Stabilization of Pattern in Complex Ginzburg Landau Equation with Spatial Perturbation Scheme

Transcription

1 Commun. Theor. Phys. (Beijing, China) 49 (2008) pp c Chinese Physical Society Vol. 49, No. 6, June 15, 2008 Stabilization of Pattern in Complex Ginzburg Landau Equation with Spatial Perturbation Scheme MA Jun, 1,2, YI Ming, 2 ZHANG Li-Ping, 1 JIN Wu-Yin, 3 and LI Yan-Long 1 1 School of Sciences, Lanzhou University of Technology, Lanzhou , China 2 Department of Physics and Institute of Biophysics, Central China Normal University, Wuhan , China 3 College of Mechano-Electronic Engineering, Lanzhou University of Technology, Lanzhou , China (Received July 13, 2007; Revised November 5, 2007) Abstract In this paper, a new scheme of spatial perturbation is proposed to stabilize the pattern in the oscillatory media, which could be described with the complex Ginzburg Landau equation. The numerical simulation results confirm that the spiral wave, antispiral wave and spatiotemporal chaos in the complex Ginzburg Landau equation could be suppressed, and the scheme is also discussed with the conservative field theory. Furthermore, the spatiotemporal noise is also introduced into the whole media, it just confirms that it is robust to the spatiotemporal noise. PACS numbers: a, r, Ck Key words: spiral wave, antispiral wave, spiral turbulence, spatial perturbation 1 Introduction Spiral wave and spatiotemporal chaos could be observed in chemical, physical and biological systems, [1 9] and it has attracted much attention since some important evidences proved that the appearance of spiral wave was linked to one kind of cardiac arrhythmia [10] and the instability [11 13] of spiral wave will induce a rapid death of heart, called as ventricular fibrillation (VF). [14 16] Therefore, it is important and interesting to find efficient schemes to prevent the appearance and breakup of the spiral wave. For example, a scheme with weakly spatially perturbation [17 19] is proposed to stabilize spatial pattern and eliminate spiral wave, then the scheme of traveling wave perturbation is further improved to control the turbulent state in the cardiac model. Zhang et al., proposed to destruct the spiral wave and spatiotemporal chaos by generating a target wave under periodical forcing in a local square area, [20 24] and then the scheme is improved to eliminate the winfree turbulence. [23,24] Ma et al., suggested that a local intermittent shock could be used to suppress the spiral wave in the excitable media, [25] and the polarized field modulated by chaotic signal is used to remove the spiral wave. [26] The local inhomogeneity and parameter perturbation scheme is introduced into the complex Ginzburg Landau equation (CGLE) to induce target waves. [27,28] Furthermore, time-delay feedback scheme [29,30] was also used to prevent breakup of spiral wave. The CGLE is the most popular equation in the physics world. Especially, it is used to describe the prototype of an oscillatory media close to the Hopf bifurcation point from a stationary state. [4,31] Spiral wave, antispiral wave (inwardly rotating spiral wave) and spiral turbulence or spatiotemporal chaos could be observed in the CGLE, [31 35] then the target scheme generated by local self-coupling is used to kill spiral wave in CGLE [36] under no-flux and periodical boundary condition, respectively. In Refs. [37] and [38], it is suggested that the spiral wave in the excitable media could be eliminated by introducing spatial force into the whole media. As far as we known, there is much difference between the excitable and oscillatory media. Therefore, it is more interesting to check its effectiveness on oscillatory media and study this scheme further from the point of spatial field. In this paper, we prefer to use the spatial scheme to suppress the spiral wave, antispiral wave and spatiotemporal chaos in the CGLE under periodical boundary instead of the common no-flux boundary conditions. Finally, the scheme will be checked in presence of spatiotemporal noise which could induce the spiral dynamics transition. [39 40] The homogeneous complex Ginzburg Landau equation usually is defined as t A = A (1 + iα) A 2 A + (1 + iβ) 2 A, (1) where A is complex variable, α and β are real numbers, i is pure imaginary unit. The spatiotemporal dynamics, for two-dimensional case 2 = 2 / x / y 2, is investigated. It is shown that spiral wave could be generated at β = 1.4, α = 0.34, then it begins to step into turbulence at α = 0.8, [20] and antispiral wave could be observed at β = 0, α = 1.5, [34,35] which are shown in Fig. 1, respectively. In this paper, the system is decentralized into The project supported partially by National Natural Science Foundation of China under Grant Nos , , and and the Natural Science Foundation of Lanzhou University of Technology under Grant No. Q hyperchaos@lut.cn; hyperchaos@163.com

2 1542 MA Jun, YI Ming, ZHANG Li-Ping, JIN Wu-Yin, and LI Yan-Long Vol grids number and the space step x = y = 1, time step size h = The periodical boundary condition is defined as A(x + nl 1, y + ml 2, t) = A(x, y, t), n, m = 0, ±1, ±2, ±3,..., for simplicity, L 1 = L 2 = 128. Fig. 1 Stable spiral wave of Eq. (1) (a), spatiotempory chaos (b), and antispiral wave (c), which are generated from perpendicular-gradient initial values after t = 500 time units (the snapshot of Re(A) is shown in gray). 2 Scheme of Control and Numerical Simulation Results It is different from the target scheme by imposing periodical signal in a local area in the media [22 24] and the spatial perturbation scheme in Refs. [17] and [18] by adjusting the critical parameter in space, here, we propose another spatial perturbation scheme to suppress the spiral wave, spatiotemporal chaos and antispiral wave in the CGLE under the appropriate parameter area. In the case of two-dimensional problem, the controller could be described as the following Eq. (2), G = F(x, y). (2) In our comprehension, it could be an external gradient field or force imposed on the media. Therefore, an actual form of the controller is suggested and its effectiveness is investigated in presence and absence of spatiotemporal noise. For example, G = kr. (3) where k is the intensity of spatial perturbation, r is the function of grid coordinate x, y, r 2 = (x x 0 ) 2 +(y y 0 ) 2, x = (i 1) x, y = (j 1) y, x 0, y 0 is the origin of coordinate and i, j are integers. For simplicity, in our numerical simulation tests, the origin of coordinate is set with x 0 = 0, y 0 = 0. The controller defined as Eq. (3) is added to the right side of the formula in Eq. (1), and the controlled system is shown in Eq. (4) t A = A (1 + iα) A 2 A + (1 + iβ) 2 A + kr. (4) Now, it is interesting to discuss the principle of this scheme. Firstly: external centric force-induced pattern formation. The position of each grid or cell in the media is changed when the external centric force is introduced into the media globally, thus the self-organization of pattern comes into being as the spatial force begins to work on the media. In fact, new pattern formation and selforganization in the media result from the interference and mutual coupling between cells and the external force. We can see that the intensity of the external force is mainly determined by the coefficient k. In a practical way, the coefficient k could not be set much too big. In our numerical simulation, k is endowed with small value (k 1) to check the transient period to suppress the pattern, including spiral wave, spatiotemporal chaos and antispiral wave in the CGLE. Clearly, the bigger k will cost a shorter transient period to remove the useless patterns. Secondly: external polarized field induced self-organization in the media. The appearance of the amplitude and phase excursion results from the position alteration when the external field is introduced into the media. For the problem with CGLE which relates to the complex field, the characteristic of the complex filed will response to the external field interference, thus the distribution of the field (amplitude and the phase) is changed. According to the controller in Eq. (3), it is necessary to prove that the above standpoints are consentient with each other, both of them could explain the new pattern formation in the CGLE. Clearly, conservative force lies in the conservative field, centrifugal force or centripetal force could be described as the following Eq. (5) F = F(r) = ±kr, (5) where symbol ± marks the direction of the field force. The relation of the potential V and conservative force F(r) could be described as the following Eq. (6) F = F(r) = V. (6)

3 No. 6 Stabilization of Pattern in Complex Ginzburg Landau Equation with Spatial Perturbation Scheme The complex A(x, y, t) describes the complex field well, when the perturbation as in Eq. (3) is introduced into the media by adding the controller to the right side of Eq. (1), the effect of the external force as controller (3) could be replaced by the centric gradient field as the following Eq. (7), t A = A (1 + iα) A 2 A + E A + (1 + iβ) 2 A, (7) where E describes the intensity of the polarized field, it is better to regard E as a complex variable than a real one in the two-dimensional space. In orthogonal coordinate, the gradient field could be simplified as the following Eq. (8), E A = E cos θ x Ai + E sin θ y (A)j, (8) where i and j stand for the unit along x- and y-axes, and the variable θ is the angel of grid vs. the positive x-axis Based on the conservative field theory, the centric gradient field-induced dynamics in the CGLE could be equivalent with the effect of conservative centric force. The ideal could be described as the following Eq. (9) according to Eq. (4) and Eq. (6), E A = kr. (9) Therefore, it is equivalent to induce pattern transition no matter whether the gradient field or an appropriate centric force is imposed on the media globally. Then the pattern in the CGLE could be stabilized and suppressed when the formula in Eq. (9) is satisfied, which the complex field A is just a function of space. It seems intricate and difficult to reach the approximate solution of Eq. (9) providing that the E is in real value. Therefore, it is more appropriate to regard E as a complex variable rather than a real one. Fig. 2 Final state at t = 300 time units for the stable spiral wave of Eq. (1), the initial state is as Fig. 1(a), k = 0.01, noise intensity D0 = 0.0(a), D0 = 0.01(b); k = 0.1, noise intensity D0 = 0.0(c), D0 = 0.01(d); k = 0.5, noise intensity D0 = 0.0(e), D0 = 0.01(f) (the snapshot of Re(A) is shown in gray). For simplicity, according to the criterion in Eq. (9), a p special solution could be A2 (r) = kr 2, or A(r) = (k)r providing that E equals A. Perhaps, it could be more intelligible under the criterion in Eq. (9) for other reactiondiffusion systems with two or three real variables Then the formula in Eq. (9) could be extended as E u = kr, where u is one real variable of the reaction-diffusion system and the polarized field E is a real number. In this case, the external polarized field E is suggested as a complex vari- able so that it could be easy to understand the controller described with Eq. (3) as a spatial perturbation. Clearly, it is interesting to investigate the effect of scheme by imposing the external polarized field on the whole media, in numerical simulation, E A or field force kr is often added to the right side of Eq. (1) so that the external perturbation imposed on the media could be considered. In this paper, the figure 1(a,b,c) is the initial state, our aim is to suppress the spiral wave (Fig. 1(a)), spatiotem-

4 1544 MA Jun, YI Ming, ZHANG Li-Ping, JIN Wu-Yin, and LI Yan-Long poral chaos (Fig. 1(b)) and antispiral wave in Fig. 1(c). Then the scheme will be checked in presence of spatiotemporal noise, which is defined as (x, y, t), h(x, y, t)i = 0, and h(x, y, t)(xb, yb, tb )i = D0 δ(x xb )δ(y yb )δ(t tb ), D0 is the intensity of the noise. In our numerical simulation, the noise intensity D0 = The controller in Eq. (3) is added to the right side of the formula in Eq. (1) and the spatiotemporal noise is also introduced into the whole media by adding it to the right side of formula in Eq. (1) as well. The origin of coordinate lies in the corner under periodical boundary condition. To investigate the Vol. 49 final state of the controlled CGLE in different parameter area, we prefer to observe the state at t = 300 time units so that the final state is stable, and other time is good as well. The coefficient k = 0.01, k = 0.1, k = 0.5 and noise intensity D0 =0.01 in consideration. The snapshots with gray are listed in Fig. 2 to show the final state of the controlled spiral wave in CGLE, respectively. The snapshots are listed in Fig. 3 to show the final state of the controlled spatiotemporal chaos in CGLE, respectively. The snapshots are listed in Fig. 4 to show the final state of the controlled antispiral wave in CGLE, respectively. Fig. 3 Final state at t = 300 time units for the spatiotemporal chaos of Eq. (1), the initial state is as in Fig. 1(b), k = 0.01, noise intensity D0 = 0.0(a), D0 = 0.01(b); k = 0.1, noise intensity D0 = 0.0(c), D0 = 0.01(d); k = 0.5, noise intensity D0 = 0.0(e), D0 = 0.01(f) (the snapshot of Re(A) is shown in gray). According to the snapshots of Fig. 2, it is confirmed that the spiral wave in the CGLE could be suppressed drastically in a few time units, and it is robust to the spatiotemporal noise. Furthermore, we can conclude from the snapshots in Fig. 3 that the scheme is effective to suppress the spatiotemporal chaos as well. Finally, in view of the stable final state in Fig. 4, it is confirmed that the scheme is also effective to suppress the antispiral wave even when the spatiotemporal noise is in consideration. Furthermore, the pattern in the CGLE, including the spiral wave, spatiotemporal chaos and antispiral wave, could be easier to be suppressed when bigger coefficient k in the field force described in Eq. (3) is imposed on the media globally. In fact, appropriate threshold is required for the spatial perturbation coefficient or intensity k, it is confirmed that the spiral turbulence cannot be suppressed at k < 0.01, for example, k = 0.005, the spiral turbulence is not removed no matter wether the spatiotemporal is introduced to the whole media or not. For simplicity, to check the equivalence of centric field and spatial force as described in Eq. (9), the complex E in Eq. (7) is endowed with the value A (that is E = A), the coefficient k is approached based on Eq. (9) by monitoring and sampling final state of an arbitrarily selected grid, then the solution k is imputing into Eq. (4), it is confirmed that the snapshot of final state is the same as the gradient field-induced results by comparing with the gray scale of sampling grids. Finally, it is more important to distinguish this scheme from other schemes. It is well known that there is much

5 No. 6 Stabilization of Pattern in Complex Ginzburg Landau Equation with Spatial Perturbation Scheme relation between the amplitude equation (CGLE) and the reaction-diffusion system. The amplitude equation (CGLE) could be derived from the reaction system by using the reductive perturbation method.[2,41,42] Therefore, it is reasonable to apply this scheme to stabilize pattern or remove useless patterns in the excitable media and/or the oscillatory media. To check its model independence, the scheme is used to eliminate the useless patterns in the well-known modified FitzHugh Nagumo model.[20,21,25,26] In our numerical simulation, the parameters a = 0.84, 1545 b = 0.07, = 0.07 correspond to the excitable media, and a = 0.84, b = 0.045, = describe the oscillatory media (b > 0 describes the excitable media and b < 0 describes the oscillatory media).[43] It just confirmed that the spiral turbulence could be eliminated quickly. In fact, it is the spatial perturbation that radial gradient field is generated, and the radial gradient field is different from the ordinary field induced by a parallel plate capacitor which the gradient field effect is described with x A or y A. Fig. 4 Final state at t = 300 time units for the spatiotemporal chaos of Eq. (1), the initial state is as in Fig. 1(c), k = 0.1, noise intensity D0 = 0.0(a), D0 = 0.01(b); k = 0.5, noise intensity D0 = 0.0(c), D0 = 0.01(d) (the snapshot of Re(A) is shown in gray). Up to our knowledge, the scheme could be explained and practiced in experiments. For example, for chemical wave in excitable or oscillatory media, the spatial perturbation described in Eq. (3) could be generated by circumrotating the reactor vs. an arbitrary axis, as a result, the spatial inertial centrifugal (force) could be made. Clearly, the inertial force could be described as F (r) = m2 r = kr, where m is understood as the mass of the cell or grid, is the circumrotation frequency induced by the external mechanical perturbation or force, and r is the distance from the cell to the axis. Therefore, it is interesting and challenging to check and practise this scheme on the chemical wave in the chemical system in experiments. In addition, the scheme is also checked by using smaller space step for x = y = 0.4, and time step t = 0.01, it just proved its effectiveness to eliminate the useless patterns as well. Furthermore, the scheme is used to stabilize the patterns in the oscillatory and excitable media in considering the parameter fluctuation induced by some uncertain factors, such as temperature, intrinsic noise and inhomogeneity and so on. But it just proved its effectiveness as well. 3 Conclusion In this paper, a spatial perturbation scheme is used to suppress the spiral wave, spatiotemporal chaos and antispiral wave in the complex Ginzburg Landau equation under periodical boundary conditions. It is different from the scheme in Refs. [17] and [18] by varying parameter perturbation with a spatial space function. The centric field force is introduced into the media, thus a centric gradient field appeared and the pattern in the CGLE is stabilized greatly. As an example, the centric field is advised to

6 1546 MA Jun, YI Ming, ZHANG Li-Ping, JIN Wu-Yin, and LI Yan-Long Vol. 49 be made by circumgyrating the reactor for chemical wave vs. an arbitrary axis with external mechanical force. It is theoretically proved that the centric force-induced pattern transition is equivalent to the results induced by the centric gradient field based on the conservative field. In fact, the dynamics and evolution of the system subjected to a Coulomb field or other conservative field could be equivalent to those subjected to the external spatial force based on the the relation between the conservative force and the potential energy, which is described as Eq. (6). Furthermore, the scheme proved its robustness to spatiotemporal noise, independence to models, especially, it is still effective to remove the useless pattern even if the parameter fluctuation of the media is in consideration. Therefore, we wish it could give some useful clue to stabilize the patterns in other reaction-diffusion systems, and it calls for new proofs in experiments. Acknowledgments We would give great thanks to professors Y. Jia and H.P. Ying for useful discussions. References [1] M.C. Cross and P. Hohenberg, Rev. Mod. Phys. 65 (1993) 851. [2] Y. Kuramoto, Chemical Oscillations, Waves, and Turbulence, Springer-Verlag, New York (1984). [3] A.T. Winfree, Science 175 (1972) 634. [4] V.K. Vanag and I.R. Epstein, Science 294 (2001) 835. [5] Q. Ouyang and J.M. Flesselles, Nature (London) 379 (1996) 143. [6] O. Steinbock, V. Zykov, and S.C. Müller, Nature (London) 366 (1993) 322. [7] K.J. Lee, E.C. Cox, and R.E. Goldstein, Phys. Rev. Lett. 76 (1996) [8] K.J. Lee, E.C. Cox, and R.E. Goldstein, Phys. Rev. Lett. 87 (2001) [9] A.V. Panfilov, S.C. Müller, and V.S. Zykov, Phys. Rev. E 61 (2000) [10] A.T. Winfree, Chaos 8 (1998) 1. [11] M. Bär, L. Brusch, and M. Or-Guil, Phys. Rev. Lett. 92 (2004) [12] L. Zhou and Q. Ouyang, Phys. Rev. Lett. 85 (2000) [13] Q. Ouyang, H.L. Swinney, and G. Li, Phys. Rev. Lett. 84 (2000) [14] S. Sinha, A. Pande, and R. Pandit, Phys. Rev. Lett. 86 (2001) [15] M.P. Nash and A.V. Panfilov, Prog. in Biophy. & Molecular Biology 85 (2004) 501. [16] A.T. Winfree, Science 266 (1994) [17] P.Y. Wang, Phys. Rev. Lett. 80 (1998) [18] P.Y. Wang, Phys. Rev. E 61 (2000) [19] P.Y. Wang, P. Xie, and H.W. Xin, Chin. Phys. 12 (2003) 674. [20] B.B. Hu and H. Zhang, Int. J. Mod. Phys. B 17 (2003) [21] H. Zhang, B.B. Hu, and G. Hu, Phys. Rev. E 68 (2003) [22] Y.Q. Fu, H. Zhang, Z.J. Cao, et al., Phys. Rev. E 72 (2005) [23] H. Zhang, Z.J. Cao, N.J. Wu, et al., Phys. Rev. Lett. 94 (2005) [24] N.J. Wu, H. Zhang, H.P. Ying, et al., Phys. Rev. E 73 (2006) [25] J. Ma, H.P. Ying, and Y.L. Li, Chin. Phys. 16 (2007) 955. [26] J. Ma, Y. Chen, and W.Y. Jin, Commun. Theor. Phys. (Beijing, China) 47 (2007) 675. [27] M. Hendery, K. Nam, P. Guzdar, et al., Phys. Rev. E 62 (2000) [28] M. Hendery, E. Ott, and T.M. Artonsen, Phys. Rev. Lett. 82 (1999) 859. [29] J.H. Xiao, G. Hu, and H. Zhang, Europhys. Lett. 69 (2005) 29. [30] V.S. Zykov, H. Engel, Phys. Rev. E 66 (2002) [31] I.S. Aranson and L. Kramer, Rev. Mod. Phys. 74 (2002) 99. [32] J.Z. Yang and M. Zhang, Phys. Lett. A 352 (2006) 69. [33] Y.F. Gong and D.J. Christini, Phys. Rev. Lett. 90 (2003) [34] L. Brusch, E.M. Nicola, and M. Bär, Phys. Rev. Lett. 92 (2004) [35] E.M. Nicola, L. Brusch, and M. Bär, J. Phys. Chem. B 108 (2004) [36] J. Ma, J.H. Gao, C.N. Wang, et al., Chaos, Solitons and Fractal 38 (2008) 521. [37] J. Ma, C.N. Wang, Z.S. Pu, et al., Commun. Theor. Phys. (Beijing, China) 45 (2006) [38] J. Ma, Z.S. Pu, W.J. Feng, et al., Acta. Phys. Sin. 54 (2005) 4602 (in Chinese). [39] H.L. Wang and Q. Ouyang, Phys. Rev. E 65 (2002) [40] B. Lindner, J. Garcia-Ojalvo, and A. Neiman, et al., Phys. Rep. 392 (2004) 321. [41] X. Shao, Y. Ren, and Q. Ouyang, Chin. Phys. 15 (2006) [42] G. Nicolisg, Introduction to Nonlinear Science, Cambridge University Press, Cambridge (1984). [43] M. Hildebrand, M. Bär, and M. Orguil, Phys. Rev. Lett. 82 (1999) 1160.