Suppression of Spiral Waves and Spatiotemporal Chaos Under Local Self-adaptive Coupling Interactions

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1 Commun. Theor. Phys. (Beijing, China) 45 (6) pp c International Academic Publishers Vol. 45, No. 1, January 15, 6 Suppression of Spiral Waves and Spatiotemporal Chaos Under Local Self-adaptive Coupling Interactions MA Jun, 1,2, WU Ning-Jie, 2 YING He-Ping, 2 and YUAN Li-Hua 1 1 Department of Physics, School of Science, Lanzhou University of Technology, Lanzhou 7300, China 2 Zhejiang Institute of Modern Physics, Zhejiang University, Hangzhou 327, China (Received April 1, 5) Abstract In this paper, a close-loop feedback control is imposed locally on the Fitzhugh Nagumo (FHN) system to suppress the stable spirals and spatiotemporal chaos according to the principle of self-adaptive coupling interaction. The simulation results show that an expanding target wave is stimulated by the spiral waves under dynamic control period when a local area of 5 5 grids is controlled, or the spiral tip is driven to the board of the system. It is also found that the spatiotemporal chaos can be suppressed to get a stable homogeneous state within time units as two local grids are controlled mutually. The mechanism of the scheme is briefly discussed. PACS numbers: Gg, a, r, Ck Key words: spiral waves, spatiotemporal chaos, close-loop feedback, Fitzhugh Nagumo (FHN) equation The targets and spirals are distinct wave patterns that have been observed in a variety of physical, chemical, and biological systems. [1,2] A target pattern is produced by concentric waves that travel away from rhythmic sources while a spiral wave is generated by a rotating source and it can turn to be chaos as well. At present, much evidence has been approved that chaos is found in normal heart while spiral is linked to cardiac arrhythmia, and the breakup of spiral causes rapid death of heart. Instability of the spirals will induce the spatiotemporal chaos and ventricular fibrillation (VF). [3 5] As a result, it is an interesting problem to suppress spiral or to prevent a breakup of spiral. Up to date, many models [6 10] have been proposed, where most of the control schemes used to suppress spiral and spatiotemporal chaos are classified as closed-loop or open-loop controls. For the open-loop control, light illumination, electric field, electric signal, and chaotic signal are imposed on the system locally or globally. For example, Zhang et al. [11] proposed a scheme to suppress spiral and chaos via a target wave by inputting periodical signal locally on the system. Wang et al. [12] used a traveling wave to kill spiral. In Ref. [13], the authors of the present paper have succeeded in control of the spiral and spatiotemporal chaos by driving the whole system with a weak Lorenz chaotic signal within time units. For practical experiments, close-loop control owns more advantages than the open-loop control because the former seldom changes the dynamics of original system. Time-delayed method has been tested to control chaotic system [14] as well as spiral waves successfully. [15 17] Now we hope to characterize efficiency of a close-loop scheme by adaptive self-coupling interaction. The aim of this paper is to convert stable spirals to target waves and/or to change chaos to be a homogeneous state by a two-step controller: to sample a few grids of the system and to feed back the error of sampling grids into the system. Finally we will consider the ability of the scheme against the noise. From the theoretical point of view, Fitzhugh Nagumo (FHN) equation is an ideal model to describe VF, as the model consists of both the spatiotemporal chaos and spiral waves under right parameters and initial values. The efficiency of the controller will be testified by our numerical simulations on this model, defined by the FHN equation, [7,8] u ( t = ε 1 u(1 u) u v + b ) + D 2 u, a v = f(u) v, (1) t 0, 0 u < 1/3, f(u) = u(u 1) 2, 1/3 u 1, (2) 1, u > 1. The system (1) is an activator controller two-variable reaction-diffusion model for CO oxidation on Pt (110), where variables u and v describe the activator and inhibitor respectively. The spatiotemporal dynamics, for two-dimensional case 2 = 2 / x / y 2, is investigated by varying ε with fixing a = 0.84 and b = The numerical simulation, for system size and grids number used in this paper, shows that the spiral is stable when 0.01 < ε < 0.06 under suitable initial values (a stable spiral is observed for parameter ε = 0.02, as illustrated in Fig. 1(a)). Furthermore, the spiral becomes unstable for ε > 0.06 and it finally steps into turbulence after ε > For example, at ε = 0.073, the system (1) emerges spatiotemporal chaos, corresponding to ventricular fibrillation, as shown in Fig. 1(b). In this paper, The project supported partially by National Natural Science Foundation of China under Grant No Corresponding author, hyperchaos@163.com

2 122 MA Jun, WU Ning-Jie, YING He-Ping, and YUAN Li-Hua the parameters are set for the FHN equation to generate a stable rotating spiral as following: ε = 0.04, time step size h = 0.02 and the diffusion coefficient D = 1. The numerical algorithm applied for the simulation is the Euler forward difference algorithm with the non-flux boundary conditions being used. Fig. 1 (a) A Stable spiral at ε = 0.02; (b) Spatiotemporal chaos developed at ε = The first problem to be considered is how to suppress spatiotemporal chaos which corresponds to ventricular fibrillation, developed from a stable spiral of Eq. (1). A controller is then introduced as follows: G = k(u(m, n) u(i, j)), (3) Vol. 45 where k is the feedback coefficient, m, n, i, j are integers to denote the grid positions, u(m, n) and u(i, j) are system variables defined in Eq. (1), which can be selected arbitrarily on the whole system. For the practical point, we choose the coefficient k = 1, which means any two grids are connected directly and their error is fed back without interfering or disturbing outside. The system evolution is presented by the activator variable u as displayed in Figs. 2 4 for spirals and spatiotemporal chaos, respectively. In the simulation, the controller (3) is imposed on the system by including it on the right hand side of the first formula in Eq. (1). A stable rotating spiral state emerges while initial values are chosen as above. At a time point t = 0 when the parameter ε jumps to 0.073, the stable spiral begins to breakup and steps into turbulence, as illustrated in Figs. 2(a) 2(d) at t = 20, 40,, 60 times units. For t = 60 time units when the controller (3) begins to work on the system, we set k = 1, u(m, n) = u(10, 10) and u(i, j) = u(126, 126) in a random choice. The evolution of the spiral to spatiotemporal chaos and the generated new states are shown in Figs. 2(e) 2(l) for t = 65, 70, 80, 90,, 105, 106, 107 times units, successively (a) 1 2 (b) 1 2 (c) (e) 1 2 (f) 1 2 (g) (i) 2 1 (j) 2 1 (k) (h) 2 1 (d) (l) Fig. 2 The evolution of a stable spirals after tuning ε = at t = 0. The spiral begins to step into turbulence in a period of t = 20 (a), 40 (b), (c), 60 (d) time units. At time t = 60, controller (3) G = 1.0(u(10, 10) u(126, 126)) works and the system states develop accordingly for t = 65(e), 70(f), 80(g), 90(h), (i), 105(j), 106(k), 107(l) time units, respectively.

3 No. 1 Suppression of Spiral Waves and Spatiotemporal Chaos Under Local Self-adaptive Coupling Interactions (a) (d) (g) (f) 2 (e) 2 1 (c) 2 (b) 2 (h) 1 (i) Fig. 3 Evolutions of stable spirals, after tuning ε = at t = 0, to the states at t = 180 (a) and t = (b). After t = putting the controller (3), G = 1.0(u(10, 10) u(126, 126)), the states of the system develop successively under the time sequence of t = + t for t = 3 until t = 221 time units in (c) (i) (a) (d) (g) 2 1 (h) (f) (e) 2 1 (c) 2 (b) 2 1 (i) Fig. 4 Evolutions of stable spirals with ε = 0.04 in a time sequence of t = 40 (a), t = (b), t = 60 (c), and t = 60 + t for t = 3 until t = 78 in (d) (i), when the controller (3) G = 1.3(u(10, 10) u(126, 126)) is turned on at time t =

4 124 MA Jun, WU Ning-Jie, YING He-Ping, and YUAN Li-Hua Vol. 45 From the numerical study, it shows that this scheme works with high efficiency to suppress transitional spirals and spatiotemporal chaos within time units for the self-coupling interaction and the feedback. When the whole system reaches a homogeneous state (the gray scale is the final state), the controller (3) will finish the control process automatically. Then we begin to consider how the controller works on the whole spatiotemporal chaos. A stable spiral state and ε = are set as the initial value to produce a whole spatiotemporal chaos on the system (1) after an enough long period evolution (e.g. time units). Those states are displayed in Figs. 3(a) and 3(b) at the time of t = 180 and t =. At the time point of t =, the controller (3) begins to be included and its influence on the system is displayed in Figs. 3(c) 3(i) in a time period of t = + t with t = 3 until t = 221, respectively. The numerical results prove that the present form of the close-loop feedback scheme works well to kill the spatiotemporal chaos quickly (within 21 time units), and the whole system subsequently becomes stable and homogeneous, where u(i, j) = 0, v(i, j) = 0 (the gray scale is the final state) even for feedback coefficient k = 1. In other studies, it gives evidences that the stronger the k is, the shorter the transient period to reach the homogeneous state is. Here, we focus on the case of k = 1 to emphasize its physical dynamics. In experiments, it can be realized that a simple metal pin to connect the two grids or cells of u(126, 126) and u(10, 10) directly and the error between them is fed back into the whole system adaptively. Further the scheme is testified on stable spiral waves under a stronger control by amplifying the coefficient k. It is found that from the simulation, plotted as in Fig. 4, the spiral is killed and the whole system becomes gradually homogeneous within 78 time units for k = 1.3. As we expected, the transient period can be shorter when a stronger gain coefficient k is introduced. On the other hand, high gain by k may call for much disturbance from outside and it may cause difficulties in practice. So far, the numerical results give evidence that the scheme for controller (3) is effective although the transient period to homogeneous states differs between the control processes for spiral or spatiotemporal chaos. To compare with the suppressions of transitional spatiotemporal chaos (Fig. 2) and whole spatiotemporal chaos (Fig. 3), it now takes a longer time period or stronger amplitude to suppress stable spiral (for example, the controller (3) fails to kill the spiral at k = 1 within 0 time units, or it requires about 80 time units as k = 1.3). Of course, the scheme is useful, in the experiments or practices, as the controller stops work automatically as soon as the system reaches the homogeneous states. In all simulations described above, the controller (3) interacts with the global grids of the system. Now we will focus on the characters when controller is imposed locally. Unfortunately, by our experience it is in vain to kill the spiral wave when controller is only imposed on the system locally (for details, see Eq. (4)). However, a new pattern of the target wave emerging from spiral wave is observed under such local control scheme. To check the problem in detail, we test the scheme by modifying the controller (3) to involve more grids (about 5 5) to be controlled. In such a case, the controller can be re-defined as G = k(u(m, n) u(i, j))δ(x x i )δ(y y j ), (4) where the function δ(x) = 1 for x = 0 and δ(x) = 0 for x 0, x(y) is grid position and x( y) is grid distance: x i = i x (y i = i y), u(m, n) and u(i, j) are the same as in the system (1). In addition, u(m,n) is sampled randomly on the system. For example, a local area of 125 < i, j < 131 with k = 1 and u(m, n) = u(10, 10) is selected in our study. From the dynamics point of view, we predict that a wave target is in birth under the work of controller (4). The error between the arbitrary sampling grid u(10, 10) and the local controlled grids (i, j) (125 < i, j < 131) can produce a pre-periodical or periodical force. It is the source to induce a strong target wave, as proved by our numerical results. Finally, a necessary calculation is paid to confirm whether this scheme is robust against white noise. The state behaviors of the system (1) under the spatiotemporal noise are investigated with the noise σ(x, y, t), where σ(x, y, t) = 0, ( σ(x, y, t)σ(x, y, t ) = Dδ(x x )δ(y y )δ(t t ), where D is an amplitude. In our study, we take D = 0.002, and choose k = 4 on the grids of u(m, n) = u(10, 10). The results are presented in Fig. 5, where state evolution for the sequence of t = t with t = 0 or until t = 40 is plotted successively. As seen in Fig. 5, the spiral and target wave compete with each other dynamically, and the target wave finally dominates the system. The spiral tip is then turned to and located on the border corner of the system. What happens as we increase the noise intensity (e.g. D = 0.008) to check the strategy still effective or not? As the noise is not included, the scheme seems to work with high efficiency. When the noise introduced, it takes a longer period to convert

5 No. 1 Suppression of Spiral Waves and Spatiotemporal Chaos Under Local Self-adaptive Coupling Interactions 125 spiral wave to the target wave if k = 1. Fig. 5 Convection of spirals to target wave for D = 0.002, k = 4, u(m, n) = u(10, 10) in the time period of t = 3000 (a), 4 (b), 4 (c), 4300 (d), 4400 (e), 40 (f) respectively. The controller (4) G = 4(u(10, 10) u(126, 126)) with white noise is turned on at time t = 0. Finally, the principle of the controller can be understood from the dynamics. In fact, the controller (3) imposes a pre-stochastic force on the whole space of the system because function G = k(u(10, 10) u(126, 126)) varies in random on the whole system. The system indeed reaches homogeneous state when enough strong amplitude of driving is involved. Based on the target wave character, we can conclude that a target will appear as the controller (3) is locally imposed on the system. In conclusion, we proposed a new scheme to suppress spiral wave and spatiotemporal chaos developed from spiral breakup and to convert the spiral into target waves or prevent it from breakup based on self-organization theory. It is also found that the system develops adaptively under a self-coupling interaction between a few grids of the system, as well as the present method is robust against noise. Specially, our strategy takes a shorter transient period and weaker gain than other schemes to change the spiral waves and spatiotemporal chaos into the homogeneous states, compared with many recent results. We hope that some hints may be provided for the study of defibrillation as well as suppression of spirals and spatiotemporal chaos in experiments. Acknowledgments We would like to thank H. Zhang for valuable discussions. References [1] M.W. Mueller and W.D. Arnett, Astrophys. 210 (1976) 670; Waves and Patterns in Chemical and Biological Media, eds. H.L. Swinneyand and V.I. Krinsky, MIT, North- Holland, Cambridge, MA (1992); S.K. Scott, Oscillations, Waves, and Chaos in Chemical Kinetics, Oxford University Press, Oxford, England (1994). [2] A.T. Winfree, When Time Breaks Down: The Three- Dimensional Dynamics of Electrochemical Waves and Cardiac Arrhythmias, Princeton, Princeton, NJ (1987). [3] Faramarz H. Samie and Jose Jalife, Cardiovascular Research (1) 242. [4] V.N. Biktashev, A.V. Holden, and S.F. Mironov, Chaos, Soliton and Fractals 13 (2) [5] Martyn P. Nash and Alexander V. Panfilov, Profess in Biophysics & Molecular Biology 85 (4) 1. [6] S. Sinha, A. Pande, and R. Pandit, Phys. Rev. Lett. 86 (1) [7] M. Bär and M. Eiswirth, Phys. Rev. E 48 (1993) R1635. [8] M. Hildebrand, M. Bär, and M. Eiswirth, Phys. Rev. Lett. 75 (1995) 13.

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