Chaos, Solitons and Fractals 33 (2007) 965–970

www.elsevier.com/locate/chaos

Suppression of spiral waves in light-sensitive mediausing chaotic signal modulated scheme q

Jun Ma *, Chun-Ni Wang, Yan-Long Li, Shi-Rong Li

School of Science, Lanzhou University of Technology, Lanzhou 730050, China

Accepted 17 January 2006

Abstract

Evolution of spiral waves in light-sensitive media described with the two variable oregonator model is investigated.The intensity of external illumination is modulated by a weak chaotic signal, which is introduced into the whole system,the additional bromide production is influenced and the dynamics thus changed. The results are confirmed within ournumerical simulation and it may give useful information in pattern formation and suppression of spiral waves. It can bean example for anti-control of chaos.� 2006 Elsevier Ltd. All rights reserved.

1. Introduction

Spiral waves are probably the most intriguing patterns in the spatial extended systems [1–6], and they could be com-monly observed in oscillatory and excitable media [1–10]. Spiral waves have been widely investigated for several rea-sons, one of which is their potential clinical relevance to cardiac arrhythmias, especially ventricular fibrillation thatregarded as a result of instability and break up of spiral waves, which can cause sudden clinical death and is the leadingcause of sudden heart death in industrialized countries. Up to date, many schemes are proposed to suppress spiral andspatiotemporal chaos, e.g. feedback schemes [11] are testified and confirm its effectiveness, then Zhang et al. [12] pro-posed to kill spiral waves using target waves by inputting periodic signal in local area, Wang et al propose to kill spiralwaves with traveling wave [13]. The authors of this paper proposed to suppress spiral waves and spatiotemporal chaosunder chaotic signal driving and chaotic-modulated gradient electric field [7] and by introducing an external centric fieldinto the whole system [6]. Here, we will investigate the dynamics of light-sensitive media defined as the oregonatormodel for Belousov–Zhabotinsky reaction. Our scheme is to adjust the illumination intensity with a weak Lorenz cha-otic signal instead of a periodical and/or constant signal, and wish the whole system could reach homogeneous state assoon as possible. As a result, it could be give some new evidence for anti-control of chaos.

0960-0779/$ - see front matter � 2006 Elsevier Ltd. All rights reserved.doi:10.1016/j.chaos.2006.01.058

q Supported partially by the National Natural Science Foundation of China Under Grant Nos. 90303010, 10572056 and the NaturalScience Foundation of Gansu Province in China Under Grant No. 3ZS042-B25-021.

* Corresponding author.E-mail addresses: hyperchaos@lut.cn, hyperchaos@163.com (J. Ma).

966 J. Ma et al. / Chaos, Solitons and Fractals 33 (2007) 965–970

The two-variable oregonator model is widely used to simulate the light-sensitive version of the Belousov–Zhabotin-sky reaction [1–5].

Fig. 1.units (

otu ¼ Dur2uþ ðu� u2 � ðfvþ UÞðu� qÞ=ðuþ qÞÞ=e; otv ¼ u� v ð1Þ

Here the variable u and v represent the dimensionless concentrations of the autocatalytic HBrO2 and catalyst, respec-tively. The scale parameters e = 0.05, q = 0.002, diffusion coefficient Du = 1 and the adjustable stoichiometry f = 1.4were fixed. The term U = U(t) describes the additional bromide production that is induced by the external illuminationof the system [14].

Compared to the results in [1–5], here the evolution of the variables and dynamics of the system (1) is investigated byadjusting the term U = U(t) with a weak chaotic signal. The system size is 51.2 · 51.2, grids number256 · 256(Dx = Dy = 51.2/256 show the distance between the two adjacent grids), the time step size h = 0.001 andthe algorithm is performed with the Euler forward method and non-flux condition is in consideration.

Our objective is to generate and suppress a rotating spiral wave with the scheme we proposed in [7]. At first, theappropriate initial values are set as [2] so that a rotating spiral wave could be generated.

u ¼ 0 for 0 < h < 0:5; u ¼ qðf þ 1Þ=ðf � 1Þ elsewhere; v ¼ qðf þ 1Þ=ðf � 1Þ þ h=8pf ð2Þ

where h is the angle (in radians) with respect to the origin of coordinates measured counterclockwise form the positivex-axis. The spiral wave is generated and rotates counterclockwise at the requirements in Eq. (2), and the result is plottedin Fig. 1 for different U.

The modulated signal is generated form the Lorenz chaotic system defined as the following Eq. (3).

dx=dt ¼ 16ðy � xÞ; dy=dt ¼ 45:92x� y � xz; dz=dt ¼ xy � 4z ð3Þ

Counterclockwise rotating spiral is generated at the initial values in Eq. (2) after a duration about 30 time units (a) and 40 timeb) in Eq. (1) for U = 0.01; 30 time units (c) and 40 time units (d) in Eq. (1) for U = 0.05.

Fig. 2. Evolution of u vs. time when the whole system is influenced by the chaotic-modulated U(t) = 0.01 + 0.01x(t) for time t = 31 (a),32 (b), 34 (c), 35 (d) time units (the final state is shown in gray scale), and the scale value in (d) is 0.00207, and (e) illustrated theevolution of grid u(20,20) vs. time.

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Now we confirm the scheme by adjusting the intensity of U(t) in Eq. (1) with the three variables x, y and z in Eq. (3),and the relevant and corresponding results are plotted in Figs. 2–4, respectively.

The numerical simulation results in Fig. 2 and Fig. 3 show that the whole system reached homogeneous state within5 time units as the U(t) = 0.01 + 0.01x(t) and/or U(t) = 0.01 + 0.01y(t) is introduced into the system at time t > 30 timeunits, furthermore numerical simulation results proved that the whole system keeps homogeneous even though the scalevalue could change in a small scope. For example, in Fig. 3(e), the time series of sampling variable u(20,20) changes in ascope about [0,0.9] which means other grids variable would change with the same pace or step.

The numerical simulation results in Fig. 4 show that the whole system reached homogeneous state within in 7 timeunits as the U(t) = 0.01 + 0.01z(t) is introduced into the system at time t > 30 time units, furthermore numerical sim-ulation results proved that the whole system keeps homogeneous even though the scale value could change in a small

Fig. 3. Evolution of u vs. time when the whole system is influenced by the chaotic-modulated U(t) = 0.01 + 0.01y(t) for time t = 31 (a),32 (b), 34 (c), 35 (d) time units (the final state is shown in gray scale),and the scale value in (d) is 0.00204, and (e) illustrated theevolution of grid u(20,20) vs. time.

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scope. For example, in Fig. 4(g), the time series of sampling variable u(20,20) changes in a scope about [0,1.05] whichmeans other grids variable would change with the same pace or step.

An interesting phenomena occurs for t = 32 time units when the U(t) = 0.01 + 0.01z(t) is introduced into the systemat time t > 30 time units, which corresponds to a transient homogeneous state. Compared to the results of Fig. 2 andFig. 3, there is some difference in the power spectrum when the sampling variable u(20,20) is investigated by a FastFourier Transform Algorithm (FFT). In our comprehension, the third variable z is positive while the first two variablesx and y could be negative and positive when they are used to adjust the rates of bromide production or the intensity ofthe external irradiation. In all, the chaotic-modulated scheme is effective to suppress the spiral in the Eq. (1) and thewhole system is influenced and kept pace with the external chaotic signal perturbation.

Fig. 4. Evolution of u vs. time when the whole system is influenced by the chaotic-modulated U(t) = 0.01 + 0.01z(t) for time t = 31 (a),32 (b), 34 (c), 35 (d), 36 (e), 37 (f) time units (the final state is shown in gray scale), and the scale value is 0.00204 in (b) and 0.00205 (f),and (g) illustrated the evolution of grid u(20,20) vs. time.

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2. Conclusions

The light-sensitive two-variable oregonator model, which is used to represent the Belousov–Zhabotinsky reaction, isinvestigated. The intensity of illumination is adjusted by the weak Lorenz chaotic signal, and thus the rates of bromideproduction from irradiation U(t) is influenced ,therefore, the whole dynamics of the system is changed. The numericalsimulation results confirm that the whole system could become stable homogeneous within 10 time units. Though stron-ger intensity of illumination is used to control the spiral waves in BZ reaction so that the whole system could reachhomogeneous state within few time units, here we intend to prove that weak chaotic could work as well, and it couldbe an application of anti-control chaos. It may give useful information for suppression of spiral and pattern formationand expects the validation in experiments.

Acknowledgements

We would like to give great thanks to Professor Zhang H and Ying HP for useful discussion, and it is supportedpartially by the National Natural Science foundation of China Under Grant No 90303010 and 10472039.

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