted as a function of vp.

(b) The basin volume ratio for attractors in eachvelocity band calculated from the data for (a).Fig.4 Velocities of attractors versus asymmetry s, obtained with the algorithm in thetext, applied per 32 steps, over 32x5000 steps, after discarding 50000 initial transients.Velocities from randomly chosen 500 initial conditions are overlayed. N = 100.

Data fora = 1.66, 1.67 · · ·, 1.85, are overlayed, for ǫ = .5. ( additional data are included from fig.5in [1]).Fig.5 The absolute values of velocities |vp| of attractors, plotted as a function of sizeN.

The velocities are computed with the algorithm in the text, applied per 32 steps, over32x5000 steps, after discarding 50000 initial transients. Velocities from randomly chosen50 initial conditions are overlayed.

a = 1.73, and ǫ = .5.Fig.6 Basin ratio for traveling wave as a function of ǫ, for a = 1.69.Velocity ofattractors from randomly chosen 50 initial conditions are examined, to count the numberof attractors with v ̸= 0.Fig.7 Space-Amplitude plot of xn(i). a = 1.72, ǫ = .5, and N = 64.

100 steps areoverlayed after discarding 10000 initial transients. This wave pattern is traveling to theleft direction.

(b) The same plot, shown per 4 steps.Arrows indicate the phase ofoscillation of the corresponding domains.Fig.8:Spatial Return Map:{xn(1), xn(2)} are plotted over the time steps n =10001, 10002, · · ·210000. a = 1.70, ǫ = .5, and N = 64.Fig.9: Lyapunov spectra of our model with ǫ = .5, starting with random initial con-ditions, discarding 50000 initial transients.

The calculation is carried out through theproducts of Jacobi matrices over 16384 time steps. N=64.

a = 1.72; for attractors with2 phase slips (solid line) and one phase slip (two examples; dotted lines), and frozenattractors without a phase slip (two examples: broken lines)Fig.10: Amplitude-space plot of xn(i) with a shift of time steps. 200 sequential pat-terns xn(i) are displayed with time (per 128 time steps), after discarding 10240 initialtransients, starting from a random initial condition.

a = 1.69, ǫ = .5, and N = 92.Fig.11: Spatial Return Map: {xn(1), xn(2)} are plotted over the time steps n =

10001, 10002, · · ·210000. a = 1.69, ǫ = .5, and N = 100.Fig.12: Lyapunov spectra of our model with ǫ = .5, starting with a random initialcondition, discarding 50000 initial transients.

The calculation is carried out through theproducts of Jacobi matrices over 32768 time steps. N=100.

a=1.69: for attractors withv = v1 (solid or dotted line) and v = 0 (broken line).Fig.13: Amplitude-space plot of xn(i) with a shift of time steps. 200 sequential pat-terns xn(i) are displayed with time (per 1024 time steps), after discarding 1024000 initialtransients, starting from a random initial condition.

a = 1.69, ǫ = .5, and N = 51.Fig.14: Spatial Return Map: {xn(1), xn(2)} are plotted over the time steps n =10001, 10002, · · ·210000. a = 1.69, ǫ = .5, and N = 51.Fig.15 Residence time distribution for a state with v ≈vk in the chaotic itinerancyof traveling wave.

The distribution is taken over 819200 time steps after 20000 initialtransients, and sampled over 500 initial conditions. a = 1.69, ǫ = .5, and N = 51.

(a)k = 0 ( staying at a frozen state) (b) k = 1 ( residence at a one-phase-slip-state) (c)k = −1 ( residence at a one-negative-phase-slip-state) (d) k = 2 ( residence at a two-phase-slip-state)Fig.16: Lyapunov spectra of our model with ǫ = .5, starting with a random initialcondition, discarding 50000 initial transients. The calculation is carried out through theproducts of Jacobi matrices over 32768 time steps.

N=51. a=1.69: Spectra from threedifferent initial conditions are overlayed.Fig.

17: Co-moving mutual information flow for the logistic lattice (1), obtained withthe algorithm in the text. I(m, tc × j) is plotted for −10 ≤m ≤10 and 0 ≤j ≤15 withcoarsegraining time tc = 256.

The probability is calculated using 64 bins and sampledover 5000xtc steps over the whole lattice. (a) a = 1.69, ǫ = .5 and N = 100: for a chaoticattractor with vp = v1.

(b) a = 1.69, ǫ = .5 and N = 100: for an attractor with vp = −3v1. (c) a = 1.69, ǫ = .5 and N = 100: for an attractor with vp = 0.Fig.

18: Co-moving mutual information flow for the logistic lattice (1), obtained withthe algorithm in the text. I(m, tc × j) is plotted for −10 ≤m ≤10 and 0 ≤j ≤16 withcoarsegraining time tc = 256.

The probability is calculated using 64 bins and sampledover 5000xtc steps over the whole lattice. a = 1.69, ǫ = .5 and N = 51Fig.19: Space-time diagram for the coupled logistic lattice (1), with ǫ = 0.5, andstarting with a random initial condition.

If xn(i) is larger than x∗(unstable fixed pointof the logistic map) , the corresponding space-time pixel is painted as black (if x > x∗2 ≡(√1 + 7a −1)/(2a) painted darker), while it is left blank otherwise. (a) a = 1.71, N = 300.

Every 128th step is plotted. (b) a = 1.73, N = 300.

Every256th step is plotted. (c) a = 1.76, N = 200.

Every 2048th step is plotted.Fig.20: Lyapunov spectra of our model with ǫ = .5, starting with a random initialcondition: Comparison of quasistationary states with an attractor. In the former, two setsof spectra in the transients states are calculated after discarding 10000 initial transients,for two different initial conditions.

They are overlayed, but agree within the linewidth ofthe figure. For the latter, the data after 1000000 steps are adopted.

The calculation iscarried out through the products of Jacobi matrices over 32768 time steps. N = 50.a =

1.88Fig.21: Switching process; Amplitude-space plot of xn(i) with a shift of time steps.200 sequential patterns xn(i) are displayed with time (per 128 time steps), after discarding40960 initial transients, starting from a random initial condition. At the time steps andlattice point indicated by the arrows, external input is applied to change the value of xn(i)at the corresponding i and n .

ǫ = .5, and N = 64 (a)a = 1.7 (b)a = 1.75.Fig.22: Amplitude-space plot of xn(i) with a shift of time steps. 200 sequential pat-terns xn(i) are plotted with time (per 1024 time steps), after discarding 20480 initialtransients, starting from a random initial condition.

ǫ = .5, and N = 100. (a)a = 1.47(b)a = 1.52.Fig.

23: Spatial return maps with the corresponding space amplitude plots: (xn(1), xn(2)),(xn(13), xn(14)), (xn(25), xn(26)), and (xn(37), xn(38)) are plotted over the time steps n =12800, 12801, · · ·, 64000, while xn(i) is plotted with time per 256 steps. a = 1.5, ǫ = .5,and N = 50; (a)without any phase slip.

(b)with one phase slip.Fig. 24: Spatial power spectra S(k) obtained from the Foureir transform of patternxn(i).

Calculated from the average over 100000 time streps after discarding initial 10000steps. ǫ = .5, and N = 2048; (a)a = 1.5 (b)a = 1.65.Fig.25: Lyapunov spectra of our model with ǫ = .5, starting with a random initialcondition.

Three examples of calculation are overlayed starting from 3 randomly choseninitial conditions, after discarding 10000 initial transients. Calculated over 32768 timesteps.

N = 50. a = 1.53.Fig. 26: Co-moving mutual information flow for the logistic lattice (1), obtained withthe algorithm in the text.

I(m, tc × j) is plotted for −10 ≤m ≤10 and 0 ≤j ≤16 withcoarsegraining time tc = 256. The probability is calculated using 64 bins and sampledover 5000xtc steps over the whole lattice.

a = 1.5, ǫ = .5 and N = 100Fig. 27 Amplitude-space plot of xn(i) (for a moving kink) with a shift of time steps.200 sequential patterns xn(i) are depicted with time (per 1024 time steps), after discarding4096 initial transients, starting from a random initial condition.

At the time steps andlattice points indicated by the arrows, the value of xn(i) at the corresponding i and n isshifted to 0 by an input. a = 1.4, ǫ = .5, and N = 64.

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