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Suppression of spiral waves in light-sensitive media using chaotic signal modulated scheme
Chaos, Solitons and Fractals 33 (2007) 965–970 www.elsevier.com/locate/chaos
Suppression of spiral waves in light-sensitive media using 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 our numerical simulation and it may give useful information in pattern formation and suppression of spiral waves. It can be an 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 commonly observed in oscillatory and excitable media [1–10]. Spiral waves have been widely investigated for several reasons, one of which is their potential clinical relevance to cardiac arrhythmias, especially ventricular fibrillation that regarded as a result of instability and break up of spiral waves, which can cause sudden clinical death and is the leading cause of sudden heart death in industrialized countries. Up to date, many schemes are proposed to suppress spiral and spatiotemporal chaos, e.g. feedback schemes [11] are testified and confirm its effectiveness, then Zhang et al. [12] proposed to kill spiral waves using target waves by inputting periodic signal in local area, Wang et al propose to kill spiral waves with traveling wave [13]. The authors of this paper proposed to suppress spiral waves and spatiotemporal chaos under chaotic signal driving and chaotic-modulated gradient electric field [7] and by introducing an external centric field into the whole system [6]. Here, we will investigate the dynamics of light-sensitive media defined as the oregonator model for Belousov–Zhabotinsky reaction. Our scheme is to adjust the illumination intensity with a weak Lorenz chaotic signal instead of a periodical and/or constant signal, and wish the whole system could reach homogeneous state as soon as possible. As a result, it could be give some new evidence for anti-control of chaos.
q
Supported partially by the National Natural Science Foundation of China Under Grant Nos. 90303010, 10572056 and the Natural Science Foundation of Gansu Province in China Under Grant No. 3ZS042-B25-021. * Corresponding author. E-mail addresses: [email protected], [email protected] (J. Ma). 0960-0779/$ - see front matter 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2006.01.058
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–Zhabotinsky reaction [1–5]. ot u ¼ Du r2 u þ ðu u2 ðfv þ UÞðu qÞ=ðu þ qÞÞ=e;
ot v ¼ u v
ð1Þ
Here the variable u and v represent the dimensionless concentrations of the autocatalytic HBrO2 and catalyst, respectively. The scale parameters e = 0.05, q = 0.002, diffusion coefficient Du = 1 and the adjustable stoichiometry f = 1.4 were fixed. The term U = U(t) describes the additional bromide production that is induced by the external illumination of the system [14]. Compared to the results in [1–5], here the evolution of the variables and dynamics of the system (1) is investigated by adjusting the term U = U(t) with a weak chaotic signal. The system size is 51.2 · 51.2, grids number 256 · 256(Dx = Dy = 51.2/256 show the distance between the two adjacent grids), the time step size h = 0.001 and the 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, the appropriate initial values are set as [2] so that a rotating spiral wave could be generated. u¼0
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 positive x-axis. The spiral wave is generated and rotates counterclockwise at the requirements in Eq. (2), and the result is plotted in 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
Fig. 1. Counterclockwise rotating spiral is generated at the initial values in Eq. (2) after a duration about 30 time units (a) and 40 time units (b) in Eq. (1) for U = 0.01; 30 time units (c) and 40 time units (d) in Eq. (1) for U = 0.05.
J. Ma et al. / Chaos, Solitons and Fractals 33 (2007) 965–970
967
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 the evolution of grid u(20, 20) vs. time.
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 within 5 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 time units, furthermore numerical simulation results proved that the whole system keeps homogeneous even though the scale value could change in a small scope. For example, in Fig. 3(e), the time series of sampling variable u(20, 20) changes in a scope 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 time units as the U(t) = 0.01 + 0.01z(t) is introduced into the system at time t > 30 time units, furthermore numerical simulation results proved that the whole system keeps homogeneous even though the scale value could change in a small
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J. Ma et al. / Chaos, Solitons and Fractals 33 (2007) 965–970
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 the evolution of grid u(20, 20) vs. time.
scope. For example, in Fig. 4(g), the time series of sampling variable u(20, 20) changes in a scope about [0, 1.05] which means 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 system at time t > 30 time units, which corresponds to a transient homogeneous state. Compared to the results of Fig. 2 and Fig. 3, there is some difference in the power spectrum when the sampling variable u(20, 20) is investigated by a Fast Fourier Transform Algorithm (FFT). In our comprehension, the third variable z is positive while the first two variables x and y could be negative and positive when they are used to adjust the rates of bromide production or the intensity of the external irradiation. In all, the chaotic-modulated scheme is effective to suppress the spiral in the Eq. (1) and the whole system is influenced and kept pace with the external chaotic signal perturbation.
J. Ma et al. / Chaos, Solitons and Fractals 33 (2007) 965–970
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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.
970
J. Ma et al. / Chaos, Solitons and Fractals 33 (2007) 965–970
2. Conclusions The light-sensitive two-variable oregonator model, which is used to represent the Belousov–Zhabotinsky reaction, is investigated. The intensity of illumination is adjusted by the weak Lorenz chaotic signal, and thus the rates of bromide production from irradiation U(t) is influenced ,therefore, the whole dynamics of the system is changed. The numerical simulation results confirm that the whole system could become stable homogeneous within 10 time units. Though stronger intensity of illumination is used to control the spiral waves in BZ reaction so that the whole system could reach homogeneous state within few time units, here we intend to prove that weak chaotic could work as well, and it could be an application of anti-control chaos. It may give useful information for suppression of spiral and pattern formation and 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 supported partially by the National Natural Science foundation of China Under Grant No 90303010 and 10472039.
References [1] Zhang H, Wu NJ, Ying HP, et al. Drift of rigidly rotating spirals under periodic and noisy illuminations. J Chem Phys 2004;121:7276. [2] Ramos JI. Pattern formation in two-dimensional reactive-diffusive media with straining. Chem Phys Lett 2002;365:260. [3] Hildbrand M, Cu JXI, Mihaliuk E, et al. Synchronization of spatiotemporal patterns in locally coupled excitable media. Phys Rev E 2003;68:026205. [4] Zykov VS, Bodiougov G, Brandsta¨dter H, et al. Periodic forcing and feedback control of non-linear lumped oscillators and meandering spiral waves. Phys Rev E 2003;68:016214. [5] Schebesch I, Engel H. Interacting spiral waves in the oregonator model of the light-sensitive Belousov–Zhabotinskii reaction. Phys Rev E 1999;60:6429. [6] Ma J, Pu ZS, Li WX, et al. A new scheme of suppression of spiral and spatiotemporal chaos in centric field. Acta Phys Sin 2005;54(10):4602. [7] Ma J, Ying HP, Pu ZS. An anti-control scheme for spiral under lorenz chaotic signals. Chin Phys Lett 2005;22(5):1065; Ma J, Ying HP, Pan GW, et al. Evolution of spiral waves under modulated electric fields. Chin Phys Lett 2005;22(9):2176. [8] Nash MP, Panfilov AV. Electromechanical model of excitable tissue to study reentrant cardiac arrhythmias. Prog Biophys Mole Biol 2004;85:501. [9] Zhan M, Wang XG, Gong XF, et al. Phase synchronization of a pair of spiral waves. Phys Rev E 2005;71:036212. [10] Marts B, Hagberg A, Meron E, et al. Bloch-front turbulence in a periodically forced Belousov–Zhabotinsky reaction. Phys Rev Lett 2004;93:108305. [11] Piwonski T, Houlihan J, Busch T, et al. Delay-induced excitability. Phys Rev Lett 2005;95:040601. [12] Zhang H, Cao ZJ, Wu NJ, et al. Suppress winfree turbulence by local forcing excitable systems. Phys Rev Lett 2005;94:188301; Zhang H, Hu B, Hu G. Suppression of spiral waves and spatiotemporal chaos by generating target waves in excitable media. Phys Rev E 2003;68:026134. [13] Wang PY, Xie P, Yi WH. Control of spiral waves and turbulent states in a cardiac model by traveling-wave perturbation. Phys Chin 2003;12:674; Yuan GY, Zhang GC, Wang GR, et al. Chin Phys Lett 2005;22:291. [14] Krug HJ, Pohlmann L, Kuhnert L. Analysis of the modified complete oregonator. J Phys Chem 1990;94:4862.
Suppression of spiral waves in light-sensitive media using 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 our numerical simulation and it may give useful information in pattern formation and suppression of spiral waves. It can be an 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 commonly observed in oscillatory and excitable media [1–10]. Spiral waves have been widely investigated for several reasons, one of which is their potential clinical relevance to cardiac arrhythmias, especially ventricular fibrillation that regarded as a result of instability and break up of spiral waves, which can cause sudden clinical death and is the leading cause of sudden heart death in industrialized countries. Up to date, many schemes are proposed to suppress spiral and spatiotemporal chaos, e.g. feedback schemes [11] are testified and confirm its effectiveness, then Zhang et al. [12] proposed to kill spiral waves using target waves by inputting periodic signal in local area, Wang et al propose to kill spiral waves with traveling wave [13]. The authors of this paper proposed to suppress spiral waves and spatiotemporal chaos under chaotic signal driving and chaotic-modulated gradient electric field [7] and by introducing an external centric field into the whole system [6]. Here, we will investigate the dynamics of light-sensitive media defined as the oregonator model for Belousov–Zhabotinsky reaction. Our scheme is to adjust the illumination intensity with a weak Lorenz chaotic signal instead of a periodical and/or constant signal, and wish the whole system could reach homogeneous state as soon as possible. As a result, it could be give some new evidence for anti-control of chaos.
q
Supported partially by the National Natural Science Foundation of China Under Grant Nos. 90303010, 10572056 and the Natural Science Foundation of Gansu Province in China Under Grant No. 3ZS042-B25-021. * Corresponding author. E-mail addresses: [email protected], [email protected] (J. Ma). 0960-0779/$ - see front matter 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2006.01.058
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–Zhabotinsky reaction [1–5]. ot u ¼ Du r2 u þ ðu u2 ðfv þ UÞðu qÞ=ðu þ qÞÞ=e;
ot v ¼ u v
ð1Þ
Here the variable u and v represent the dimensionless concentrations of the autocatalytic HBrO2 and catalyst, respectively. The scale parameters e = 0.05, q = 0.002, diffusion coefficient Du = 1 and the adjustable stoichiometry f = 1.4 were fixed. The term U = U(t) describes the additional bromide production that is induced by the external illumination of the system [14]. Compared to the results in [1–5], here the evolution of the variables and dynamics of the system (1) is investigated by adjusting the term U = U(t) with a weak chaotic signal. The system size is 51.2 · 51.2, grids number 256 · 256(Dx = Dy = 51.2/256 show the distance between the two adjacent grids), the time step size h = 0.001 and the 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, the appropriate initial values are set as [2] so that a rotating spiral wave could be generated. u¼0
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 positive x-axis. The spiral wave is generated and rotates counterclockwise at the requirements in Eq. (2), and the result is plotted in 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
Fig. 1. Counterclockwise rotating spiral is generated at the initial values in Eq. (2) after a duration about 30 time units (a) and 40 time units (b) in Eq. (1) for U = 0.01; 30 time units (c) and 40 time units (d) in Eq. (1) for U = 0.05.
J. Ma et al. / Chaos, Solitons and Fractals 33 (2007) 965–970
967
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 the evolution of grid u(20, 20) vs. time.
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 within 5 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 time units, furthermore numerical simulation results proved that the whole system keeps homogeneous even though the scale value could change in a small scope. For example, in Fig. 3(e), the time series of sampling variable u(20, 20) changes in a scope 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 time units as the U(t) = 0.01 + 0.01z(t) is introduced into the system at time t > 30 time units, furthermore numerical simulation results proved that the whole system keeps homogeneous even though the scale value could change in a small
968
J. Ma et al. / Chaos, Solitons and Fractals 33 (2007) 965–970
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 the evolution of grid u(20, 20) vs. time.
scope. For example, in Fig. 4(g), the time series of sampling variable u(20, 20) changes in a scope about [0, 1.05] which means 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 system at time t > 30 time units, which corresponds to a transient homogeneous state. Compared to the results of Fig. 2 and Fig. 3, there is some difference in the power spectrum when the sampling variable u(20, 20) is investigated by a Fast Fourier Transform Algorithm (FFT). In our comprehension, the third variable z is positive while the first two variables x and y could be negative and positive when they are used to adjust the rates of bromide production or the intensity of the external irradiation. In all, the chaotic-modulated scheme is effective to suppress the spiral in the Eq. (1) and the whole system is influenced and kept pace with the external chaotic signal perturbation.
J. Ma et al. / Chaos, Solitons and Fractals 33 (2007) 965–970
969
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.
970
J. Ma et al. / Chaos, Solitons and Fractals 33 (2007) 965–970
2. Conclusions The light-sensitive two-variable oregonator model, which is used to represent the Belousov–Zhabotinsky reaction, is investigated. The intensity of illumination is adjusted by the weak Lorenz chaotic signal, and thus the rates of bromide production from irradiation U(t) is influenced ,therefore, the whole dynamics of the system is changed. The numerical simulation results confirm that the whole system could become stable homogeneous within 10 time units. Though stronger intensity of illumination is used to control the spiral waves in BZ reaction so that the whole system could reach homogeneous state within few time units, here we intend to prove that weak chaotic could work as well, and it could be an application of anti-control chaos. It may give useful information for suppression of spiral and pattern formation and 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 supported partially by the National Natural Science foundation of China Under Grant No 90303010 and 10472039.
References [1] Zhang H, Wu NJ, Ying HP, et al. Drift of rigidly rotating spirals under periodic and noisy illuminations. J Chem Phys 2004;121:7276. [2] Ramos JI. Pattern formation in two-dimensional reactive-diffusive media with straining. Chem Phys Lett 2002;365:260. [3] Hildbrand M, Cu JXI, Mihaliuk E, et al. Synchronization of spatiotemporal patterns in locally coupled excitable media. Phys Rev E 2003;68:026205. [4] Zykov VS, Bodiougov G, Brandsta¨dter H, et al. Periodic forcing and feedback control of non-linear lumped oscillators and meandering spiral waves. Phys Rev E 2003;68:016214. [5] Schebesch I, Engel H. Interacting spiral waves in the oregonator model of the light-sensitive Belousov–Zhabotinskii reaction. Phys Rev E 1999;60:6429. [6] Ma J, Pu ZS, Li WX, et al. A new scheme of suppression of spiral and spatiotemporal chaos in centric field. Acta Phys Sin 2005;54(10):4602. [7] Ma J, Ying HP, Pu ZS. An anti-control scheme for spiral under lorenz chaotic signals. Chin Phys Lett 2005;22(5):1065; Ma J, Ying HP, Pan GW, et al. Evolution of spiral waves under modulated electric fields. Chin Phys Lett 2005;22(9):2176. [8] Nash MP, Panfilov AV. Electromechanical model of excitable tissue to study reentrant cardiac arrhythmias. Prog Biophys Mole Biol 2004;85:501. [9] Zhan M, Wang XG, Gong XF, et al. Phase synchronization of a pair of spiral waves. Phys Rev E 2005;71:036212. [10] Marts B, Hagberg A, Meron E, et al. Bloch-front turbulence in a periodically forced Belousov–Zhabotinsky reaction. Phys Rev Lett 2004;93:108305. [11] Piwonski T, Houlihan J, Busch T, et al. Delay-induced excitability. Phys Rev Lett 2005;95:040601. [12] Zhang H, Cao ZJ, Wu NJ, et al. Suppress winfree turbulence by local forcing excitable systems. Phys Rev Lett 2005;94:188301; Zhang H, Hu B, Hu G. Suppression of spiral waves and spatiotemporal chaos by generating target waves in excitable media. Phys Rev E 2003;68:026134. [13] Wang PY, Xie P, Yi WH. Control of spiral waves and turbulent states in a cardiac model by traveling-wave perturbation. Phys Chin 2003;12:674; Yuan GY, Zhang GC, Wang GR, et al. Chin Phys Lett 2005;22:291. [14] Krug HJ, Pohlmann L, Kuhnert L. Analysis of the modified complete oregonator. J Phys Chem 1990;94:4862.