Zig-zag chaos - a new spatiotemporal pattern in nonlinear dynamics

Zig-zag chaos - a new spatiotemporal pattern in nonlinear dynamics Iuliana Oprea 1, 2 1 Faculty of Mathematics, University of Bucharest, Romania 2 Department of Mathematics, Colorado State University, US Abstract In this paper we study a complex spatiotemporal pattern, identified as spatiotemporal chaos, bifurcating at the onset from a spatially uniform solution of a system of globally coupled complex Ginzburg Landau equations governing the weakly nonlinear evolution of four travelling wave envelopes. The Ginzburg Landau system is derived directly from the weak electrolyte model for electroconvection in nematic liquid crystals when the primary instability is a Hopf bifurcation to oblique travelling rolls. The chaotic nature of the pattern and the resemblance with the observed experimental spatiotemporal chaos in the nematic electroconvection of the nematic I52 are confirmed through a combination of techniques that include time series analysis of the amplitudes of the dominant modes, statistical descriptions and normal form theory, showing good agreement between theory and experiments. 1 Introduction Spatiotemporal chaos (STC), a deterministic pattern with unpredictable, irregular spatial and temporal variation, occurs in a wide variety of physical systems when many spatial degrees of freedom contribute to the dynamics (see [13] and references therein for a recent review). While there is nowadays a concise description of chaos in systems with few degrees of freedom, the problem of finding general approaches for the characterization of complex dynamics in high dimensional chaotic systems, as well as the existence of universal organizing principles for STC are still unsolved [1, 12, 13]. Due to the short characteristic times and large aspect ratios [2] nematic electroconvection is a well suited experimental system to find and study periodic and more complex spatiotemporal patterns like worms, defects and spatiotemporal chaos. It has been predicted by experimentalists [4] that, since STC evolves continuously from the uniform state, it is possible to use coupled complex Ginzburg Landau equations, one for each mode, with coefficients derived from the equations of motion for the nematic electroconvection, to elucidate and characterize its dynamics. A theoretical treatment is now available, via the weak electrolyte model(wem) [9, 5]. We found numerically[11], in the analysis of a system of four globally coupled complex Ginzburg Landau equations (GCCGLE) derived from WEM in [5], a complex spatiotemporal pattern very similar to the experimentally observed STC described in [3, 14], with complex spatiotemporal behavior preserving portions of structures similar to zig and zag families of travelling waves, connected by chaotic excursions in space and time. We will refer to it as zig-zag-chaos. In this paper we present a quantitative and qualitative analysis of the computed pattern. The paper is structured as follows. In Sec. 2 we present the derivation of the GCGLE; in Sec. 3 we describe the principal characteristics of the zig-zag-chaos. Sec. 4 is dedicated to conclusions and open questions. 1

2 The system of four globally coupled complex Ginzburg Landau equations The recently developed mathematical model for the nematic electroconvection, the weak electrolyte model (WEM) [9], consists of partial differential equations for the velocity field, the director, the electric potential and charge, derived from the Navier-Stokes equations for an anisotropic electrically conducting fluid, the conservation of charge, Poisson s law, and a partial differential equation for the conductivity. The WEM equations are extremely complicated, therefore a weakly nonlinear analysis near the onset is particularly useful. In our previous work [5, 10], we have analyzed stable basic wave patterns predicted by the WEM near a Hopf bifurcation of the basic state. In our Ginzburg-Landau analysis of the WEM equations, the complex amplitudes of these waves are considered as slowly varying wave envelopes A i (x i, t), i = 1, 2, 3, 4, that modulate the travelling waves in the four oblique directions. These envelopes depend on characteristic wave variables x 1 = (x +, y + ), x 2 = (x, y + ), x 3 = (x, y ), x 4 = (x +, y ), where x ± = x ± v x t, y ± = y ± v y t with (v x, v y ) the critical group velocities. The rescaled form of the dynamical equations for the envelopes results in a system of four globally coupled complex Ginzburg-Landau equations (GCCGLE) [6] for the envelopes A i. These equations are formulated in terms of slow variables ξ ± = εx ±, η ± = εy ±, τ = ε 2 t, where the small parameter ε 2 measures the distance of the bifurcation parameter from the onset value. The equation for A 1 has the form A 1t (x +, y +, t) = {a 0 + D( x+, y+ ) + a 1 A 1 (x +, y +, t) 2 + a 2 < A 2 (s, y +, t) 2 > + a 3 < A 3 (x + + s, y + + s, t) 2 + a 4 < A 4 (x +, s, t) 2 >}A 1 (x +, y +, t) + (1) a 5 < A 2 (x + + s, y +, t)a 3 (x + + s, y + + s, t)a 4 (x +, y + + s, t) >, where D is a second order differential operator, D( x+, y+ ) = D 1 x 2 + + D 2 x+ y+ + D 3 y 2 +, and the brackets denote averages over s. The equations for A 2, A 3, A 4 follow from Eq. 1 by reflection operations, see [5, 10] for details. The linear and nonlinear coefficients occurring in (1) are computed numerically from the underlying WEM equations for various choices of the parameters. To any solution of the GCCGLE we associate a spatiotemporal pattern by setting u(x, t) = Re{A 1 (x +, y +, t)e i(ω ct+p c x+q c y) + A 2 (x, y +, t)e i(ω ct p c x+q c y) + A 3 (x, y, t)e i(ωct pcx qcy) + A 4 (x +, y, t)e i(ωct+pcx qcy) }. (2) If spatial variations are ignored, the globally coupled system for the A j reduces to the normal form for a Hopf bifurcation with O(2) O(2) symmetry, that has six basic solutions corresponding to six basic wave patterns: travelling (TW) and standing waves, two types of travelling rectangles, standing rectangles, and alternating waves, which alternate periodically between differently oriented standing waves. Each of these waves can occur with different orientations, for example, a TW can propagate in any of the four directions (±p c, ±q c ), and their stability in the ODE setting is classified in terms of the nonlinear coefficients a j (see [6]). The basic periodic solutions of the above normal form induce wave solutions of the original system (1) with critical wave numbers (p c, q c ). The globally coupled system of complex Ginzburg Landau equations (1) allows to extend these solutions to families of wave solutions with nearby critical wave numbers (p c + εp, q c + εq). 3 Characteristics of the zig-zag chaos The first experiments on zig-zag spatiotemporal chaos at the onset, in the weakly nonlinear regime, for the nematic liquid crystal I52, have been reported by Dennin, Cannel and Ahlers in [3]. Similar 2

patterns have been observed in the nematic electroconvection for the nematic I52 in [14]. The zigzag spatiotemporal chaos observed in [3] consists of a superposition of four degenerate modes, namely right - and left- relatively fast travelling waves with wave vectors at angles Θ and π Θ with respect to the director (the average molecular alignment). Several statistics tools have been used in [3, 4] in the analysis of the demodulated video images to reveal the chaotic aspects of the observed pattern. The authors have suggested that, since STC evolves continuously from the uniform state, in the weakly nonlinear regime, an appropriate system of four globally coupled Ginzburg Landau equations which describe the dynamics of the envelopes of the four modes could be derived from the equations of motion, allowing quantitative and qualitative comparison between theory and experiment. They are available now in the form of the four GCCGLE described in the previous section. In the numerical simulation of the four GCCGLE (2) we found numerically, for various sets of parameters, different types of STC and other localized structures evolving at the onset. Among them we found a complex spatiotemporal pattern very similar to the experimentally observed STC described in [3, 14], with complex spatiotemporal behavior preserving portions of structures similar to left- and right- families of travelling waves, connected by chaotic excursions in space and time, which we called the zig-zag chaos. Details of the computation and values of the nonlinear coefficients a i, 1 i 5 can be found in [11]. One of the most striking characteristics of this pattern, which can be clearly seen in a movie, is an alternation between phases of irregular, chaotic behavior and phases of regular behavior. This alternation is illustrated in the three pattern snapshots of Figure 1. The pattern shown in Figure 1a was recorded at t = 2700, which is during a chaotic phase. Here we can see an irregular distribution of zig and zag waves, as well as transitions between them, and regions of low and high intensity. The patterns shown in Figures 1b,c, recorded at t = 3120 and t = 3127, occur during a regular phase. During these phases the dynamics is approximately that of an alternating wave that extends throughout the spatial domain, which is clearly apparent in the snapshots. The time series of the three most energetic mode amplitudes a j (m, n, t) of the spectral decomposition of unknowns in (2) are shown in Figure 2. The sharp downward peaks of the dominant central mode, combined with upward peaks of one or both of the other modes, signify the chaotic phases. To confirm the chaotic dynamics we have computed the Lyapunov exponents of these time series using a three dimensional embedding technique[11]. All time series turned out to have one positive and two negative Lyapunov exponents. Although the average energy content of the non central mode amplitudes is small, they still have a significant effect on the shapes of the pattern during the irregular phases. The envelope patterns A j (x, y, t) in physical space show an interesting and aesthetic spatiotemporal variation on the slow time (T/ε 2 ) and spatial (λ x /ε,λ y /ε) scales. Most of the times two envelopes exhibit regular structures, whereas one or sometimes two of the other envelopes appear spatially irregular. Both the regular and irregular structures are changing permanently, and there are regular irregular transitions from one envelope to another. A snapshot of the A j corresponding to the pattern of Figure 1a is shown in Figure 3. Here we see that A 1, A 2 appear spatially irregular, whereas A 3 and A 4 exhibit a D 2 D 4 and an O(2) D 4 translation invariance, respectively. The alternation between regular and irregular phases strongly suggests that the dynamics of our pattern can be described as chaotic excursions from a torus. In general, a system with three continuous symmetries such as our 2D GCCGLE can have structurally stable invariant tori of dimension up to five, however, the dominance of the (0, 0) modes indicates that the torus is the AW torus that occurs already in the O(2) O(2) normal form. The nature of the excursions from the torus is not yet clear. One possibility is that the parameters are close to a homoclinic bifurcation and we see a higher dimensional version of Shilnikov type chaos. Another possibility is that there exists a structurally stable homoclinic orbit in higher dimensions that is not present in 3

the O(2) O(2) normal form. To clearly decide about the nature of the pattern dynamics requires further investigation, including parameter variations and numerical studies as well as the analysis of the modulational stability of the family of AW solutions. In the case of a homoclinic bifurcation we can expect that variations of the parameters may cause the disappearance of the intermittent regular phases and lead to a fully spatiotemporally chaotic dynamics. We note that, when comparing movies, the pattern dynamics of u(x, t) during the irregular phases appears quite similar to the near onset electroconvection pattern recorded recently at Kent State University for the nematic liquid crystal I52 [14], after applying the demodulation techniques described in [3] to the images. To be able to clearly diagnose spatiotemporal chaos and to distinguish a chaotic spatiotemporal pattern from other types of complex spatiotemporal behavior, we have also use the method of Karhunen-Loève (KL) decomposition[8] combined with time-series analysis of the resulting mode amplitudes. The spectrum of eigenvalues of this decomposition, as well as the amplitudes of these spatial modes, give insight in the character of the solution and can be used to diagnose chaotic behavior. Results of the characterization of the zig-zag chaos based on the KL decomposition, a long used technique in the signal analysis and processing, will be presented elsewhere[11]. Since the KL mode decomposition distinguishes between temporal behavior, reflected in the KL modes amplitudes, and spatial behavior reflected in the KL modes themselves, this method seems to fit very well the type of spatiotemporal pattern we consider here. 4 Discussion and conclusions The overall characterization of the zig-zag chaos we develop in this paper offer deeper insight into the spatiotemporal structure of the pattern than a simple statistical analysis as performed in the experimental case. Our example shows a richness of spatial structures essential to the dynamics, since 10 KL modes are needed to fully describe the solution[11]. The dynamics of the zig-zag chaos is complex and the correlation analysis shows that even if we are not in the presence of an extensive STC, however the temporal and spatial complexity are not independent of each other, as it might happen in systems where the system size is comparable with the typical correlation length. The availability of the amplitude equations (1) allows us to find new approaches in the characterization of the zig-zag chaos. A preliminary bifurcation analysis shows that two cases are possible: either the zig-zag chaos represents a structurally stable homoclinic orbit, with families of alternating waves as saddle, or the system is close to a homoclinic bifurcation generating chaotic dynamics, like a higher dimensional Shilnikov saddle-node bifurcation dynamics. This conjecture deserves further studies: an Eckhaus stability analysis for the structurally stable homoclinic orbit and/or the addition of higher order terms in (1) or its ODE associated normal form, combined with numerical simulations, might give a definite answer. Preliminary numerical computations for other sets of parameters indicate that the GCGL equations (1) seem to be a promising way to investigate more complex spatiotemporal structures as grain boundaries, worms and other localized patterns [10]. Overall, (1) seems to be an appropriate phenomenological description of highly dimensional complex spatiotemporal phenomena. An exciting task for future research is to compare the quantitative, experimentally determined statistical measures of these states with the calculations based on the equations of motion and the tools developed in this paper. Such work, relevant to real experimental data, is in process[7] 4

(a) (b) (c) Figure 1: Three pattern snapshots. a: Typical pattern observed in a chaotic phase. b,c: Two patterns in a time range in which the dynamics is approximately an alternating wave. In this and the other 2D patterns shown, maxima and minima are mapped to red and blue, respectively. 5

0.45 0.45 0.4 0.4 0.35 0.35 0.3 0.3 a 1 (m,n) 0.25 0.2 a 2 (m,n) 0.25 0.2 0.15 0.15 0.1 0.1 0.05 0.05 0 0 50 100 150 200 250 300 (a) t/t 0 0 50 100 150 200 250 300 t/t (b) 0.5 0.5 a 3 (m,n) 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 0 50 100 150 200 250 300 t/t (c) a 4 (m,n) 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 0 50 100 150 200 250 300 t/t (d) Figure 2: Time series a j (m, n, t) for the three modes (m, n) with the highest energy content. Blue: dominant mode (0, 0), red and green: second and third mode ( 1, 0) and (0, 1) for a 1, (0, 1) and (0, 1) for a 2, (0, 4) and (0, 4) for a 3, and ( 1, 0) and (1, 0) for a 4. As time unit we use the basic period T = 2π/ω c corresponding to 32 time steps. 6

Figure 3: Envelope snapshot A j (x, y), j = 1..4, associated with the pattern shown in Figure 1a. 7

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