全局耦合不连续映象系统的集体动力学研究
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摘要
本文通过数值模拟方法研究了分段线性不连续不可逆全局耦合映象系统的集体动力学。选取单映象处于周期吸引子与混沌吸引子共存区域、周期吸引子共存区域、“映孔危机”区域、混沌吸引子区域等几个具有代表性的参数域展开研究,计算了同步序参量、最大李雅普诺夫指数,空间振幅变化图、时空行为发展图等。本文的主要结论如下：(1)不连续不可逆全局耦合映象系统具有丰富的动力学行为；(2)由于全局耦合映象系统的格点间具有长程的相互作用,不论单映象处于何种动力学状态,当耦合强度超过某个阈值后,系统都能达到同步状态,出现冻结化随机图案。同步时系统的状态由单映象所在区域的行为决定。当单映象处于单个吸引子状态时,系统同步后的状态与单映象的状态一致；在多吸引子共存态,系统在合适的耦合强度和初始条件下将同步到其中的一个吸引子态。(3)单映象处于混沌吸引子状态时,格点间的耦合仍能将系统调制到规则的运动,这种特征对于混沌控制具有一定的利用价值。
The collective dynamics in the globally coupled both-discontinuous-and- inconvertible-maps is studied through the numerical simulation methods in this thesis. Under the typical control parameters where the dynamics of the single map is in the states of coexistence of the periodic and the chaotic attractors, coexistence of periodic attractors, hole-induced crisis, and chaotic attractor, respectively, the order parameters of synchronization, the largest Lyapunov exponents, space-amplitude plots, and the space-time diagrams are calculated. The main conclusions are:(1) globally coupled map lattices consisting of both discontinuous and inconvertible maps exhibits rich dynamic phenomena; (2) Due to the long range interaction, the system can always reach the synchronized states, showing frozen random patterns, when the coupling strength is beyond some critical values regardless of the dynamical states of the single map. The synchronized state depends on the dynamic behavior of the single map. The system may synchronize to the state that is the same to the dynamics of the single map when it is in single attractor state. While in the coexistence states of the single map, the synchronized state may be one of the attractor at some proper coupling strengths and initial condition. (3) The system can be modulated to some regular state through the coupling among lattices even when the single map is in its chaotic state, which may have some important applications in controlling chaos.
The collective dynamics in the globally coupled both-discontinuous-and- inconvertible-maps is studied through the numerical simulation methods in this thesis. Under the typical control parameters where the dynamics of the single map is in the states of coexistence of the periodic and the chaotic attractors, coexistence of periodic attractors, hole-induced crisis, and chaotic attractor, respectively, the order parameters of synchronization, the largest Lyapunov exponents, space-amplitude plots, and the space-time diagrams are calculated. The main conclusions are:(1) globally coupled map lattices consisting of both discontinuous and inconvertible maps exhibits rich dynamic phenomena; (2) Due to the long range interaction, the system can always reach the synchronized states, showing frozen random patterns, when the coupling strength is beyond some critical values regardless of the dynamical states of the single map. The synchronized state depends on the dynamic behavior of the single map. The system may synchronize to the state that is the same to the dynamics of the single map when it is in single attractor state. While in the coexistence states of the single map, the synchronized state may be one of the attractor at some proper coupling strengths and initial condition. (3) The system can be modulated to some regular state through the coupling among lattices even when the single map is in its chaotic state, which may have some important applications in controlling chaos.
引文
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[51]A. R. Sonawane. Directed Percolation Criticality due to Stochastic Switching Between Attractive and Repulsive Coupling in Coupled Circle Maps[J]. Physical Review E,2010,81:05620601-05620609.
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[58]F. Bellido, J. B. R. Malo. Periodic and Chaotic Dynamics of a Sliding Driven Oscillator with Dry Friction[J]. Nonlinear Mechanics,2006,41:860-871.
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[63]S. X. Qu, Y. Z. Lu, L. Zhang, D. R. He. Discontinuous Bifurcation and Coexistence of Attractors in a Piecewise Linear Map with a Gap[J]. Chinese Physics B,17(12): 4418-4423.
[64]L. M. Pecora, T. L. Carroll. Synchronization in Chaotic Systems[J]. Physical Review Letters,1990,64:821-824.
[65]M. G. Rosenblum, A. S. Pilovsky, J. Kurths. From Phase to Lag Synchronization in Coupled Chaotic Systems[J]. Physical Review Letters,1990,78:4193-4196.
[66]L. Kocarev, U. Parlitz. Generalized Synchronization, Predictability, and Equivalence of Unidirectionally Coupled Dynamic Systems [J]. Physical Review Letters,1996,76:1816-1819.
[67]K. Pyragas. Weak and Strong Synchronization of Chaos[J]. Physical Review E, 1996,54:R4508-R4511.
[68]C. S. Zhou, J. Kurth, I. Z. Kiss, J. L. Hudson. Noise-Enhanced Phase Synchronization in Coupled Chaotic Oscillator [J]. Physical Review Letters,2002, 89:014101-014109.
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[2]D. Ruelle, F. Tankens. On the Nature of Turbulence[J]. Communation Mathematical Physics,1971,20:167-192.
[3]R. M. May. Simple Mathematical Models with Very Complicated Dynamics[J]. Nature,1976,261:459-467.
[4]M. J. Feigenbaurm. Quantitative Universality for a Class of Nonlinear Transforma-Tions[J]. Journal of the Statistical Physics,1978,19(1):25-52.
[5]刘宗华.混沌动力学基础及其应用[M].北京：高等教育出版社,2006：4-5.
[6]L. M. Pecora, T. L. Carroll. Synchronization of Chaotic Systems[J]. Physical Review Letters,1990,6(4):821-832.
[7]E. Ott, C. Grebogi, J. Yorke. Controlling Chaos[J]. Physical Review Letters,1990, 64(8):1196-1199.
[8]G.Hu, Z. L. Qu. Controlling Spatiotemporal Chaos in Coupled Map Lattice Systems[J]. Physical Review Letters,1994,72:68-71.
[9]Y. Li, X. Zhang. Controlling Localized Spatiotemporal Chaos Using Feedback Control Method[J]. Physical Letters A,2006,357:209-212.
[10]刘秉正,彭建华.非线性动力学[M].北京：高等教育出版社,2004：403-414.
[11]吴振兴,陈忠.混沌学导论[M].上海：上海科学技术文献出版社,1996：60-63.
[12]K. Kanoke. Overview of Coupled Map Lattices[J]. Chaos,1992,2(3):279-282.
[13]R. Kaparal. Pattern Formation in Two Dimetional Arrays of Coupled Discrete-time Oscillators[J]. Physical Review A,1985,31(6):3868-3879.
[14]K. Kanoke. Chaotic but Regular Posi-Nega Switching Among Coded Attractors by Cluster-Size Variation[J]. Physical Review Letters,1989,63(3):219-223.
[15]K. Kanoke. Period-Doubling of Kink-Antikink Patterns, Quasiperiodicity in Antifero-Like Structures and Spatial Intermittency in Coupled Logistic Lattices-Towards a Prelude of a "Field theory Of Chaos"[J]. Progress theoretical Physics,1984,72(3):480-486.
[16]I. Waller, R. Kaparal. Spatial and Temporal Structure in System of Coupled Nonlinear Oscillators[J]. Physical Review A,1984,30:2047-2055.
[17]A. Scala, L. A. N. Amaral, M. Barthelemy. Small Word Networks and the Conformation Space of a Short Lattice Polymer Chain[J]. Europhysics Letters, 2001,55(4):594-600.
[18]J. Byl. Self-Reproduction in Small Cellular Automata[J]. Physica D,34:295-299.
[19]R. Bidaux, N. Boccara, H.Chate. Order of the Transition versus Space Dimesion in a Family of Cellular Automata[J]. Physical Review A,1989,39:3094-3105.
[20]K. Kanoke. Spatiotemporal Intermittency in Coupled Map Lattices[J]. Progress Theoretical Physics,1985,74(5):1033-1044.
[21]D. L. Youngs. Modeling Turbulent Mixing by Rayleigh-Taylor Instability [J]. Physica D,1989,37:270-287.
[22]C. Tang, K. Wensenfeld, P. Bak, S. Coppersmith, P. Littilewood. Phase Organization[J]. Physical Review Letters,1987,58:1161-1164.
[23]J. F. Heagy, T. L. Carroll, L. M. Pecora. Synchronous Chaos in Coupled Oscillator Systems[J]. Physical Review E,1994,50:1874-1885.
[24]G. L. Oppo, K. Kapral. Discrete Models for the Formation and Evolution of Spatial Structure in Dissipative Systems [J]. Physical Review A,1986,33:4219-4231.
[25]杨维明.时空混沌和耦合映像格子[M].上海：上海教育科技出版社,1994：17-18.
[26]K. Konishi, H. Kokame, K. Hirata. Spatiotemporal Stability in One-Way Open Coupled Nonlinear Systems[J]. Physical Review E,2000,62:6383-6387.
[27]K. Kaneko. Spatial Period-doubling in Opened Flow[J]. Physics Letters A,1985, 111(7):321-325.
[28]F. H. Willeboordse, K. Kaneko. Bifurcations and Spatial Chaos in an Open Flow Model[J]. Physical Review Letters,1994,73:533-536.
[29]F. H. Willeboordse, K. Kaneko. Pattern Dynamics of a Coupled map Lattices for Open Flow[J]. Physica D,1995,86(3):428-425.
[30]K. Konishi, M. Hirai, K. Kaneko. Decentralized Delayed-feedback Control of a Coupled Map Model for Open Flow[J]. Physical Review E,58:3055-3059.
[31]J. M. Houlrik, I. Webman, M. H. Jensen. Mean-Field Theory and Critical Behavior of Coupled Map Lattice[J]. Physical Review A,1990,41:4210-4222.
[32]A.S. Pikovsky, J. Kurths. Do Globally Coupled Maps Really Violate the Law of Large Numbers?[J]. Physical Review Letters,1994,72:1644-1646.
[33]K. Kaneko. Clustering, Coding, Switching, Hierarchaical Ordering, and Control in a Network of Chaotic Elements[J]. Physica D,1990,41(2):137-172.
[34]T. Shibata, K. kaneko. Collective Chaos[J]. Physical Review Letters,1998,81: 4116-4119.
[35]K. Kaneko. Globally Coupled Chaos Violates the Law of Large Numbers but not the Central-limit theorem[J]. Physical Review Letters,65:1391-1394.
[36]K. Wiesenfeld, P. Hadley. Attactor Crowding in Osillator Arrays[J]. Physical Review Letters,1989,62:1335-1338.
[37]S. Watanabe, S. H. Strogatz. Integrability of a Gloablly Coupled Oscillator Array[J]. Physical Review Letters,70:2391-2394.
[38]S. H. Strogatz, R. E. Mirollo. Splay States in Globally Coupled Josephson Arrays: Analytical Prediction of Floquet Multipliers [J]. Physical Review E,1993,47: 220-227.
[39]H. G. Winful, L. Rahman. Synchronized Chaos and Spatiotemporal Chaos in Arrays of Coupled Lasers[J]. Physical Review Letters,65:1575-1578.
[40]K. Wiesenfeld, J. W. Swift. Averaged Equation for Josephson Junction Series Arrays[J]. Physical Review E,51:1020-1025.
[41]D. Dominguez,H. A. Cerdeira. Order and Turbulence in rf-driven Josephson Junction Series Arrays [J]. Physical Review Letters,71:3359-3362.
[42]D. Dominguez, H. A. Cerdeira. Spatiotemporal Chaos in rf-driven Josephson Junction Series Arrays [J]. Physical Review B,1995,52:513-526.
[43]J. Levy, M. S. Sherwin. Unified Model of Switching and Nonswitching Charge-Density-Wave Dynamics[J]. Physical Review Letters,1992,68(19): 2968-2971.
[44]S. H. Strogatz, C. M. Marcus, R. M. Westervelt, R. E. Mirollo. Collective Dynamics of Coupled Oscillators with Random Pinning[J]. Physica D,1989, 36(1-2):23-50.
[45]K. Wiesenfeld, C. Bracikowski, G. James, R. Roy. Observation of Antiphase State in a Multimode Laser [J], Physical Review Letters,1990,65:1749-1752.
[46]H. Sompolinsky, D. Golomb, D. Kleinfeld. Cooperative Dynamics in Visual Processing[J]. Physical Review A,1991,43:6990-7011.
[47]T. Wang, K. J. Wang, N. Jia. Chaos Control and Associative Memory of a Time-Delay Globally Coupled Neural Network Using Symmetric Map[J]. Neural Computing,2011,74:1673-1680.
[48]K. Kaneko. Relevance of Dynamic Clustering to Biological Networks [J]. Physica D, 1994,75(1-3):55-73.
[49]K. Kaneko. Information Cascade with Marginal Stability in a Network of Chaotic elements[J]. Physica D,1994,77(4):456-472.
[50]A. Sharma, N. Gupte. Spatiotemporal Intermittency AndScaling Laws in Inhomogeneous Coupled Map Lattices [J]. Physical Review E,2002,66: 0362101-0362110.
[51]A. R. Sonawane. Directed Percolation Criticality due to Stochastic Switching Between Attractive and Repulsive Coupling in Coupled Circle Maps[J]. Physical Review E,2010,81:05620601-05620609.
[52]K. Kaneko. Globally Coupled Circle Maps[J]. Physica D,1991,54:5-19.
[53]G. V. Osipov. Regular and Chaotic Phase Synchronization of Coupled Circle Maps[J]. Physical Review E,2001,65:01621601-01621612.
[54]W. Just. On the Collective Motion in Globally Coupled Chaotic Systems [J]. Physics Reports,1997,290:101-110.
[55]C. J. Budd, P. T. Piiroinenb. Corner Bifurcation in Non-Smoothly Forced Impact Oscillators[J]. Physica D,2006,220:127-145.
[56]J. Wang, X. L. Ding, B. Hu, B. H. Wang, J. S. Mao, D. R. He. Characteristic of a Piecewise Smooth Area-Preserving Map[J]. Physical Review E,2001,64: 02620201-02620209.
[57]K. Grudzinski, J. J. Zebrowski. Modeling Cardiac Pacemakers with Relaxation Osillators[J]. Physica A,2004,336:153-162.
[58]F. Bellido, J. B. R. Malo. Periodic and Chaotic Dynamics of a Sliding Driven Oscillator with Dry Friction[J]. Nonlinear Mechanics,2006,41:860-871.
[59]F. Ji, D. R. He. Experimental Evidence and Further Numerical Study of Type V intermittency [J]. Physical Letters A,1993,177(2):125-129.
[60]Z. T. Zhusubaliyev, E. Mosekilde. Bifurcation and Chaos in Piecewise-Smooth Dynamical Systems[M]. Singapore:World Scientific,2003:113-115.
[61]H. P. Ren, D. Liu. Bifurcation Behaviors of Peak Current Controlled PFC Boost Converter[J]. Chinese Physics,2005,14:1352-1358.
[62]S. X. Qu, B. Cristiansen, D. R. He. Hole-Induces Crisis in a Piece-Wise Linear Map[J]. Physics Letters A,1995,201:413-418.
[63]S. X. Qu, Y. Z. Lu, L. Zhang, D. R. He. Discontinuous Bifurcation and Coexistence of Attractors in a Piecewise Linear Map with a Gap[J]. Chinese Physics B,17(12): 4418-4423.
[64]L. M. Pecora, T. L. Carroll. Synchronization in Chaotic Systems[J]. Physical Review Letters,1990,64:821-824.
[65]M. G. Rosenblum, A. S. Pilovsky, J. Kurths. From Phase to Lag Synchronization in Coupled Chaotic Systems[J]. Physical Review Letters,1990,78:4193-4196.
[66]L. Kocarev, U. Parlitz. Generalized Synchronization, Predictability, and Equivalence of Unidirectionally Coupled Dynamic Systems [J]. Physical Review Letters,1996,76:1816-1819.
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