可激发介质螺旋波和时空混沌耦合控制研究
|
摘要
斑图是在空间或时间上具有某种规律性的非均匀宏观结构，普遍存在于自然界中。螺旋波是非平衡斑图中最常见的一种，在自然界中广泛存在，可以在振荡系统、激发系统和双稳系统中观察到，包括许多物理、生物、化学等系统。由于在很多实际系统中螺旋波及其破碎是有害的，例如，在心肌中，螺旋波是导致心动过速的主要原因；螺旋波的自发破碎会导致心室失颤。因此，开展对螺旋波的失稳机制以及螺旋波和时空混沌的控制研究具有十分重要的意义。本文重点研究了螺旋波和时空混沌的控制，特别是通过互耦合及驱动耦合的方法来研究其对螺旋波和时空混沌的控制效果。
论文的第一章重点介绍了什么是可激发系统、一些可激发系统模型以及目前成熟的关于螺旋波动力学行为和螺旋波破碎的理论。
第二章重点讨论了螺旋波和时空混沌的控制和同步。我们从引入反馈控制法、外力控制法和调整参数控制法对可激发系统中的螺旋波和时空混沌控制进行归类。对于时空斑图的同步讨论，主要集中在双层耦合可激发模型。
第三章主要研究了两个相互耦合的可激发系统时空混沌的同步行为。我们通过相互耦合的方法实现了两个可激发系统时空混沌的完全同步。给出了确定耦合系数的方法，得出了能使两个可激发系统时空混沌同步所需要的耦合系数的阈值。基于相互耦合法，将时空混沌态系统与静息态系统进行相互耦合，实现了将时空混沌控制成静息态；将时空混沌态系统与外周期力控制系统进行相互耦合，在一定的参数区域内(0.065≤ε≤0.090)，实现了只需微调已知外周期力参数w，便可将该参数区域内时空混沌控制成靶波态。
第四章主要研究了驱动耦合控制法对可激发介质中螺旋波的控制效果。研究结果显示，这种控制方法具有教高的控制效率及鲁棒性，即既能控制CO在铂(Pt110)表面氧化反应中的螺旋波，又能控制FHN模型中的螺旋波。对于不同的模型，我们发现最小控制时步n与耦合系数c之间存在形如n～c~(-k)的幂指数关系。
Pattern is a kind of inhomogeneous, spatiotemporal macro-structure with some laws, to exist in a large number of systems in nature. Spiral wave is a kind of nonequilibrium pattern, which can be found in excitable media, oscillatory media, and bistable(more precisely double metastable) media, including a lot of physical, biological and chemical systems. In some cases spiral waves and spatiotemporal chaos are undesirable because of their harmfulness. Such as the spiral waves in cardiac muscle can be a cause of tachycardia. Repetitious breakup of spiral waves can lead to spatiotemporal chaos, which is believed as a mechanism of ventricular fibrillation (VF). Therefore, effective control methods are needed to control spiral waves and spatiotemporal chaos. In this master thesis, we focus on the study of spiral waves and spatiotemporal chaos control in excitable media by using mutual coupling method and unidirectional coupling method.
The first chapter mainly reviews the excitable systems, some models and the recent development about the dynamics and the breakup of spiral waves.
The second chapter discusses the control and synchronization of spiral waves and spatiotemporal chaos. Three control methods have been mentioned, such as feedback control method, outside force control method and parameter adjustment control method. The synchronization of spatiotemporal pattern also has been discussed especially focused in two layer coupled excitable model.
The third chapter gives a systematic research of the effects of the mutual coupling method on the spatiotemporal chaos synchronization behavior. Complete synchronization can be achieved by using this method. The method to calculate the coupling coefficient is given and the suitable coupling coefficient which complete synchronization can be achieved has been attained. Spatiotemporal chaos is suppressed in two ways by coupling with two systems in different states.
The effects of unidirectional coupling method to suppress spiral waves in excitable media are studied in the forth chapter. It is found that this control method has high control efficiency and robust. It adapts to control spiral waves for catalytic CO oxidation on platinum as well as for the FHN model. And the power law n～c~(-k) of control time steps n versus the coupling strength c for different model has been obtained.
论文的第一章重点介绍了什么是可激发系统、一些可激发系统模型以及目前成熟的关于螺旋波动力学行为和螺旋波破碎的理论。
第二章重点讨论了螺旋波和时空混沌的控制和同步。我们从引入反馈控制法、外力控制法和调整参数控制法对可激发系统中的螺旋波和时空混沌控制进行归类。对于时空斑图的同步讨论，主要集中在双层耦合可激发模型。
第三章主要研究了两个相互耦合的可激发系统时空混沌的同步行为。我们通过相互耦合的方法实现了两个可激发系统时空混沌的完全同步。给出了确定耦合系数的方法，得出了能使两个可激发系统时空混沌同步所需要的耦合系数的阈值。基于相互耦合法，将时空混沌态系统与静息态系统进行相互耦合，实现了将时空混沌控制成静息态；将时空混沌态系统与外周期力控制系统进行相互耦合，在一定的参数区域内(0.065≤ε≤0.090)，实现了只需微调已知外周期力参数w，便可将该参数区域内时空混沌控制成靶波态。
第四章主要研究了驱动耦合控制法对可激发介质中螺旋波的控制效果。研究结果显示，这种控制方法具有教高的控制效率及鲁棒性，即既能控制CO在铂(Pt110)表面氧化反应中的螺旋波，又能控制FHN模型中的螺旋波。对于不同的模型，我们发现最小控制时步n与耦合系数c之间存在形如n～c~(-k)的幂指数关系。
Pattern is a kind of inhomogeneous, spatiotemporal macro-structure with some laws, to exist in a large number of systems in nature. Spiral wave is a kind of nonequilibrium pattern, which can be found in excitable media, oscillatory media, and bistable(more precisely double metastable) media, including a lot of physical, biological and chemical systems. In some cases spiral waves and spatiotemporal chaos are undesirable because of their harmfulness. Such as the spiral waves in cardiac muscle can be a cause of tachycardia. Repetitious breakup of spiral waves can lead to spatiotemporal chaos, which is believed as a mechanism of ventricular fibrillation (VF). Therefore, effective control methods are needed to control spiral waves and spatiotemporal chaos. In this master thesis, we focus on the study of spiral waves and spatiotemporal chaos control in excitable media by using mutual coupling method and unidirectional coupling method.
The first chapter mainly reviews the excitable systems, some models and the recent development about the dynamics and the breakup of spiral waves.
The second chapter discusses the control and synchronization of spiral waves and spatiotemporal chaos. Three control methods have been mentioned, such as feedback control method, outside force control method and parameter adjustment control method. The synchronization of spatiotemporal pattern also has been discussed especially focused in two layer coupled excitable model.
The third chapter gives a systematic research of the effects of the mutual coupling method on the spatiotemporal chaos synchronization behavior. Complete synchronization can be achieved by using this method. The method to calculate the coupling coefficient is given and the suitable coupling coefficient which complete synchronization can be achieved has been attained. Spatiotemporal chaos is suppressed in two ways by coupling with two systems in different states.
The effects of unidirectional coupling method to suppress spiral waves in excitable media are studied in the forth chapter. It is found that this control method has high control efficiency and robust. It adapts to control spiral waves for catalytic CO oxidation on platinum as well as for the FHN model. And the power law n～c~(-k) of control time steps n versus the coupling strength c for different model has been obtained.
引文
[1] A. M. Turning, Phil. Tran. R. Soc. London B, 237, 37, 1952
[2] G.. Nicolis, I. Pringogine, Self-Organization in Non-equilibrium Systems, John Wiley &Sons, Inc., 1997
[3] A, N. Zhaikin, A.M. Zhabotinsky, Concentration Wave Propagation in Two-dimensionalLiquid-phase Self-oscillating System, Nature, 225,535, 1970
[4] 欧阳颀，反应扩散系统中的斑图动力学，上海科学技术出版社，2000
[5] M. C. Cross, P. C. Hohenberg, Pattern formation outside of equilibrium, Rev. Mod. Phys.,65, 851, 1993
[6] J. M. Davidenko, A. V. Pertosov, P. Salomonsz, et al, Stationary and drifting spiral waves of excitation in isolated cardiac muscle, Nature, 355,349, 1992
[7] F. Siegert, C. Wdijer, Digital image Processing of optical density wave propagation in Dictyostelium discoideum and analysis of the effects of caffeine and ammonia, J. Cell. Sci., 93, 325, 1989
[8] N. A. Gorrlova, J. Bures, Spiral waves of spreading depression in the isolated chicken retina, J. Neuro, 14, 353, 1983
[9] J. Lechleiter, S. Girard, E. Peralta, Spiral calcium wave propagation and annihilation in Xenopus laevis oocytes, D. Clapham, Science, 252, 123, 1991
[10] M. Bar, M. Eiswirth, H. H. Rotemund, G. Ertl, Solitary-wave phenomena in an excitable surface reaction, Phys. Rev. Lett., 69, 945, 1992
[11] A. T. Winfree, Spiral Waves of Chemical Activity, Science, 175,634, 1972
[12] A. T. Winfree, Scl. Am., 231, 1974
[13] G. Li, Q. Quyang, v. Petrov, H. L. Swinney, Transition from simple rotating chemical spirals to meandering and traveling spirals, Phys. Rev. Lett., 77, 2105, 1996
[14] G. Veser, F. Esch, R. Imbihl, Regular and irregular spatial patterns in the catalytic reduction of NO with NH 3 on Pt (100), Catal. Lett., 13,371, 1992
[15] G. Veser, F. Mertens, A. S. Mikhailov, R. Imbihl, Global coupling in the presence of defects: Synchronization in an oscillatory surface reaction, Phys. Rev. Lett., 71,935, 1993
[16] J. Bures, O. Buresova, J. Kxivanek, The mechanism and applications of Leao's Spreading Deppression of electroencephalographic activity, Academia, Prague, 1974
[17] G. Maetins-Ferreira, G. Deoliveira Castro, C. J. Struchiner, P. S. Rodrigues, Circling spreading depression in isolated chick retina, J. Neurophysiol, 4, 773, 1974
[18] K. Tomchik, P. Devreotes, Adenosine 3',5'-monophosphate waves in Dictyostelium discoideum: a demonstration by isotope dilution—fluorography, Science, 212, 443, 1981
[19] F. Alcantara, M. Monk, J. Gen. Microbiol., Signal propagation during aggregation in the slime mould Dictyostelium discoideum, 85, 321, 1974
[20] M. J. Berrldge, R. F. Irvine, Inositol phosphates and cell signalling, Nature, 341, 197, 1989
[21] M. J. Berrldge, P. H. Cobbold, K. S. R. Cuthbertson, Spatial and Temporal Aspects of Cell Signalling [and Discussion], Philos. Trans. R. Soc. London Ser., B 320, 325, 1988
[22] 曹周键，非线性介质中的驱动响应及时空斑图和时空混沌控制。北京：北京师范大学博士论文，2006
[23] A. T. Winfree, When Time Breaks Down, Princeton University Press, Princeton, NJ,1987
[24] http://cardio.rned.stu.edu.cn/RelativeCourses/ICD, 2004
[25] J. L. Prevost, F. Battelli, Sur quel ques effets des dechanges electriques sur le coer mammifres, Comptes Rendus Seances Acad. Sci., 129, 1267,1899
[26] C. S. Beck, W. H. Pritchard, and H. S. Feil, Ventricular fibrillation of long duration abolished by electric shock, J. Am. Med. Assoc., 135, 985,1947
[27] P. M. Zoll, A. J. Linethal, W. Gibson, and M. H. Paul, Termination of ventricular fibrillation in man by externally applied electric countershock, N. Engl. J. Med., 254, 727,1956
[28] M. Allessie, C. Kirchhof, G. J. Schffer, F. Chorro, and J. Brugada, Regional control of atrial fibrillation by rapid pacing in conscious dogs, Circulation, 84, 1689, 1991
[29] A. Cappucci, F. Ravelli, G. Nollo, A. S. Montenero, M. Biffi, G. Q. Villani, Capture window in human atrial fibrillation: evidence of an excitable gap, J. Cardio-vasc, Electrophysiol., 10, 319, 1999
[30] E. G. Daoud, B. Pariseau, M. Niebauer, F. Bogun, R. Goyal, M. Harvey, K. C. Man, S. A. Strickberger, F. Morady, Response of Type I Atrial Fibrillation to Atrial Pacing in Humans, Circulation, 94, 1036, 1996
[31] F. A. Hatib, E. Trendafilova and I. Daskalov, Transthoracic electrical impedance during external defibrillation: comparison of measured and modelled waveforms, Physiol. Meas., 21, 145, 2000
[32] C. Kirchhof, F. Chorro, G. J. Schffer, J. Brugada, K. Konings, Z. Zetelaki, and M. Allessie, Regional entrainment of atrial fibrillation studied by high-resolution mapping in open-chest dogs, Circulation, 88, 736, 1993
[33] J. M. Kalman, J. E. Olgin, M. R. Karch and M. Lesh, Regional entrainment of atrial fibrillation in man, J. Cardiovasc. Electrophysiol., 7, 867, 1996
[34] P. S. Chen, A. Garfinkel, J. N. Weiss, H. S. Karagueuzian, Computerized mapping of fibrillation in normal ventricular myocardium, Chaos, 8, 127, 1998
[35] F. X. Witkowski, L. J. Leon, P. A. Penkoske, W. R. Giles, M. L. Spano, W. L. Ditto and A. T. Winfree, Spatiotemporal evolution of ventricular fibrillation, Nature, 392, 78, 1998
[36] M. Cross and P. Hohenberg, Spatiotemporal Chaos, Science, 263, 1569, 1994
[37] H. Zhang, B. Hu, G. Hu, Suppression of spiral waves and spatiotemporal chaos by generating target waves in excitable media, Phys. Rev. E, 68, 026134, 2003
[38] 马军、蒲忠胜、冯旺军、李维学，旋转中心力场消除螺旋波和时空混沌，物理学报，54，4602，2005
[39] P. Y. Wang and P. Xie, Eliminating spatial chaos and spiral waves by weak spatial perturbations, Phys. Rev. E, 61, 5120, 2000
[40] P. Pamananda and J. L, Hudson, Controlling spatiotemporal chemical chaos using delayed feedback, Phys. Rev. E, 64, 037201, 2001
[41] H. Zhang, Z. J. Cao, N. J. Wu, H. P. Ying, G. Hu, Suppress Winfree Turbulence by Local Forcing Excitable Systems, Phys. Rev. Lett., 94, 188301, 2005
[42] 朱振明，化学体系螺旋波的噪声动力学。合肥：中国科学技术大学博士论文，2001
[43] B. Lindner, J. Garcia-Ojalva, A. Neiman, et al, Effects of noise in excitable systems, Phys. Rep, 392, 321, 2004
[44] R. FitzHugh, Impulses and Physiological States in Theoretical Models of Nerve Membrane, Biophys. J, 1,445, 1961
[45] J. S. Nagumo, S. Arimoto, S. Yoshizawa, An Active Pulse Transmission Line Simulating Nerve Axon, Proc. IRE 50, 2061, 1962
[46] D. Barkley, Mark Kness, Laurette S. Tuckerman, Spiral-wave dynamics in a simple model of excitable media: The transition from simple to compound rotation, Phys. Rev. A, 42, R2489, 1990
[47] M. Bar, M. Eiswirth, Turbulence due to spiral breakup in a continuous excitable medium, Phys. Rev. E, 48, R1635, 1993
[48] M. Bar, N. Gottschlk, M. Eiswith, Spiral waves in a surface reaction: Model calculations, J. Chem. Phys, 100(2), 1202, 1994
[49] D. Barkley, A model for fast computer-simulation of waves in excitable media., Physica D (Amsterdam), 49, 61, 1991
[50] V. Hakim, A. Karma, Phys. Rev. E, Theory of spiral wave dynamics in weakly excitable media: Asymptotic reduction to a kinematic model and applications, 60, 5073, 1999
[51] 袁国勇，双层可激系统的动力学行为及螺旋波的控制。北京：中国工程物理研究院博士论文，2005
[52] A. T. Winfree, Scroll-shaped wave of chemical activity in three dimensions, Science, 1973, 181:937
[53] D. Barkley, Phys. Rev. Lett, Euclidean symmetry and the dynamics of rotating spiral waves, 72, 164, 1994
[54] Q. Ouyang and J. M. Flesselles, Transition from spirals to defect turbulence driven by a convective instability, Nature, 379, 143, 1996
[55] Q. Ouyang, H. L. Swinney, and G. Li, Transition from Spirals to Defect-Mediated Turbulence Driven by a Doppler Instability, Phys. Rev. Lett., 84, 1047, 2000
[56] 陈式刚，映象与混沌，国防工业出版社，2000
[57] 王光瑞，混沌的控制、同步与利用，国防工业出版社，2001
[58] 胡岗，萧井华，郑志刚，混沌控制，上海科技教育出版社，2000
[59] H. Fujisaka, T. Yama, Stability Theory of Synchronized Motion in Coupled-Oscillator Systems, Prog. Theor. Phys., 69, 32, 1983
[60] V. S. Afraimovich, N. N. Verichev and M.. Rebinovich, et al, Izvestiya Vysshkh Uchebnykh Zavedenii Radiofizika., 59, 1050, 1986
[61] L. K. Pecora and T. L. Carroll, Synchronization in chaotic systems, Phys. Rev. Lett., 64, 821,1990
[62] L. M. Rosenblum, A. S. Pikovsky and J. Kurths, Phase Synchronization of Chaotic Oscillators, Phys. Rev. Lett., 76,1804,1996
[63] E. R. Rosa, E. Ott and M. H. Hess, Transition to Phase Synchronization of Chaos, Phys. Rev. Lett., 80,1642,1997
[64] L. M. Rosenblum, A. S. Pikovsky and J. Kurths, From Phase to Lag Synchronization in Coupled Chaotic Oscillators, Phys. Rev. Lett., 78,4193,1997
[65] N. F. Rullkov, M. M. Sushchik and L. S. Tsimring, et al, Generalized synchronization of chaos in directionally coupled chaotic systems, Phys. Rev. E., 51, 980,1995
[66] L. Kocarev and U. Parlitz, Generalized Synchronization, Predictability, and Equivalence of Unidirectionally Coupled Dynamical Systems, Phys. Rev. Lett., 76,1816,1996
[67] S. Boccaletti and D. L. Valldares, Characterization of intermittent lag synchronization, Phys. Rev. E, 62, 7497,2000
[68] M. A. Zaks, E. H. Park, M. G. Rosenblum and J. Kurths, Alternating Locking Ratios in Imperfect Phase Synchronization, Phys. Rev. Lett., 82,4228,1999
[69] R. Femat and G. Solis-Perales, Laplace domain controllers for chaos control, Phys. Rev. A, 262, 50,1999
[70] W. Lu, D. Yu and R. G. Harrison, Control of Patterns in Spatiotemporal Chaos in Optics, Phys. Rev. Lett., 76, 3316,1996
[71] K. Pyragas, Continuous control of chaos by self-controlling feedback, Phys. Lett. A, 170, 421, 1992
[72] R. Martin, A. J. Scroggie and G. L. Oppo, Stabilization, Selection, and Tracking of Unstable Patterns by Fourier Space Techniques, Phys. Rev. Lett., 77, 4007, 1996
[73] D. Hochheiser, J. V. Moloney and J. Lega, Controlling optical turbulence, Phys. Rev. A, 55.R4011,1997
[74] A. Pikovsky, M. Rosenblum and J. Kurths, Synchronization in a population of globally coupled chaotic oscillators, Europhys. Lett., 34, 165, 1996
[75] D. H. Zanette, Dynamics of globally coupled bistable elements, Phys. Rev. E, 55, 5315, 1997
[76] D. H. Zanette and A. S. Mikhailov, Mutual synchronization in ensembles of globally coupled neural networks, Phys. Rev. E, 58, 872, 1998
[77] E S. de San Reman, S. Boccaletti, D. Maza, and H. Mancini, Weak Synchronization of Chaotic Coupled Map Lattices, Phys. Rev. Lett., 81, 3639, 1998
[78] D. Maza, S. Boccaletti and H. Mancini, Phase clustering and collective behaviors in globally coupled map lattices due to mean field effects, Int. J. Bifurc. Chaos., 4, 829, 2000
[79] W. J. Rappel, F. Fenton and A. Karma, Spatiotemporal Control of Wave Instabilities in Cardiac Tissue, Rhys. Rev. Lett., 83, 456, 1999
[80] H. Sakaguchi and T. Fujimoto, Elimination of spiral chaos by periodic force for the Aliev-Panfilov model, Phys. Rev. E, 67, 067202, 2003
[81] R Y. Wang, R W Xie and H. W. Yin, Control of spiral waves and turbulent states in a cardiac model by travelling-wave perturbations, Chin. Phys., 12, 674, 2003
[82] M. Woltering and M. Markus, Control of Spatiotemporal Disorder in an Excitable Medium, ScienceAsia, 28, 43, 2002
[83] 袁国勇、杨世平、王光瑞、陈式刚，物理学报，两个延迟耦合FitzHugh-Nagumo系统的动力学行为，54，1510，2005
[84] M. Valderrabano, M. H. Lee, T. Ohara, A. C. Lai, M. C. Fishbein, S. F. Lin, H. S. Karagueuzian, and E S.Chen, Dynamics of Intramural and Transmural Reentry During Ventricular Fibrillation in Isolated Swine Ventricles, Circulation Research, April 27; 83, 2001
[85] J. Kneller, R. Zou, E. J. Vigmond, Zhiguo Wang, L. J. Leon, and S. Nattel, Circulation Research, May 90, 73, 2002
[86] R. F. Gilmour and D. R. Chialvo, Electrical restitution, critical mass, and the fiddle of fibrillation, J. Cardiovasc Electrophysiol, 10, 1087, 1999
[87] M. Hildebrand, J. X. Cui, E. Mihaliuk, J.Wang, and K. Showalter, Synchronization of spatiotemporal patterns in locally coupled excitable media, Phys. Rev. E, 68, 026205, 2003
[88] L. F. Yang, I. R. Epstein, Symmetric, asymmetric, and antiphase Turing patterns in a model system with two identical coupled layers, Phys. Rev. E, 69, 026211, 2004
[89] J. Z. Yang, E G. Xie, Z. L. Qu and A. Garfinkel, Mechanism for Spiral Wave Breakup in Excitable and Oscillatory Media, Phys. Rev. Lett., 91, 148302, 2003
[90] G. Y. Yuan, G. R. Wang and S. G. Chen, Control of spiral waves and spatiotemporal chaos by periodic perturbation near the boundary, Europhys. Lett. 72, 908, 2005
[2] G.. Nicolis, I. Pringogine, Self-Organization in Non-equilibrium Systems, John Wiley &Sons, Inc., 1997
[3] A, N. Zhaikin, A.M. Zhabotinsky, Concentration Wave Propagation in Two-dimensionalLiquid-phase Self-oscillating System, Nature, 225,535, 1970
[4] 欧阳颀，反应扩散系统中的斑图动力学，上海科学技术出版社，2000
[5] M. C. Cross, P. C. Hohenberg, Pattern formation outside of equilibrium, Rev. Mod. Phys.,65, 851, 1993
[6] J. M. Davidenko, A. V. Pertosov, P. Salomonsz, et al, Stationary and drifting spiral waves of excitation in isolated cardiac muscle, Nature, 355,349, 1992
[7] F. Siegert, C. Wdijer, Digital image Processing of optical density wave propagation in Dictyostelium discoideum and analysis of the effects of caffeine and ammonia, J. Cell. Sci., 93, 325, 1989
[8] N. A. Gorrlova, J. Bures, Spiral waves of spreading depression in the isolated chicken retina, J. Neuro, 14, 353, 1983
[9] J. Lechleiter, S. Girard, E. Peralta, Spiral calcium wave propagation and annihilation in Xenopus laevis oocytes, D. Clapham, Science, 252, 123, 1991
[10] M. Bar, M. Eiswirth, H. H. Rotemund, G. Ertl, Solitary-wave phenomena in an excitable surface reaction, Phys. Rev. Lett., 69, 945, 1992
[11] A. T. Winfree, Spiral Waves of Chemical Activity, Science, 175,634, 1972
[12] A. T. Winfree, Scl. Am., 231, 1974
[13] G. Li, Q. Quyang, v. Petrov, H. L. Swinney, Transition from simple rotating chemical spirals to meandering and traveling spirals, Phys. Rev. Lett., 77, 2105, 1996
[14] G. Veser, F. Esch, R. Imbihl, Regular and irregular spatial patterns in the catalytic reduction of NO with NH 3 on Pt (100), Catal. Lett., 13,371, 1992
[15] G. Veser, F. Mertens, A. S. Mikhailov, R. Imbihl, Global coupling in the presence of defects: Synchronization in an oscillatory surface reaction, Phys. Rev. Lett., 71,935, 1993
[16] J. Bures, O. Buresova, J. Kxivanek, The mechanism and applications of Leao's Spreading Deppression of electroencephalographic activity, Academia, Prague, 1974
[17] G. Maetins-Ferreira, G. Deoliveira Castro, C. J. Struchiner, P. S. Rodrigues, Circling spreading depression in isolated chick retina, J. Neurophysiol, 4, 773, 1974
[18] K. Tomchik, P. Devreotes, Adenosine 3',5'-monophosphate waves in Dictyostelium discoideum: a demonstration by isotope dilution—fluorography, Science, 212, 443, 1981
[19] F. Alcantara, M. Monk, J. Gen. Microbiol., Signal propagation during aggregation in the slime mould Dictyostelium discoideum, 85, 321, 1974
[20] M. J. Berrldge, R. F. Irvine, Inositol phosphates and cell signalling, Nature, 341, 197, 1989
[21] M. J. Berrldge, P. H. Cobbold, K. S. R. Cuthbertson, Spatial and Temporal Aspects of Cell Signalling [and Discussion], Philos. Trans. R. Soc. London Ser., B 320, 325, 1988
[22] 曹周键，非线性介质中的驱动响应及时空斑图和时空混沌控制。北京：北京师范大学博士论文，2006
[23] A. T. Winfree, When Time Breaks Down, Princeton University Press, Princeton, NJ,1987
[24] http://cardio.rned.stu.edu.cn/RelativeCourses/ICD, 2004
[25] J. L. Prevost, F. Battelli, Sur quel ques effets des dechanges electriques sur le coer mammifres, Comptes Rendus Seances Acad. Sci., 129, 1267,1899
[26] C. S. Beck, W. H. Pritchard, and H. S. Feil, Ventricular fibrillation of long duration abolished by electric shock, J. Am. Med. Assoc., 135, 985,1947
[27] P. M. Zoll, A. J. Linethal, W. Gibson, and M. H. Paul, Termination of ventricular fibrillation in man by externally applied electric countershock, N. Engl. J. Med., 254, 727,1956
[28] M. Allessie, C. Kirchhof, G. J. Schffer, F. Chorro, and J. Brugada, Regional control of atrial fibrillation by rapid pacing in conscious dogs, Circulation, 84, 1689, 1991
[29] A. Cappucci, F. Ravelli, G. Nollo, A. S. Montenero, M. Biffi, G. Q. Villani, Capture window in human atrial fibrillation: evidence of an excitable gap, J. Cardio-vasc, Electrophysiol., 10, 319, 1999
[30] E. G. Daoud, B. Pariseau, M. Niebauer, F. Bogun, R. Goyal, M. Harvey, K. C. Man, S. A. Strickberger, F. Morady, Response of Type I Atrial Fibrillation to Atrial Pacing in Humans, Circulation, 94, 1036, 1996
[31] F. A. Hatib, E. Trendafilova and I. Daskalov, Transthoracic electrical impedance during external defibrillation: comparison of measured and modelled waveforms, Physiol. Meas., 21, 145, 2000
[32] C. Kirchhof, F. Chorro, G. J. Schffer, J. Brugada, K. Konings, Z. Zetelaki, and M. Allessie, Regional entrainment of atrial fibrillation studied by high-resolution mapping in open-chest dogs, Circulation, 88, 736, 1993
[33] J. M. Kalman, J. E. Olgin, M. R. Karch and M. Lesh, Regional entrainment of atrial fibrillation in man, J. Cardiovasc. Electrophysiol., 7, 867, 1996
[34] P. S. Chen, A. Garfinkel, J. N. Weiss, H. S. Karagueuzian, Computerized mapping of fibrillation in normal ventricular myocardium, Chaos, 8, 127, 1998
[35] F. X. Witkowski, L. J. Leon, P. A. Penkoske, W. R. Giles, M. L. Spano, W. L. Ditto and A. T. Winfree, Spatiotemporal evolution of ventricular fibrillation, Nature, 392, 78, 1998
[36] M. Cross and P. Hohenberg, Spatiotemporal Chaos, Science, 263, 1569, 1994
[37] H. Zhang, B. Hu, G. Hu, Suppression of spiral waves and spatiotemporal chaos by generating target waves in excitable media, Phys. Rev. E, 68, 026134, 2003
[38] 马军、蒲忠胜、冯旺军、李维学，旋转中心力场消除螺旋波和时空混沌，物理学报，54，4602，2005
[39] P. Y. Wang and P. Xie, Eliminating spatial chaos and spiral waves by weak spatial perturbations, Phys. Rev. E, 61, 5120, 2000
[40] P. Pamananda and J. L, Hudson, Controlling spatiotemporal chemical chaos using delayed feedback, Phys. Rev. E, 64, 037201, 2001
[41] H. Zhang, Z. J. Cao, N. J. Wu, H. P. Ying, G. Hu, Suppress Winfree Turbulence by Local Forcing Excitable Systems, Phys. Rev. Lett., 94, 188301, 2005
[42] 朱振明，化学体系螺旋波的噪声动力学。合肥：中国科学技术大学博士论文，2001
[43] B. Lindner, J. Garcia-Ojalva, A. Neiman, et al, Effects of noise in excitable systems, Phys. Rep, 392, 321, 2004
[44] R. FitzHugh, Impulses and Physiological States in Theoretical Models of Nerve Membrane, Biophys. J, 1,445, 1961
[45] J. S. Nagumo, S. Arimoto, S. Yoshizawa, An Active Pulse Transmission Line Simulating Nerve Axon, Proc. IRE 50, 2061, 1962
[46] D. Barkley, Mark Kness, Laurette S. Tuckerman, Spiral-wave dynamics in a simple model of excitable media: The transition from simple to compound rotation, Phys. Rev. A, 42, R2489, 1990
[47] M. Bar, M. Eiswirth, Turbulence due to spiral breakup in a continuous excitable medium, Phys. Rev. E, 48, R1635, 1993
[48] M. Bar, N. Gottschlk, M. Eiswith, Spiral waves in a surface reaction: Model calculations, J. Chem. Phys, 100(2), 1202, 1994
[49] D. Barkley, A model for fast computer-simulation of waves in excitable media., Physica D (Amsterdam), 49, 61, 1991
[50] V. Hakim, A. Karma, Phys. Rev. E, Theory of spiral wave dynamics in weakly excitable media: Asymptotic reduction to a kinematic model and applications, 60, 5073, 1999
[51] 袁国勇，双层可激系统的动力学行为及螺旋波的控制。北京：中国工程物理研究院博士论文，2005
[52] A. T. Winfree, Scroll-shaped wave of chemical activity in three dimensions, Science, 1973, 181:937
[53] D. Barkley, Phys. Rev. Lett, Euclidean symmetry and the dynamics of rotating spiral waves, 72, 164, 1994
[54] Q. Ouyang and J. M. Flesselles, Transition from spirals to defect turbulence driven by a convective instability, Nature, 379, 143, 1996
[55] Q. Ouyang, H. L. Swinney, and G. Li, Transition from Spirals to Defect-Mediated Turbulence Driven by a Doppler Instability, Phys. Rev. Lett., 84, 1047, 2000
[56] 陈式刚，映象与混沌，国防工业出版社，2000
[57] 王光瑞，混沌的控制、同步与利用，国防工业出版社，2001
[58] 胡岗，萧井华，郑志刚，混沌控制，上海科技教育出版社，2000
[59] H. Fujisaka, T. Yama, Stability Theory of Synchronized Motion in Coupled-Oscillator Systems, Prog. Theor. Phys., 69, 32, 1983
[60] V. S. Afraimovich, N. N. Verichev and M.. Rebinovich, et al, Izvestiya Vysshkh Uchebnykh Zavedenii Radiofizika., 59, 1050, 1986
[61] L. K. Pecora and T. L. Carroll, Synchronization in chaotic systems, Phys. Rev. Lett., 64, 821,1990
[62] L. M. Rosenblum, A. S. Pikovsky and J. Kurths, Phase Synchronization of Chaotic Oscillators, Phys. Rev. Lett., 76,1804,1996
[63] E. R. Rosa, E. Ott and M. H. Hess, Transition to Phase Synchronization of Chaos, Phys. Rev. Lett., 80,1642,1997
[64] L. M. Rosenblum, A. S. Pikovsky and J. Kurths, From Phase to Lag Synchronization in Coupled Chaotic Oscillators, Phys. Rev. Lett., 78,4193,1997
[65] N. F. Rullkov, M. M. Sushchik and L. S. Tsimring, et al, Generalized synchronization of chaos in directionally coupled chaotic systems, Phys. Rev. E., 51, 980,1995
[66] L. Kocarev and U. Parlitz, Generalized Synchronization, Predictability, and Equivalence of Unidirectionally Coupled Dynamical Systems, Phys. Rev. Lett., 76,1816,1996
[67] S. Boccaletti and D. L. Valldares, Characterization of intermittent lag synchronization, Phys. Rev. E, 62, 7497,2000
[68] M. A. Zaks, E. H. Park, M. G. Rosenblum and J. Kurths, Alternating Locking Ratios in Imperfect Phase Synchronization, Phys. Rev. Lett., 82,4228,1999
[69] R. Femat and G. Solis-Perales, Laplace domain controllers for chaos control, Phys. Rev. A, 262, 50,1999
[70] W. Lu, D. Yu and R. G. Harrison, Control of Patterns in Spatiotemporal Chaos in Optics, Phys. Rev. Lett., 76, 3316,1996
[71] K. Pyragas, Continuous control of chaos by self-controlling feedback, Phys. Lett. A, 170, 421, 1992
[72] R. Martin, A. J. Scroggie and G. L. Oppo, Stabilization, Selection, and Tracking of Unstable Patterns by Fourier Space Techniques, Phys. Rev. Lett., 77, 4007, 1996
[73] D. Hochheiser, J. V. Moloney and J. Lega, Controlling optical turbulence, Phys. Rev. A, 55.R4011,1997
[74] A. Pikovsky, M. Rosenblum and J. Kurths, Synchronization in a population of globally coupled chaotic oscillators, Europhys. Lett., 34, 165, 1996
[75] D. H. Zanette, Dynamics of globally coupled bistable elements, Phys. Rev. E, 55, 5315, 1997
[76] D. H. Zanette and A. S. Mikhailov, Mutual synchronization in ensembles of globally coupled neural networks, Phys. Rev. E, 58, 872, 1998
[77] E S. de San Reman, S. Boccaletti, D. Maza, and H. Mancini, Weak Synchronization of Chaotic Coupled Map Lattices, Phys. Rev. Lett., 81, 3639, 1998
[78] D. Maza, S. Boccaletti and H. Mancini, Phase clustering and collective behaviors in globally coupled map lattices due to mean field effects, Int. J. Bifurc. Chaos., 4, 829, 2000
[79] W. J. Rappel, F. Fenton and A. Karma, Spatiotemporal Control of Wave Instabilities in Cardiac Tissue, Rhys. Rev. Lett., 83, 456, 1999
[80] H. Sakaguchi and T. Fujimoto, Elimination of spiral chaos by periodic force for the Aliev-Panfilov model, Phys. Rev. E, 67, 067202, 2003
[81] R Y. Wang, R W Xie and H. W. Yin, Control of spiral waves and turbulent states in a cardiac model by travelling-wave perturbations, Chin. Phys., 12, 674, 2003
[82] M. Woltering and M. Markus, Control of Spatiotemporal Disorder in an Excitable Medium, ScienceAsia, 28, 43, 2002
[83] 袁国勇、杨世平、王光瑞、陈式刚，物理学报，两个延迟耦合FitzHugh-Nagumo系统的动力学行为，54，1510，2005
[84] M. Valderrabano, M. H. Lee, T. Ohara, A. C. Lai, M. C. Fishbein, S. F. Lin, H. S. Karagueuzian, and E S.Chen, Dynamics of Intramural and Transmural Reentry During Ventricular Fibrillation in Isolated Swine Ventricles, Circulation Research, April 27; 83, 2001
[85] J. Kneller, R. Zou, E. J. Vigmond, Zhiguo Wang, L. J. Leon, and S. Nattel, Circulation Research, May 90, 73, 2002
[86] R. F. Gilmour and D. R. Chialvo, Electrical restitution, critical mass, and the fiddle of fibrillation, J. Cardiovasc Electrophysiol, 10, 1087, 1999
[87] M. Hildebrand, J. X. Cui, E. Mihaliuk, J.Wang, and K. Showalter, Synchronization of spatiotemporal patterns in locally coupled excitable media, Phys. Rev. E, 68, 026205, 2003
[88] L. F. Yang, I. R. Epstein, Symmetric, asymmetric, and antiphase Turing patterns in a model system with two identical coupled layers, Phys. Rev. E, 69, 026211, 2004
[89] J. Z. Yang, E G. Xie, Z. L. Qu and A. Garfinkel, Mechanism for Spiral Wave Breakup in Excitable and Oscillatory Media, Phys. Rev. Lett., 91, 148302, 2003
[90] G. Y. Yuan, G. R. Wang and S. G. Chen, Control of spiral waves and spatiotemporal chaos by periodic perturbation near the boundary, Europhys. Lett. 72, 908, 2005