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International Journal of Modern Physics B Vol. 17, Nos. 22, 23 & 24 (2003) 4139–4148 c World Scientific Publishing Company

HIERARCHICAL STRUCTURES IN SPATIALLY EXTENDED SYSTEMS

JIAN LIU† and ZHEN-SU SHE∗,†,‡ † State

‡

Key Laboratory for Turbulence and Complex Systems and Department of Mechanics and Engineering Science, Peking University, Beijing 100871, P.R. China Department of Mathematics, UCLA, Los Angeles, CA 90095, USA QI OUYANG

School of Physics and State Key Laboratory for Mesoscopic Physics Peking University, Beijing, 100871, P.R. China X. T. HE Institute of Applied Physics and Computational Mathematics, P.O. Box 8009, Beijing 100088, P.R. China

Received 8 August 2002 Hierarchical Structure (HS) analysis is applied to numerical solutions of two-dimensional complex Ginzberg–Landau equation (CGLE). We show that both the ordered spiral waves and the spatiotemporal chaotic states of the CGLE satisfy the She-Leveque hierarchical symmetry [She and Leveque, Phys. Rev. Lett. 72, 336 (1994)]. The HS parameter β is measured for the CGLE solutions, which characterizes the statistical similarity between fluctuation structures of different intensities at different scales. The value of β is consistent with that recently obtained from an analysis of experimental spatiotemporal fluctuations. The analysis is able to reveal two different mechanisms of breaking-up: far-field and core breaking-up in the developement of spatiotemporal chaos.

Complex spatiotemporal dynamics in spatially extended systems has received considerable attention over the past three decades. Typical examples include hydrodynamic turbulence, nonlinear optics, biological systems and oscillatory chemical systems. Spiral patterns are one of the most intriguing spatiotemporal structures in macroscopic systems driven far from thermodynamic equilibrium. Despite extensive theoretical and experimental effort with fruitful results documented in literature,1 our understanding is yet limited to the initial phase of the instability which results in breaking spiral wave and the early transition to disorder states.2 When spontaneous spatiotemporal disorder is developed from spiral instabilities, appropriate methods ∗ Correspondance:

[email protected] 4139

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of analysis remain to be elucidated. It is not trivial to describe the structure and statistical properties of turbulent states beyond the development of spatiotemporal chaos.2 In this paper, we develop a statistical method for the analysis of spatiotemporal chaos, which is derived from a hierarchical structure model of turbulence.3 The method is for the first time applied to analyze numerically simulated spatiotemporal chaotic field. The numerical computation is carried out for the two-dimensional complex Ginzburg–Landau equation with parameters that give rise to spatiotemporal patterns of ordered and turbulent states. We demonstrate that the She-Leveque hierarchical symmetry3 is satisfied for the solutions of GL equations, with a measurable HS parameter β that characterizes similarities among fluctuations of various intensities. We conclude that spatiotemporal chaotic fields are at a self-organized state and are intermittent in nature. The method of analysis originated from a scaling model for hydrodynamic turbulence3 which is later developed to a hierarchical structure (HS) model4 with a formulation of the method of HS analysis.5– 7 According to the HS model, during the cascade process, turbulence generates a wide range of small eddies and, at the same time, a wide range of strong eddies (of high intensity). The later is often referred to as the intermittency effects. Hence, small-large eddies come along with strong-weak eddies, their description should be unified. Under a suitable definition of a hierarchy of intensities, such a unified description is made possible. The intensities of eddies of successive order are assumed to have an invariant relation, called hierarchical symmetry.4 The She-Leveque hierarchical symmetry3 is one of such generalized similarity law governing self-organized multiscaling fluctuation fields such as fully developed hydrodynamic turbulence. The hierarchical symmetry is an extension of the Kolmogorov complete scale-invariant self-similarity, which have been proved to describe correctly intermittent effects and anomalous scaling behavior in turbulence. The hierarchical symmetry can be tested for any experimentally detected fluctuation signal or for any numerically simulated fluctuation field. The method of β-test has been developed in Ref. 6 and applied to turbulent Couette–Taylor flows,5 flows in rapidly rotating annulus,8 and several other numerically simulated turbulent flows.7 When the β-test passes, the fluctuation field is said to posses the hierarchical symmetry property and the HS parameter β can be measured, which describes how intermittent the field is. When β = 1, the field is not intermittent, either extremely ordered or extremely disordered. When β = 0, the field is extremely intermittent which is dominated by one kind of intermittent structure. A variety of spatiotemporally turbulent fields have been shown to satisfy the hierarchical structure model including diffusion-limited aggregation (DLA),13 natural image,14 geophysical and geological system,15 etc. In this work, the system of interest — spatially extended system of chemical reactions — does not have the same cascade process of energy from large to small scales leading to the generation of a wide range of small eddies as in a high Reynolds

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number hydrodynamic turbulence. Nevertheless, when spontaneous spatiotemporal chaos are developed, spatial patterns display a wide range of irregularities with variation in size and in magnitude. Our analysis is suitable to study correlations among such variable structures. We use similar concepts of small and large eddies, and weak and strong eddies to denote fluctuation structures of various kinds. The complex Ginzburg–Laudau equation (CGLE) is written as: ∂A = A + (1 + ic1 )4A − (1 + ic3 )|A|2 A, ∂t

(1)

where A(r, t) is a complex function of time t and space r and the real parameters, c1 and c3 are coefficients characterizing linear and nonlinear dispersion. Although the CGLE is supposed to be relevant only near the threshold of a supercritical Hopf bifurcation, it has been found to be capable to reproduce correct qualitative behavior over a much larger range of the parameters. The two-dimensional CGLE [Eq. (1)] describes the time evolution of a twodimensional pattern. This equation appears in a wide class of spatially extended systems when oscillations and waves are present. Indeed, our computation gives rise to both simple spiral waves with a fixed core and turbulent states in which one finds many disordered spatiotemporal oscillations. The numerical study of Eq. (1) is performed using an Euler algorithm with no-flux boundary on a 256 × 256 square lattice and a time step of 4t = 0.03. A few snapshots of typical spatial pattern are shown in Figs. 1 and 2, which is a gray-scale picture of the real part f of the complex amplitude field A = f + ig. As in some experiments,9 we use simple spiral waves as the initial field which are stable at the parameters c1 = −1.4 and c3 = 0.5 [Fig. 1(a)]. While keeping c1 unchanged, we gradually vary the value of c3 to obtain other states. At c3 = 0.8, the pattern becomes quite chaotic both spatially and temporally [Fig. 1(c)]. In our simulation, convective instability occurs at c3 = 0.69, where the spiral wave breaks up far from the tip, then grows towards the center, and finally dominate the whole computational box. When c3 = 0.75, a coexistence of the spiral wave and spatiotemporal chaos occur [Fig. 1(b)]. This spiral-breaking process, due to the long wavelength, has been observed in experiments.10,11 The complexity of the spiral wave dynamics in excitable medium is such that there may be more than one turbulent state. It was found experimentally16 that a different instability mechanism may be at work when spiral waves become unstable, namely a Doppler instability can be observed in which spiral waves break near the spiral core/tip. The explanation is that when a Hopf bifurcation takes place, the spiral core may meander and the meandering motion may develop a chaotic slow dynamics. We have attempted to find such Doppler-instability induced chaos in the CGLE. But instead, we find something different. At c3 = 0.8, we find such initial spiral wave fields which undergo a process of core breaking-up without meandering and then subsequently develop to a complete spatiotemporal chaotic pattern. In Fig. 2, three snapshots of the spatial pattern are shown at successive times: (a) at t = 330, a single spiral wave is observed [Fig. 2(a)] with super arms; (b) at t = 420,

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Fig. 1. Snapshots of spatial patterns of the real part f (r) at three different parameters of the two-dimensional CGLE. (a) stable spiral, c3 = 0.60; (b) coexistence of a spiral and spatiotemporal chaos, c3 = 0.75; (c) full spatiotemporal chaos, c3 = 0.80. Note that the development of a chaotic state goes through a so-called far-field breaking-up scenario.

Fig. 2. Snapshots of spatial patterns of the real part f (r) at three times showing the development of a chaotic state through a so-called core breaking-up scenario. In the two-dimensional CGLE, c3 = 0.80 is fixed. (a): t = 300; (b): t = 420; (c) t = 600.

defects are generated at the center of the spirals [Fig. 2(b)]; (c) at t = 600, full chaotic state is developed in space and time [Fig. 2(c)]. Defect mediated turbulence17 is the most chaotic state of the two-dimensional CGLE, where the density of defects varies with the parameter c1 and c3 . Such state has been characterized by an exponential decay of correlations with small correlations lengths and times. The defects are points in space and time where A = 0 and the phase can not be defined, so the medium is also called “amplitude turbulence”.18 Although defects represent an important feature of the CGLE, they often do not act as particle-like excitations; for example, they do not emit waves. They behave more like passive objects advected by the surrounding chaotic fluctuations.2 Using finite-time Lyapunov dimension, Egolf found the each defect “carries” only a few degrees of freedom.19 It is hence suggested to use the number of defects as a convenient way to characterize spatiotemporal chaos. However, due to the fact that the relation between the finite-time Lyapounov dimension and the number of defects has not been examined for different system sizes and duration intervals, the method has not been fully established. Instead of the defects, we focus on characterizing the statistical structures of chaotic fluctuations at multiple scales.

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These fluctuations occur in many other systems. We use the basic principles of the hierarchical structure model3 to examine the symmetry across scales and intensities for spatiotemporal patterns of the CGLE. In this work, we focus on analyzing the property of the real part of the amplitude field f (x, y). By symmetry, the imaginary part field has similar statistical structure, which we have computationally verified. We are particularly interested in analyzing the spatial variation, so in analogy to the analysis of the energy dissipation in turbulence, we propose to analyze G(x, y) defined as "   2 #1/2 2 ∂f ∂f + . (2) G(x, y) = ∂x ∂y The field G(x, y) can be called the total variation field. The large amplitudes of G(x, y) correspond to strong changes in f (x, y), while the small amplitudes correspond to weak variations. Their multiscaling properties can be analyzed by constructing a series of local two-dimensional integral with size `: Z Z 1 x+` y+` ε` (x, y) = 2 G(u, v)dudv , (3) ` x y and then examine the variation of the moments of ε` as ` varies. The moments Sp (`) = h|ε` |p i are calculated by a space-average of 256 × 256 pixels; when a temporally steady state is attained (e.g. in the stable spiral regime of Fig. 1(a) or in the stable chaotic regime of Fig. 1(c)), space-time averages were performed. When in turbulent states, an efficient way to examine the statistical state of fluctuations at length scale ` is to plot the probability density functions (PDF) of

Fig. 3. Probability distribution functions of ε` at two length scales (` = 75 and ` = 15) for two chaotic states: far-field breaking-up (Fig. 1(c)) and core breaking-up (Fig. 2(c)). Both chaotic states are obtained with the same parameters in the CGLE: c1 = −1.4, c3 = 0.8.

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fluctuation variable ε` . The PDFs of ε` at the two turbulent states arrived from different initial data [Figs. 1(c) and 2(c)] are shown in Fig. 3. From this figure, one notices that the two PDFs are very similar [Fig. 3(inset)] at small scale (` = 15), but different at large scale (` = 75), specially at the left wing. The left wing of the PDF correspond to probabilities at small amplitude of ε` , or weak variation of the field f (x, y) (dark parts in the grey plot of Figs. 1(c) and 2(c). We pursue to quantitatively measure this difference by the HS analysis. The HS analysis4 begins with the calculation of the moments Sp (`) mentioned above and introduce a family (hierarchy) of scaling functions: Fp (`) = Sp+1 (`)/Sp (`), each of which describes the intensity of fluctuations of certain order p. As p increases (decreases), Fp (`) is associated with higher (lower) intensity fluctuations. She and Leveque3 have proposed an invariant relation between Fp (`) and Fp+1 (`), which is referred to as a hierarchical symmetry relation (a symmetry with respect to a translation in p): Fp+1 (`) = Ap Fp (`)β F∞ (`)1−β ,

(4)

where 0 < β < 1, Ap are independent of ` and F∞ (`) = limp→∞ Fp (`), which characterizes the most intermittent structures. The term F∞ (`)1−β can be eliminated by considering the ratio β  Ap Fp (`) Fp+1 (`) . (5) = F2 (`) A1 F1 (`) From a log-log plot of Fp+1 (`)/F2 (`) vs. Fp (`)/F1 (`), one can examine the linearity (validity of the symmetry), and if successful, find the value of β from the slope of the line. This is called β-Test.6 – 8 Note that the β-test is performed for moments in a range of p and for scales ` in a suitable range. The range of p and ` are associated with intensities and scales of the fluctuation structures of interest. Wherever a result is presented below, we will specify both ranges. The parameter β measures the degree of intermittency of a turbulent flow. The limit β → 1 correspond to no intermittency, which may be realized at either extremely ordered state or extremely stochastic state (e.g. Kolmogorov’s complete self-similar state). We have found indeed evidence of a β = 1 state for the spiral wave in Fig. 4. However, when spatiotemporal chaos are developed, we find β < 1. That is, intermittency is developed in spatiotemporal chaos. Here, β < 1 denotes a mixture of a hierarchy of structures of different intensities, each having slightly different scaling property. The limit β → 0 correspond to the extremely intermittent case where only one kind of intermittent structure dominate, the most intermittent structures F∞ (`). One example of this limit is the black and white β model20 where only one type of structures (white) are responsible for the energy dissipation. Figures 4–7 report the preliminary results of the β-test on the two sets of spatiotemporal states. In Figs. 4 and 5, we examine properties of large intensity fluctuations by performing the β-test on moments from p = 1 to p = 6, which are dominated by the right wing in the PDFs of Fig. 3. In Figs. 6 and 7, the moments

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(a)

(b) Fig. 4. β-test for high-intensity fluctuation fields during a process of (a) far-field breaking-up [Figs. 1(a) and 1(c)] and (b) core breaking-up [Figs. 2(a) and 2(c). Moment orders p are from 1 to 6 and the averaging length scales ` are from 30 to 128 (in a box size of 256). The values of β are obtained by a least-square fit which yield straight lines shown with the data points. Both F p /F1 and Fp+1 /F2 are normalized with the values at the largest legnth scale so the data points should pass through the origin. For clarity, the second set of data points are displaced vertically up by a suitable amount. The inset in (a) shows an enlarged range for c3 = 0.6.

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(a)

(b) Fig. 5. β-test for low-intensity fluctuation fields during a process of (a) far-field breaking-up [Figs. 1(a) and 1(c)] and (b) core breaking-up [Figs. 2(a) and 2(c)]. Moment orders p are from -1 to -6 and the averaging length scales ` are from 44 to 128 (in a box size of 256). The values of β are obtained by a least-square fit which yield straight lines shown with the data points. Both F p /F−1 and Fp−1 /F−2 are normalized with the values at the largest legnth scale so the data points should pass through the origin. For clarity, the second set of data points are displaced vertically up by a suitable amount. The inset in (a) shows an enlarged range for c3 = 0.6.

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from p = −1 and p = −6 are used to explore the lower intensity fluctuation parts of the fields, which are dominated by the left wing in the PDFs of Fig. 3. Both the ordered states and spatialtemporal chaos states are studied. The value of β is close to one for the ordered spiral states of Figs. 1(a) and 2(a), confirming our earlier argument. After chaos develop, Fp for different p vary over a wider range, indicating the development of a wide range of intensity for fluctuations. Yet, the linearity still hold and the hierarchical symmetry is verified in the chaotic regime. The presence of the hierarchical symmetry is a manifestation of the selforganized multi-scale correlations. From a quick look at the graph of Figs. 1(c) and 2(c), large-scale structures may not be obvious. But the HS analysis reveals the existence of a large-scale modulation of small-scale strong fluctuations, since the averaged moments over larger `’s are related to small `’s through the HS formula. This correlation is quantitatively described by the parameter β in the model. In a recent work, we have applied the analysis to characterize spatiotemporal chaos in an excitable medium obtained experimentally, which has successfully identified two different chaotic states.9 Here, our finding is that a β around 0.9 is obtained for both high and low intensities fluctuations and for both chaotic states of far-field breaking-up [Fig. 1(c)] and of core breaking-up [Fig. 2(c)], indicating that the chaotic states arrived from two different processes seem to have the same multiple scales and multiple intensities correlation. Hence, the field of Figs. 1(c) and 2(c) are equally intermittent, although the PDFs in Fig. 3 show some degree of differences. This value of β is consistent with the previous measurement in experimental spiral wave turbulence.9 In conclusion, in this preliminary study, we demonstrate that the hierarchical symmetry is satisfied in developed spatiotemporal chaos. Instead of characterizing defects, the HS analysis captures the correlation between fluctuation structures of different intensities; the targeted structure for analysis is more general than the mere defect structures. The HS analysis shows that the far-field breaking-up and the core breaking-up process at c3 = 0.8 for the two-dimensional CGLE leads to the same spatiotemporal chaos. We have benefited from useful discussions with Weidong Su and many other students at the LTCS of Peking University. The service of the Turbulence Simulation Center of LTCS is acknowledged. This work was supported by the project 49975024 and 10225210 from the NSFC.

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