Multi-mode Spiral Wave in a Coupled Oscillatory Medium

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1 Commun. Theor. Phys. (Beijing, China) 53 (2010) pp c Chinese Physical Society and IOP Publishing Ltd Vol. 53, No. 5, May 15, 2010 Multi-mode Spiral Wave in a Coupled Oscillatory Medium WANG Qun ( ), 1,3, GAO Qing-Yu (Ôï ), 2 LÜ Hua-Ping (ùù ),1 and ZHENG Zhi-Gang (Üã ) 3 1 School of Physics and Electronic Engineering, Xuzhou Normal University, Xuzhou , China 2 School of Chemical Engineering, China University of Mining and Technology, Xuzhou , China 3 Department of Physics, Beijing Normal University, Beijing , China (Received July 8, 2009; revised manuscript received November 24, 2009) Abstract Multi-mode spiral wave and its breakup in 1-d and 2-d coupled oscillatory media is studied here by theoretic analysis and numerical simulations. The analysis in 1-d system shows that the dispersion relation curve could be nonmonotonic depending on the coupling strength. It may also lead to the coexistence of different wave numbers within one system. Direct numerical observations in 1-d and 2-d systems conform to the prediction of dispersion relation analysis. Our findings indicate that the wave grouping can also be observed in oscillatory media without tip meandering and waves with negative group velocity can occur without inhomogeneity. PACS numbers: Ck, r, Kd, Xt Key words: multi-mode spiral wave, coupling oscillatory media, wave group 1 Introduction Amplitude equation has been used in a variety of scientific researches to describe spatiotemporal modulation of reference state close to the onset of criticality. For example, in a dynamical system close to a supercritical Hopf bifurcation, pattern formation can be studied with the complex Ginzburg Landau equation (CGLE). [1 3] It can also be applied to the study of the spiral waves in a large number of chemical reaction-diffusion systems, as in the Belousov Zhabotinsky (BZ) reaction. [4] The CGLE alone, however, is insufficient for the qualitative description of realistic models in the neighborhood of a Hopf bifurcation, where other bifurcations occur nearby. [5] For example, near a supercritical Hopf bifurcation point, another stable limit cycle or a slow real mode may exist. For these systems, the CGLE cannot be used without appropriate modifications. To incorporate the dynamics of the slow real mode into a valid description close to criticality, one may derive an amplitude equation namely the distributed slow-hopf equation (DSHE). [6] Furthermore, it was argued that a situation may arise where the oscillators coupling become effectively nonlocal, and as a consequence the system will exhibit such peculiar dynamics as can never be seen in the CGLE. [7 10] These systems can be described by a nonlocally coupled complex Ginzburg Landau equation (CCGLE). [7] It was shown that the slow mode and the nonlocal coupling introduced a novel set of finite wavelength instabilities not previously presented in the CGLE. For spiral wave, these instabilities highly affect the location of regions of convective and absolute instability. [6] Recently, a new type of spiral pattern, a wave-grouped spiral, was observed in excitable media with oscillatory dispersion. [11 13] This type of spiral can be named as multi-mode spiral because it has more than one wave number. The underlying mechanism for wave grouping has been well addressed, which is caused by the coactions of the externally applied disturbance and the oscillatory dispersion relation of the excitable system. In Refs. [11] [12] the spiral tip was artificially dragged to imitate the meandering spiral waves in the experimental condition. And in Ref. [13] the source of external disturbance was the direct current or alternating current. All these disturbances are in one direction and will lead to the symmetry breaking of spiral waves. These studies prompt some interesting questions: Can we find wave grouping in an isotropy oscillatory system? Is the wave grouping a generic phenomenon? In this paper, we attempt to answer these questions by studying the dynamical behaviors of spiral wave in a CCGLE system. The spiral dynamics is determined by the medium properties such as the dispersion relation of medium, which determines the dependence of the speed of periodic wave trains on the wavelength in excitable medium and the frequency on the wave number in oscillatory medium. Usually for normal dispersion relation in oscillatory medium, [3] the frequency monotonically increases/decreases with the wave number. However, researchers have observed some other types of dispersion relation different from the normal one, such as the damped oscillatory dispersion relation curve. [14 16] Such anomalous dispersions give rise to some exotic phenomena that cannot be found in excitable medium with normal disper- Supported by the National Natural Science Foundation of China under Grant Nos , , and the Natural Science Foundation of Xuzhou Normal University under Grant No. 07PYL02 wangq@xznu.edu.cn

2 978 WANG Qun, GAO Qing-Yu, LÜ Hua-Ping, and ZHENG Zhi-Gang Vol. 53 sion. To our knowledge, however, so far little research has been done on the oscillatory medium with anomalous dispersion relation. 2 Model and Dispersion Relation Analysis In the CGLE A t = A + (1 + iβ) 2 A (1 + iα) A 2 A, (1) where the variable A being the complex oscillation amplitude, the real parameters β and α characterizing linear and nonlinear dispersion, it is convenient to calculate the dispersion relation analytically in respect to plane waves with frequency ω and wave number k, i.e. ω = α + (β α)k 2, as the corresponding eigenfunctions of the waves having the form of Fourier modes. [3,17] The dispersion relation is obviously monotonic increase while wave number k [0, 1]. The coupling oscillatory system can be described with [10] A t = (1 + iσ)a + (1 + iβ) 2 A (1 + iα) A 2 A + Cg(z, A), (2a) τ z t = h(z, A) + l2 2 z z, (2b) where the variable z denoting the additional complexvalued diffusive field. Detuning σ is determined by the natural and forcing frequencies. [18] Here, we set σ = 0. It means that the complex oscillation amplitude A has the same frequency as the additional diffusive field. g(a, z) = z, and h(a) = A are coupling functions, and C is the respective coupling constant. τ and l 2 are the characteristic time scale and diffusion scale of the coupling field z, respectively. If τ 1, l 1, the coupling field z will be an inertial and slowly changing field in space. [10] The equations (2) are brought into a dimensionless form by choosing the characteristic diffusion length in the oscillatory subsystem as the length unit and choosing the characteristic relaxation time scale of the oscillators as the time unit. In Ref. [10], the equations (2) were shown to support birhythmicity. Our concern in this paper is in a different parameter region where interesting spatiotemporal motions develop. If the spatial dimension is d, the equation (2b) can be solved formally. [7] That is t z(r, t) = (2π) d dq exp(iqr) exp ( (1 + l 2 q 2 ) t ) t h q (t ). (3) τ Among it h q (t ) is the spatial Fourier transform of the coupling function h(z, A). By substituting Eq. (3) into Eq. (2a), it is easy to find that the effect of the nonlocal coupling is temporal as well as spatial. dt τ For the CCGLE (2), we assume a 1-d plane wave solution of the form A = A 0 exp[i(kx ωt)], z = z 0 exp[i(kx ωt + θ)], (4) where A 0 and z 0 are amplitude of complex variables A and z, respectively, θ is the phase difference between A and z. Substituting (4) into (2) yields ( A 2 0 = 1 z0 ) k2 + C cosθ 1, (5) A 0 ω = k 2 β + αa 2 0 C z 0 sin θ σ, A 0 (6) A 0 sinθ = τωz 0, (7) A 0 cosθ = z 0 (1 + l 2 k 2 ). (8) From (7) and (8), one can get z0 2 A 2 0 = 1 τ 2 ω 2 + (1 + l 2 k 2 ) 2. (9) Substituting (7) into (6), (8) into (5), and considering (9) the oscillation amplitude of A is A 0 = 1 k l C + C 2 k 2 τ 2 ω 2 + (1 + l 2 k 2 ) 2, (10) and the dispersion relation between frequency and wave number is τ 2 ω 3 + [(α β)k 2 α + σ]τ 2 ω 2 + [Cτ + (1 + l 2 k 2 ) 2 ]ω + [(α β)k 2 α + σ](1 + l 2 k 2 ) 2 Cα(1 + l 2 k 2 ) = 0. (11) Fig. 1 The analysis of plane wave solution of the CC- GLE (a) Frequency ω vs. coupling strength C for different wave number k. (b) Dispersion relation curves for different coupling strength. When wave number k = 0.54 frequency ω is maximum for coupling strength C = 4.1. We set (α, β, σ) = (1.0, , 0), τ = 10 according to [10] and [19]. We set l = 3 as the difference between diffusion coefficients of reactant is seldom more than 100

3 No. 5 Multi-mode Spiral Wave in a Coupled Oscillatory Medium 979 times. The equation (11) indicates that the frequency of oscillations depends on the coupling coefficient C, as well as wave number k. In Fig. 1(a), which shows the relation of frequency in respect to coupling coefficient, the dash line shows that there is a smaller negative slope for k = 0.15 when C 4.0. It indicates that a stable spiral formed in the less coupling coefficient system (C < 4.0) may break up when C has increased (C > 4.0) due to the rapid change of frequency. The real line suggests that the frequency increment of short wave (k = 0.75) is very small with the increase of C. On the contrary, when C 4.0 there is significant frequency increment of long wave, especially for the bulk oscillations. That shows the short waves are more stable than long waves at the increase of parameter C against others constant parameters in this extended CCGLE system. In the system described by Eq. (1), plane waves will be convectively instable when wave number k > k E, according to the Eckhaus criterion. [3] In the CCGLE system (2), the coupling can give rise to finite wavelength instability of plane waves and spiral wave. [6] Phenomenally, plane waves or spiral wave may be destroyed far away from the wave source when the Eckhaus or finite wavelength instability is present. As shown in Fig. 1(a), there is an intersection point joining two different wave numbers. It means that two wave strains with different wave number can coexist within one oscillatory system. This possibility can also be found in Fig. 1(b) in a clearer way, in which the curves of dispersion relation are represented. For other constant parameters, with C > 3.64 the curve will be non-monotonic. For example, when C = 4.1 the long waves (k < 0.54) and short waves (k > 0.54) must have different group velocities with contrary signs. Therefore, the non-monotonic dispersion relation indicates that the complex spatiotemporal patterns will emerge when the coupling coefficient C > Simulations and Discussion To discuss the complex patterns in CCGLE (2), the fourth order Runge Kutta algorithm is employed to integrate the ODEs, and five points approximation is used instead of the Laplace operator. The boundary conditions are Non-flux. Time size is 0.005, and the distance between two closest grids is 0.5. Similar results can be obtained with finer space and time steps. For 1-d system, 1 < x < L, we set (A, z) = (0, 0) at x = 1, this point acts thereafter as a pacemaker which periodically generates phase slips echoing away as a kink train with phase velocity v p = ω/k. Initial values of A and Z of the others points are given randomly. Note that the sign of phase velocity is determined by frequency ω and wave number k, and no matter positive or negative the phase velocity be, one should always have positive group velocity in CGLE system, i.e. v g = ω/ k > 0 [16,20]. Fig. 2 Space-time diagrams for different coupling strength C. (a) C = 3.0; (b) C = 3.6; (c) C = 3.8; (d) C = 4.01; (e) Space-time diagrams of A corresponding to C = 4.01; (f) Modules of A(x,t) distributed in 1-d space for a period of time for C = 4.1. The ws, ns, qm, and im mean wave source, normal waves, quasi-periodic modulation and irregular modulation, respectively. The size of system is 1000 grids. Shown as Fig. 1(b), when C = 3.0 the slope of dash line is negative. Therefore, the selected wave number by the system should also be negative according to group velocity judgment in the CGLE system (v g > 0), i.e. k < 0. Because the frequency of oscillations is positive, the waves must be of abnormal pattern (anti-waves). Figure 2(a) shows the space-time plot for C = 3.0 and it indicates the wave train is propagated to the left in contrary to its group velocity. When C = 3.6, the selected wave number is almost 0 (shown as Fig. 2(b)). Increasing the parameter C to 3.8, we will find normal waves shown as Fig. 2(c). During the transition between anti-waves and normal waves,

4 980 WANG Qun, GAO Qing-Yu, LÜ Hua-Ping, and ZHENG Zhi-Gang Vol. 53 the change of wave number is continuous. This is in accordance with the transition in CGLE system. [20 21] In Fig. 2(d), the instability of wave train presents for C = 4.01 in the right part of system, i.e. near the end x = L. The front separating the normal waves from the laminar region moves towards left with Cincreasing. This is clearly visible in Figs. 2(e) and 2(f) for C = 4.01 and 4.1 respectively. Figure 2(f) indicates that there is no amplitude defect in modulation region found in this finite 1-d system, i.e. A(x, t) 0 except at x = 0. There are four types of regions in Fig. 2(d), Figs. 2(f) and 2(e) which is the corresponding amplitude of space-time diagram 2(d), i.e. wave source (ws), normal waves (nw), quasi-periodic modulation (qm), and irregular modulation (im) region. In the modulation regions there are at least two selected wave numbers. This phenomenon was reported first in Ref. [17], therein, the transition from normal waves to turbulence occurs when the wave frequency pushes the selected wave vector into the regime of absolute Eckhaus instability. In Fig. 2(f), which shows module of variable A(x, t) in a period of time, a convective instability, similar to the one that drives familiar examples like the movement of cigarette smoke or turbulent jets, is shown before quasiperiodic modulation region. However, the module of A(x, t) stops growing in the quasi-modulation region. Therefore, there must be other modes confined to the right boundary. There are other modes which can be proved by the discussion about group velocity. According to group velocity judgment (v g > 0) and the real line of dispersion relation shown in Fig. 1(b), wave train has negative (positive) wave number when k > 0.54 (< 0.54) for C = 4.1. Generally waves with negative wave number are anti-waves if frequency ω > 0, however, normal waves with k = 0.63 and antiwave trains with k = 0.08 will be found in the modulation region of the space-time diagram corresponding to Fig. 2(f) (not shown), while the normal waves in the left part of system have positive wave number, k = This suggests that the waves in the modulation region have negative group velocities respecting to wave source which is at the left end. Therefore, those waves in modulation region are partially affected by the certain wave source which is in the right part of the system. The radial dynamics of non-meandering 2-d spiral wave and target wave can be approximated by 1-d simulations, [22] however, in CGLE system the quasimodulation region named laminar region is very narrow and can not be found in 2-d system. [17] In the following part we will discuss the spiral wave with multi-mode with the CCGLE. First, a rigid rotating spiral for C = 3.8 (shown as Fig. 3(a)) is generated, then we observe the development of spiral wave with the increase of the parameter C step by step. As in the 1-d case the breakup of spiral wave appears first near the boundary of system as shown in Fig. 3(b) for C = 4.0. In contrary to the situation in CGLE, the breakup here does not lead to turbulence immediately near the boundary, and waves with bigger wave number do emerge near the boundary. If we generate a spiral in a larger system (e.g ), three regions can be found during the temporal process of generation of spiral, i.e. waves with less wave number ( k = 0.19) near spiral core, waves with bigger wave number ( k = 0.73), and bulk oscillatory region, shown as Fig. 3(c). In this finite system, the interface separating spiral waves and bulk oscillations, as well as the interface separating different wave trains are moving towards the boundary of system. These indicate that the interface acts as a moving wave resource. Fig. 3 Snapshots of normal spiral wave (a), multi-mode spiral waves (b-e), and space-time diagram (f). The coupling coefficient C is 3.8 in (a), 4.0 in (b, c), and 4.15 in (d-f). Module of A corresponding to (d) is shown in (e). The vertical and horizontal white arrows in (f) denote the time and radial directions and the left end is the spiral tip. Size of system in (c) is , in the others is In order to observe wider modulation region in spiral wave, we gradually increase the parameter C to The radius of residual spiral wave with unique wave number near the core becomes smaller, and the alternation of wave trains with small wave number and big one becomes obvious. As shown in Fig. 3(d), the whole spiral wave is very similar to those super-spiral waves caused

5 No. 5 Multi-mode Spiral Wave in a Coupled Oscillatory Medium 981 by tip meandering. [23 24] Here, however, the residual spiral wave near the core is a rigid rotating spiral wave. This can be seen clearly in amplitude diagram of A in Fig. 3(e) corresponding to Fig. 3(d). Figure 3(e) also indicates that the stable amplitude distribution (at t = 10 8 ) has circular symmetry which is absent in those researched super-spiral waves. As shown in Fig. 3(f), if we consider one-dimensional cut with the fixed spiral core position in Fig. 3(d) (x [300, 600]) and compare the results with Fig. 2(d), we will find the modulation region can also exist in a 2-d system. The vertical and horizontal white arrows denote the time and radial directions. According to the dispersion relation analysis and the group velocity judgment, those wave trains with the lowest wave number in Fig. 4(f) have negative phase and group velocities in contrast to the left residual spiral core. Fig. 4 The breakup of multi-spiral waves for different coupling strength C. C is 4.2 in (a) and (b), 4.3 in (c) and (d). (b) and (d) are snapshots of module A corresponding to (a) and (c), respectively. Size of systems is When the parameter C increases further the multimode spiral wave will break up into underdeveloped turbulence (Figs. 4(a) and 4(c)) as the prediction of CGLE and as the experimental observations. [24 25] The interfaces separating spirals (Figs. 4(b) and 4(d)) in this system are discontinuous. This is different from the case in CGLE, where the underdeveloped turbulence in named frozen state and even the development turbulence have smooth or piecewise smooth interfaces. [26] In Fig. 4(a) for C = 4.2, there are group waves in the region separating the residual normal spiral and the developing turbulence. In the amplitude distribution diagram Fig. 4(b) corresponding to Fig. 4(a), the circular symmetry disappears, even in the modulation region. Instead of the circle amplitude there emerges the multi-arm amplitude spirals. When C = 4.3, system is fully filled by small spirals, shown as Fig. 4(c). Though not being the development turbulence, there are many defects in these regions separating the spiral and others. Note that the complex interfaces between spirals remain unchanged until one or more system parameters are changed in CGLE system. [26] 4 Summary and Remarks In conclusion, we have found that wave trains with opposite sign phase velocities can coexist in a whole spiral wave with complex structure in the CCGLE system. This multi-mode spiral wave is caused by the non-monotonic dispersion relation depending on the coupling strength. When a multi-mode spiral wave breaks up into underdevelopment turbulence, the frozen state which is common in the CGLE system cannot be found in finite 2-d CCGLE system. Usually, normal spiral wave or antispiral wave has positive group velocity. When two or more wave sources coexist in one system it is possible that the negative group velocity prevails, in respect to one of the wave sources. In another word, the group velocity judgment only can be applied in respect to one wave source. For example, the dispersion relation analysis shown in Fig. 1(b) shows no sign of the possibility of presence of normal waves with wave number k > 0.54 because the group velocity is negative, i.e. v g < 0 when k > 0.54 for C = 4.1. Another example is a new type spiral wave named sinklike spiral wave in oscillatory media with a disk-shaped inhomogeneity. [27] Depending on the properties of the medium and the inhomogeneity, spiral wave with negative group velocity will be able to emerge. Obviously the interface separating the disk-shaped region and the other part of system is a special wave source. Discussions about the multi-mode spiral wave and its breakup may shed light on the understanding of the experimental and simulation results. [17,25] The presence of wave grouping in isotropy system confirms the conclusion in [13], i.e. movement of the spiral tip is not necessary for the appearance of wave grouping. Furthermore, this finding indicates the wave grouping can exist not only in excitable medium but also in oscillatory medium, and it is generic in nature. References [1] Y. Kuramoto, Chemical oscillations, waves and turbulence, Springer, Berlin (1984). [2] M.C. Cross and P.C. Hohenberg, Rev. Mod. Phys. 65 (1993) 851. [3] I.S. Aranson and L. Kramer, Rev. Mod. Phys. 74 (2002)

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