Control spiral and multi-spiral wave in the complex Ginzburg–Landau equation

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Chaos, Solitons and Fractals 38 (2008) 521–530

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Control spiral and multi-spiral wave in the complexGinzburg–Landau equation

Ma Jun a,*, Gao Ji-Hua b,c, Wang Chun-Ni a, Su Jun-Yan a

a Department of Physics, Lanzhou University of Technology, Lanzhou 730050, Chinab College of Science, Shenzhen University, Shenzhen 518060, China

c Shenzhen Key Laboratory of Special Functional Materials, Shenzhen 518060, China

Accepted 27 November 2006

Abstract

In this letter, scheme is proposed to suppress the spiral wave and multi-spiral waves in the complex Ginzburg–Landau equation (CGLE) under a local feedback control, which the perturbation is imposed on a small square areaabout 3� 3 grids in the center of the media under periodical boundary conditions and 5� 5 grids in the boundaryof the media under no-flux conditions. Starting from random and/or perpendicular-gradient initial conditions, andthe periodical boundary condition and/or no-flux boundary condition is in consideration, respectively. The numericalsimulation results show that a target wave appears as the feedback began to work and the spiral and multi-spiral wavesare overcome by the new generated target wave, furthermore, it confirms its effectiveness even if the spatiotemporalnoise is introduced into the whole media.� 2006 Elsevier Ltd. All rights reserved.

1. Introduction

Spiral wave [1,2] and antispiral wave (inwardly rotating spiral wave) [3–5] are special pattern which could beobserved in the excitable [1,2,6–8] and oscillatory media [9,10]. People began to pay more attention to the topic sincemore evidences confirmed that the appearance of spiral could be linked to cardiac arrhythmia and the instability ofspiral would induce a rapid death of heart, which defined as ventricular fibrillation (VF). Generally, the reaction–dif-fusion equation is often used to describe these problems. Therefore, it is an interesting problem to investigate theschemes how to prevent appearance and breakup of spiral waves in experiments. On the other hand, more people intendto investigate their dynamic characters based on the nonlinear science. For example, evolution of spiral wave in thepolarized field is investigated [11], transition from spiral to antispiral is discussed [4,5], and the drift of tip of spiral waveis focused on [10]. In the view of suppression of spiral waves, target scheme is proposed by periodical forcing in a localarea in the media [6,12,13], while Hendrey et al. [14] preferred to generate target waves by introducing a spatially local-ized inhomogeneity into the two-dimensional complex Ginzburg–Landau equation, and the stationary and breathingtarget wave were observed. The author of this paper proposed to convert spiral wave into target wave [8] with a locallinear feedback scheme. In this letter, we proposed to suppress the spiral and multi-spiral wave appeared in the complex

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* Corresponding author. Tel.: +86 13609306713.E-mail address: [email protected] (J. Ma).

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522 J. Ma et al. / Chaos, Solitons and Fractals 38 (2008) 521–530

Ginzburg–Landau equation (CGLE) with a local control scheme in consideration of different conditions, our aim is toconvert the spiral wave into a target wave within shorter transient period with the local control scheme under periodicalboundary condition and/or no-flux boundary condition, and its effectiveness will be testified in presence of spatiotem-poral noise in the whole media.

The CGLE is defined as Eq. (1) [9,10,15]

Fig. 1.coefficunits (after t

no-flux

@ tA ¼ Aþ ð1þ ic1Þr2A� ð1þ ic2ÞjAj2A ð1Þ

where c1 and c2 are the real parameters. CGLE is often used to describe the pattern formation in extended spatiotem-poral system. Spiral wave [10], antispiral wave [3–5] and spatiotemporal chaos [12] could be observed in the CGLE un-der appropriate parameters and initial values. Compared to our recent results in [16], which a local feedback scheme isused to suppress the meandering spiral wave in the excitable media described by the Barkley model under no-flux

Spiral appearance from perpendicular-gradient initial conditions, at c1 ¼ �1:35, c2 ¼ 0:34, time step 0.01, space distance 1, theient k ¼ 1 in controller (3), and the control area is 5� 5 grids at a ¼ 62–66, b ¼ 1–5, for t ¼ 0 time units (a), for t ¼ 100 timeb), t ¼ 300 time units (c), t ¼ 700 time units (d), t ¼ 1000 time units (e). (a) generated from perpendicular-gradient initial values= 500 time units. Competition between the spiral wave and the new generated target wave subjected to the controller (3) under

boundary conditions.

J. Ma et al. / Chaos, Solitons and Fractals 38 (2008) 521–530 523

boundary condition, here, we focus on the aim how to suppress the spiral wave in the oscillatory media when a peri-odical or no-flux boundary condition is in consideration, respectively. The periodical boundary condition is describedas

Fig. 2.coefficit ¼ 100increaswave s

Aðxþ nL1; y þ mL2; tÞ ¼ Aðx; y; tÞ; n;m ¼ 0;�1;�2; . . . ð2Þ

In our numerical simulation, L ¼ 128, time step h ¼ 0:01, space distance Dx ¼ Dy ¼ 1; L1 ¼ L2 ¼ 128, grids number128� 128 without special statement.

Spiral appearance from perpendicular-gradient initial conditions, at c1 ¼ �1:35, c2 ¼ 0:34, time step 0.01, space distance 1, theent k=1 in controller (3), intensity of noise D0 ¼ 0:008, and the control square area is 5� 5 grids at a ¼ 62–66, b ¼ 1–5, for

time units (a), t ¼ 300 time units (b), t ¼ 700 time units (c), t ¼ 1000 time units (d). The intensity of spatiotemporal noise ised to D0 ¼ 0:02 (e) and D0 ¼ 0:2 (f) at t ¼ 1000 time units. Competition between the spiral wave and the new appeared targetubjected to controller (3) under no-flux boundary conditions.


2. Scheme and numerical results

The controller defined as Eq. (3) is imposed on the right side of Eq. (1) directly

Fig. 3.coefficunits (t ¼ 50

G ¼ kðRealðAði; jÞÞ � ImagðAði; jÞÞÞdi;adj;b ð3Þ

where k is the feedback coefficient, RealðAði; jÞÞ; ImagðAði; jÞÞ is the real part and imaginary part of the variable A. a; bis integer, di;a ¼ 1; a ¼ i; dj;b ¼ 1; j ¼ b; di;a ¼ 0; a 6¼ i; dj;b ¼ 0; j 6¼ b, the perturbation in Eq. (3) on Eq. (1) is imposedon a square in the media. The controller Eq. (3) is imposed on the square area in the center of the media under peri-odical boundary condition, and the controller is imposed on the square area in boundary of the media under no-fluxboundary condition. The spatiotemporal noise fðx; y; tÞ, which satisfied with hfðx; y; tÞi ¼ 0 and hfðx; y; tÞfðx0; y0; t0Þi ¼D0dðx� x0Þdðy � y0Þdðt � t0Þ;D0 is the intensity of noise. In numerical simulation, the spatiotemporal noise fðx; y; tÞ is

Multi-spiral wave appearance from random initial conditions, at c1 ¼ �1:35, c2 ¼ 0:34, time step 0.01, space distance 1, theient k ¼ 1 in controller (3), and the control area is 5� 5 grids at a ¼ 62–66, b ¼ 1–5, for t ¼ 0 time units (a), for t ¼ 100 timeb), t = 300 time units (c), t ¼ 700 time units (d), t ¼ 1000 time units (e). (a) generated from random initial conditions after0 time units. Evolution of the multi-spiral subjected to the controller (3) under no-flux boundary conditions.


imposed on the whole media, that is to say, fðx; y; tÞ is added to the right side of Eq. (1). All the snapshots are real partof the amplitude A in Eq. (1).

3. Part I: No-flux boundary condition

Here, the parameters c1 and c2 are selected as c1 ¼ �1:35, c2 ¼ 0:34 [10] so that spiral wave could be generated fromrandom or perpendicular-gradient initial conditions.

Fig. 4. Multi-spiral wave appearance from random initial conditions, at c1 ¼ �1:35, c2 ¼ 0:34, time step 0.01, space distance 1, thecoefficient k = 1 in controller (3), intensity of noise D0 ¼ 0:008, and the control area is 5� 5 grids at a ¼ 62–66, b ¼ 1–5, for t ¼ 100time units (a), t ¼ 300 time units (b), t ¼ 700 time units (c), t ¼ 1000 time units (d). The intensity of spatiotemporal noise is increased toD0 ¼ 0:02 (e) and D0 ¼ 0:2 (f) at t ¼ 1000 time units. Evolution of the multi-spiral wave subjected to the controller (3) under no-fluxboundary conditions.


The numerical simulation results in Fig. 1 confirm that a new target appeared and the spiral was suppressed within100 time units under weak feedback gain k ¼ 1. Then its robust to spatiotemporal noise will be testified in the following,and the results will be plotted in Fig. 2 in presence of noise D0 ¼ 0:008.

It is confirmed its effectiveness in presence of spatiotemporal noise in the whole media, then the intensity of noise isincreased to D0 ¼ 0:02, it is still robust to spatiotemporal noise, furthermore, too strong spatiotemporal noise willinduce a breakup of spiral wave, and the results are plotted in Fig. 2 at t = 1000 time units for D0 ¼ 0:02 (e) andD0 ¼ 0:2 (f).

How about its effectiveness when the spiral waves are generated from random initial conditions? The results are illus-trated in Fig. 3.

It is confirmed that the multi-spiral wave could be killed by the new generated target wave soon as the controllerworks in a small square area under the no-flux boundary conditions. Furthermore, its effectiveness in presence of spa-tiotemporal noise will be testified in the following, which showed in Fig. 4, respectively.

The numerical simulation results proved that the scheme is robust to the spatiotemporal noise at the intensityD0 ¼ 0:008, furthermore, it confirms its effectiveness when the intensity of noise increased slowly (D0 ¼ 0:02 (e)),then the breakup of spiral will be observed when too strong spatiotemporal noise is introduced into the wholemedia.

4. Part II: Periodical boundary conditions

Here, the same parameters are set as c1 ¼ �1:35, c2 ¼ 0:34 to generate spiral waves, firstly, perpendicular-gradientinitial conditions are used, and the relevant results are shown in Figs. 5 and 6 in absence of noise and in presence ofspatiotemporal noise. Then random initial conditions are discussed and the relevant results are shown in Figs. 7 and8, respectively.

Fig. 5. Spiral appearance from perpendicular-gradient initial conditions, at c1 ¼ �1:35, c2 ¼ 0:34, time step 0.01, space distance 1, thecoefficient k ¼ 1 in controller (3), and the control area is 3� 3 grids at a ¼ 63–65, b ¼ 63–65, for t ¼ 0 time units (a), for t ¼ 100 timeunits (b), t ¼ 300 time units (c). (a) generated from perpendicular-gradient initial values after t ¼ 500 time units. Competition betweenthe spiral wave and the new appeared target wave subjected to the controller (3) under periodical boundary conditions.

Fig. 6. Spiral appearance from perpendicular-gradient initial conditions, at c1 ¼ �1:35, c2 ¼ 0:34, time step 0.01, space distance 1, thecoefficient k ¼ 1 in controller (3), intensity of noise D0 ¼ 0:008, and the control area is 3� 3 grids at a ¼ 63–65, b ¼ 63–35, for t ¼ 100time units (a), t ¼ 300 time units (b). The intensity of spatiotemporal noise is increased to D0 ¼ 0:02 (c) and D0 ¼ 0:2 (d) at t ¼ 300time units. Competition between the spiral wave and the new generated target wave will be controlled subjected to the controller (3)under periodical boundary conditions.


It is confirmed that the spiral wave could be suppressed by the new appeared target wave within 300 time unitsunder the periodical boundary conditions, and it proved its effectiveness when the whole media is subjected to thespatiotemporal noise at intensity D0 ¼ 0:008. The new appeared target wave overcame the spiral wave even the inten-sity of noise is increased to D0 ¼ 0:02, furthermore, breakup of spiral wave is observed when the noise is increased toD0 ¼ 0:2.

It is confirmed that the spiral and/or multi-spiral wave could be overcome and suppressed by the new generatedtarget wave under this scheme even in presence of spatiotemporal noise is in consideration. Furthermore, instabilityand breakup will be induced when strong spatiotemporal noise is introduced into the whole media. Furthermore,other space distances are verified under this scheme, the numerical simulation results just confirmed its effectivenessagain.

5. Discussion and conclusion

Compared the scheme under no-flux and periodical boundary conditions, the numerical simulation results con-firm that the scheme is more robust to noise under the no-flux boundary conditions. In our comprehension andunderstanding, it is the controller that quasi-stochastic perturbation was produced and imposed on the local squarearea in the media; thus, an outwardly traveling wave is generated as a target wave. In this letter, a closed-loopscheme is proposed to suppress the multi-spiral and/or spiral wave, which generated from the perpendicular-gradi-ent initial conditions and random initial conditions in the CGLE under periodical and/or no-flux boundary condi-tions. The spiral wave in the media is perturbed in a small square area with 3� 3 or 5� 5 grids among all thegrids 128� 128 in the center or the boundary. The feedback gain is set as k ¼ 1 in our numerical simulation (dif-ferent feedback gain k was testified, here, we prefer to select weak gain as k ¼ 1 so that the media will not be


changed greatly under strong perturbation). Differing from other target schemes or local periodical pining [6,17],our scheme in this paper may show some advantages, it asks for weaker perturbation, shorter transient periodto suppress spiral wave and the multi-spiral wave, in practice, the parameters of the media could be unknown, usu-ally, a local driving with periodical signal calls for a intrinsic frequency or eigenfrequency related with the param-eters in the media so that resonance could be occurred and thus the spiral wave could be suppressed as quick aspossible with the same driving intensity. Furthermore, our scheme proved to be more effective in presence of thespatiotemporal noise.

Fig. 7. Multi-spiral appearance from random initial conditions, at c1 ¼ �1:35, c2 ¼ 0:34, time step 0.01, space distance 1, thecoefficient k ¼ 1 in controller (3), and the control area 3� 3 grids at a ¼ 63–65, b ¼ 63–35, for t ¼ 0 time units (a), for t ¼ 100 timeunits (b), t ¼ 300 time units (c), t ¼ 700 time units (d), t ¼ 1000 time units (e). (a) generated from random initial values after t ¼ 500time units. Evolution of the multi-spiral wave subjected to the controller (3) under periodical boundary conditions.

Fig. 8. Multi-spiral appearance from random initial conditions, at c1 ¼ �1:35, c2 ¼ 0:34, time step 0.01, space distance 1, thecoefficient k ¼ 1 in controller (3), intensity of noise D0 ¼ 0:008, and the control area 3� 3 grids at a ¼ 63–65, b ¼ 63–35, for t ¼ 100time units (a), for t ¼ 300 time units (b), t ¼ 700 time units (c), t ¼ 1000 time units (d). The intensity of spatiotemporal noise isincreased to D0 ¼ 0:02 (e) and D0 ¼ 0:2 (f) at t ¼ 1500 time units. Evolution of the multi-spiral wave subjected to the controller (3)under periodical boundary conditions.


Acknowledgements

We would like to give thanks to Professor Zhan M for useful discussion, and this work is partially supported by theNNSF under the Grant Nos. 10405018, 10572056 and NSF of the Gansu Province under the Grant No. 3ZS042-B25-021.

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Control spiral and multi-spiral wave in the complex Ginzburg–Landau equation