链式与树状网络的时空混沌同步研究
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摘要
复杂网络的同步研究是目前复杂网络研究最活跃的领域之一。由于复杂网络的同步能够解释自然界的许多复杂现象,并紧密联系现实社会,所以,对复杂网络的同步研究具有一定的实际意义和广泛的应用前景。本文对复杂网络的研究进展做了概括性的介绍,包括复杂网络模型的建立、网络的性质、网络同步的意义以及国内外研究现状。同时,还介绍了复杂网络同步的两个重要判定方法：主稳定函数判据和Lyapunov函数判定法以及目前已报道的几个典型的复杂网络同步实例。在此基础上,采用单变量耦合连接方式,通过backstepping方法构造Lyapunov函数,研究了同结构链式网络的时空混沌同步问题。以Gray-Scott系统为例,仿真模拟验证了该网络同步原理的可行性。此方法只需在复杂网络节点终端加入一个控制器,便可以实现整个网络的同步,因此同步代价小,便以实际应用。另外,本文还依据主稳定函数判据,研究了异结构树状网络的时空混沌同步问题。通过确定网络的最大Lyapunov指数,得到了实现网络同步的条件。采用具有时空混沌行为的Panfilov系统、Fitzhugh—Nagumo系统以及Plankton系统作为网络节点,仿真模拟验证了该方法的有效性。
Synchronization study of complex network is the one of the most active areas of complex network study at present. The synchronization of complex network can explain many complex phenomena in nature, and close contact with the real world. Therefore, synchronization study of complex network has some practical significance and wide applications. The research progress of complex network is recapitulative introduced in this thesis, including established complex network models, the features of network, the significance of network synchronization and research status at home and abroad. At the same time, two criterions of the synchronization of complex network:Master stability function criterion and Lyapunov function criterion are described, as well as several typical reported examples of the complex network synchronization are introduced. On this basis, Lyapunov function is constructed by backstepping design for single variable coupling connections, and the spatiotemporal chaos synchronization in chain network with topology equipollence is investigated. Taking the Gray-Scott spatiotemporal chaos system as an example, the simulation results show that the synchronization principle is feasibility. In this method, only one controller is added at the terminal node of complex network and synchronization of the entire network can be achieved, meaning the low-cost synchronization and convenient practical applications. In addition, the spatiotemporal chaos synchronization in tree network with different structure is studied in our work based on the Master stability function criterion. The condition for complex network synchronization is obtained by determining the largest Lyapunov index of the network. The Panfilov, Fitzhugh-Nagumo and Plankton spatiotemporal chaos systems are taken as network nodes, respectively, numerical simulation results show the effectiveness of the method.
Synchronization study of complex network is the one of the most active areas of complex network study at present. The synchronization of complex network can explain many complex phenomena in nature, and close contact with the real world. Therefore, synchronization study of complex network has some practical significance and wide applications. The research progress of complex network is recapitulative introduced in this thesis, including established complex network models, the features of network, the significance of network synchronization and research status at home and abroad. At the same time, two criterions of the synchronization of complex network:Master stability function criterion and Lyapunov function criterion are described, as well as several typical reported examples of the complex network synchronization are introduced. On this basis, Lyapunov function is constructed by backstepping design for single variable coupling connections, and the spatiotemporal chaos synchronization in chain network with topology equipollence is investigated. Taking the Gray-Scott spatiotemporal chaos system as an example, the simulation results show that the synchronization principle is feasibility. In this method, only one controller is added at the terminal node of complex network and synchronization of the entire network can be achieved, meaning the low-cost synchronization and convenient practical applications. In addition, the spatiotemporal chaos synchronization in tree network with different structure is studied in our work based on the Master stability function criterion. The condition for complex network synchronization is obtained by determining the largest Lyapunov index of the network. The Panfilov, Fitzhugh-Nagumo and Plankton spatiotemporal chaos systems are taken as network nodes, respectively, numerical simulation results show the effectiveness of the method.
引文
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[2]Watts D J, Strongatz S H. Collective dynamics of small-world networks. Nature,1998, 393:440-442
[3]Cohen R, Havlin S. Scale-free networks are ultrasmall. Phys. Rev. Lett,2003,86: 3682-3685
[4]Fronczak A, Fronczak P, Holyst J A.. Phys. Rev. E,2003,68:046126-046129
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[6]Wang Q Y, Lu Q S, Chen G, et al. Chaos synchronization of coupled neurons with gap junction. Phys. Lett. A,2006,356:17-25
[7]Wang Q Y, Lu Q S. Phase synchronization in small world chaotic neural networks. Chin. Phys. Lett.,2005,21:1329-1332
[8]Wang Q Y, Duan Z S, Feng Z S, et al. Synchronization transition in gap-junction-coupled leech neurons. Physica A,2008,387:4404-4410
[9]Gong H, Liu H, Wang B. An asymmetric full velocity difference car-following model. Physica A,2008,387:2595-2602
[10]Cai S M, Yan G, Zhou T, et al. Scaling behavior of an artificial traffic model on scale-free networks. Phys. Lett. A,2007,366:14-19
[11]Li P, Wang B H. Extracting hidden fluctuation patterns of Hang Seng stock index from network topologies. Physica A,2007,378:519-526
[12]Palla G, Barabasi A L, Vicsek T. Quantifying social group evolution. Nature,2007,446: 664-667
[13]Dorogovtsev S N, Mendes J F F. Evolution of networks. Adv. Phys.,2002,51:1079-1187
[14]Strogatz S H. Exploring complex networks. Nature,2001,410:268-276
[15]Wang X F. Complex networks:topology, dynamics, and synchronization. J Bif. Chaos,2002, 5:885～916
[16]Albert R, Barabasi A L. Topology of evolving networks:Local events and universality. Phys. Rev. Lett,2000,85:5234～5237
[17]Watts D J. The'new'science of networks. Annual Review of Sociology,2004,30:243-270
[18]赵明、汪秉宏、蒋品群、周涛.复杂网络动力学系统同步的研究进展.物理学进展,2005,25：273-295
[19]Barabasi A L, Bonabeau E. Scale-free networks. Scientific American,2003,288:50-59
[20]Albert R, Barabasi A L. Statistical mechanics of complex network. Review of Modern Physics,2002,74:47-97
[21]Newman M E J. The structure and function of complex networks. SIAM. Rev,2003,45: 167-256
[22]Erdos P, Renyi A. On the evolution of random graphs. Publ. Math. Inst. Hung. Acad. Sci., 1960,5:17-60
[23]Solomonoff R, Rapoport A. Connectivity of random nets. Bulletin of Math. Biophys.,1951, 13:107-117
[24]Erdos P, Renyi A. On the strength of connectedness of a random graph. Acta. Math. Acad. Sci. Hung,1961,12:261-267
[25]Bollobas B. Random Graphs. Academic Press,2001
[26]Newman M E J. Models of the small world. J. Stat. Phys.,2000,101:819-841
[27]Barrat A, Weigt M. On the properties of small world networks. J. Eur. Phys. B,2000, 13:547-560
[28]Watts D J, Strongatz S H. Collective dynamics of small-world networks. Nature,1998, 393:440-442
[29]Barrat A, Weigt M. On the properties of small world networks. J. Eur. Phys. B,2000, 13:547-560
[30]Newman M E J, Moore C, Watts D J. Mean field solution of the small-world network model. Phys. Rev. Lett,2000,84:3201-3204
[31]Newman M E J, Watts D J. Renormalization group analysis of the small-world network model. Phys. Lett. A,1999,263:341-346
[32]Barabasi A L, Albert R. Emergence of scaling in random networks. science,1999,286: 509-512
[33]Pastor-Satorras R, Vespignani A. Epidemic dynamics infinite size scale-free networks. Phys. Rev. E,2002,65:035108-035111
[34]Barabasi A L, Albert R, Jeong H. Mean-field theory for scale-free random networks. Physica A,1999,272:173-187
[35]Cohen R, Havlin S. Scale-free networks are ultrasmall. Phys. Rev. Lett,2003,86: 3682-3685
[36]Huberman B A, Adamic L A. Growth dynamics of the World-Wide Web.1999, Nature,401: 130-131
[37]Maslov S, Sneppen K. Specificity and stability in topology of protein networks. Science, 2002,296:910-913
[38]Wang X F, Chen G. Synchronization in scale-free dynamical networks:robustness and fragility. IEEE,2002,49:54-62
[39]Atay F M, Jost J, Wende A. Delays, Connection Topology, and Synchronization of Coupled Chaotic Maps. Phys. Rev. Lett,2004,92:144101-144104
[40]Ichinomiya T. Frequency Synchronization in random oscillator network. Phys. Rev. E, 2004,70:026116-026120
[41]Belykh I V, Lange E N, Hasler M. Synchronization of Bursting Neurons:what matters in the network topology. Phys. Rev. Lett,2005,94:188101-188104
[42]Wiener N. Nonlinear Problems in Random Theory. Cambridge. MA. MIT Prsss,1998
[43]Winfree A T. Biological rhythms and the behavior of population ofcoupled oscillators. Journal of Theoretical Biology,1967
[44]Kuramoto Y.Chemical Oscillator. Waves and Turbulence. Springer-Verlag,1984
[45]Heagy J F, Carroll T L, Pecora L M. Synchronous chaos in coupled oscillator systems. Phys. Rev. E,1994,50:1874-1885
[46]Heagy J F, Pecora L M, Carroll T L. Short wavelength bifurcations and size instabilities in coupled oscillator systems. Phys. Rev. Lett,1995,74:4185-4187
[47]Pecora L M. Synchronization conditions and desynchronizing patterns in coupled limit-cycle and chaotic systems. Phys. Rev. E,1998,58:347-360
[48]Fink K S, Carroll T L. Three coupled oscillators as a universal probe of synchronization stability in coupled oscillator arrays. Phys. Rev. E,2000,61:5080-5090
[49]吕翎非线性动力学与混沌(大连：大连出版社)2000
[50]Pecora L M, Carroll T L. Synchronization in chaotic systems. Phys Rev Lett,1990,64: 821-824
[51]Kocarev L, Parlitz U. General approach for chaotic synchronization with application to comunication.Phys. Rev. Lett,1995,74:5028-5031
[52]Winful H G, Rahman L. Synchronized chaos and spatiotemporal chaos in arrays of coupled lasers. Phys. Rev. Lett,1990,65:1575-1578
[53]Huberman B A, Lumer E. Dynamics of adaptive systems. IEEE. Trans. Circuits. Syst,1990, 37:547-550
[54]Xie W X, Wen C Y, Li Z G. Impulsive control for the stabilization and synchronization of Lorenz systems. Phys. Lett. A,2000,275:67-72
[55]Wu X Q,Lu J H.Parameter identification and backstepping control of uncertain Lu system, Chaos. Solitons and Fractals,2003,15:897-901
[56]Wang C, Ge S S. Adaptive synchronization of uncertain chaotic systems via backstepping design. Chao,2001,12:1199-1206
[57]Li G H. Projective synchronization of chaotic system using backstepping control. Chao, 2006,29:490-494
[58]LU J H, Zhang S C. Controlling Chen's chaotic attractor using backstepping design based on parameters identification. Physica A,1999,286:148-152
[59]Yu Y G, Zhang S C. Controlling uncertain Lu system using backstepping design. Chao, 2003,15:897-902
[60]Huang L L, Wang M. Synchronization of generalized Henon map via backstepping design. Chao,2005,23:617-620
[61]Tan X H, Zhang J, Hasler M. Synchronizing chaotic systems using backstepping design. Chao,2003,16:37-45
[62]Timme M, Wolf F, Geisel T. Topological speed limits to network synchronization. Phys. Rev. Lett,2004,92:074101-074104
[63]Pecora L M, Carroll T L. Master stability functions for synchronization coupled systems. Phys. Rev. Lett,1998,80:2109-2113
[64]Hong H, Choi M Y, Kim B J. Synchronization on small-world networks. Phys. Rev. E,2002, 65:026139-0216143
[65]Park K, Lai Y C, Gupte S, Kim J W. Synchronization in complex networks with a modular structure. Chao,2006,16:015105-015115
[66]吕翎、李钢、柴元.单向耦合映象格子时空混沌同步.物理学报,2008,57：7517-7521
[67]张旭、沈柯.时空混沌的单向耦合同步.物理学报,2002,51：2702-2705
[68]吕翎、夏晓岚.非线性耦合时空混沌系统的反同步研究.物理学报,2009,58：814-818
[69]敬晓丹、吕翎.非线性耦合完全网络的时空混沌系统同步.物理学报,2009,58：7539-7543
[70]Pearson J E. Comple patters in a simple system. Science,1993,261:777-786
[71]敬晓丹、吕翎.相空间压缩实现时空混沌系统的广义同步.物理学报,2008,57：4766～4770
[72]Hildebrand M, Eiswirth M, Carroll M. Statistics of topological defects and spatiotemporal chaos in a Reaction-Diffusion system. Phys. Rev. Lett,1995,75: 1503-1506
[73]Malchow H, Radtke B. Spatio-temporal pattern formation in coupled models of plankton dynamics and fish school motion. Nonlinear. Analysis:Real world Applications,2000, 1:53-67
[2]Watts D J, Strongatz S H. Collective dynamics of small-world networks. Nature,1998, 393:440-442
[3]Cohen R, Havlin S. Scale-free networks are ultrasmall. Phys. Rev. Lett,2003,86: 3682-3685
[4]Fronczak A, Fronczak P, Holyst J A.. Phys. Rev. E,2003,68:046126-046129
[5]Shorten R, King C, Wirth F, et al. Modelling TCP congestion control dynamics in drop-tail environments. Automatica,2007,43:441-449
[6]Wang Q Y, Lu Q S, Chen G, et al. Chaos synchronization of coupled neurons with gap junction. Phys. Lett. A,2006,356:17-25
[7]Wang Q Y, Lu Q S. Phase synchronization in small world chaotic neural networks. Chin. Phys. Lett.,2005,21:1329-1332
[8]Wang Q Y, Duan Z S, Feng Z S, et al. Synchronization transition in gap-junction-coupled leech neurons. Physica A,2008,387:4404-4410
[9]Gong H, Liu H, Wang B. An asymmetric full velocity difference car-following model. Physica A,2008,387:2595-2602
[10]Cai S M, Yan G, Zhou T, et al. Scaling behavior of an artificial traffic model on scale-free networks. Phys. Lett. A,2007,366:14-19
[11]Li P, Wang B H. Extracting hidden fluctuation patterns of Hang Seng stock index from network topologies. Physica A,2007,378:519-526
[12]Palla G, Barabasi A L, Vicsek T. Quantifying social group evolution. Nature,2007,446: 664-667
[13]Dorogovtsev S N, Mendes J F F. Evolution of networks. Adv. Phys.,2002,51:1079-1187
[14]Strogatz S H. Exploring complex networks. Nature,2001,410:268-276
[15]Wang X F. Complex networks:topology, dynamics, and synchronization. J Bif. Chaos,2002, 5:885～916
[16]Albert R, Barabasi A L. Topology of evolving networks:Local events and universality. Phys. Rev. Lett,2000,85:5234～5237
[17]Watts D J. The'new'science of networks. Annual Review of Sociology,2004,30:243-270
[18]赵明、汪秉宏、蒋品群、周涛.复杂网络动力学系统同步的研究进展.物理学进展,2005,25：273-295
[19]Barabasi A L, Bonabeau E. Scale-free networks. Scientific American,2003,288:50-59
[20]Albert R, Barabasi A L. Statistical mechanics of complex network. Review of Modern Physics,2002,74:47-97
[21]Newman M E J. The structure and function of complex networks. SIAM. Rev,2003,45: 167-256
[22]Erdos P, Renyi A. On the evolution of random graphs. Publ. Math. Inst. Hung. Acad. Sci., 1960,5:17-60
[23]Solomonoff R, Rapoport A. Connectivity of random nets. Bulletin of Math. Biophys.,1951, 13:107-117
[24]Erdos P, Renyi A. On the strength of connectedness of a random graph. Acta. Math. Acad. Sci. Hung,1961,12:261-267
[25]Bollobas B. Random Graphs. Academic Press,2001
[26]Newman M E J. Models of the small world. J. Stat. Phys.,2000,101:819-841
[27]Barrat A, Weigt M. On the properties of small world networks. J. Eur. Phys. B,2000, 13:547-560
[28]Watts D J, Strongatz S H. Collective dynamics of small-world networks. Nature,1998, 393:440-442
[29]Barrat A, Weigt M. On the properties of small world networks. J. Eur. Phys. B,2000, 13:547-560
[30]Newman M E J, Moore C, Watts D J. Mean field solution of the small-world network model. Phys. Rev. Lett,2000,84:3201-3204
[31]Newman M E J, Watts D J. Renormalization group analysis of the small-world network model. Phys. Lett. A,1999,263:341-346
[32]Barabasi A L, Albert R. Emergence of scaling in random networks. science,1999,286: 509-512
[33]Pastor-Satorras R, Vespignani A. Epidemic dynamics infinite size scale-free networks. Phys. Rev. E,2002,65:035108-035111
[34]Barabasi A L, Albert R, Jeong H. Mean-field theory for scale-free random networks. Physica A,1999,272:173-187
[35]Cohen R, Havlin S. Scale-free networks are ultrasmall. Phys. Rev. Lett,2003,86: 3682-3685
[36]Huberman B A, Adamic L A. Growth dynamics of the World-Wide Web.1999, Nature,401: 130-131
[37]Maslov S, Sneppen K. Specificity and stability in topology of protein networks. Science, 2002,296:910-913
[38]Wang X F, Chen G. Synchronization in scale-free dynamical networks:robustness and fragility. IEEE,2002,49:54-62
[39]Atay F M, Jost J, Wende A. Delays, Connection Topology, and Synchronization of Coupled Chaotic Maps. Phys. Rev. Lett,2004,92:144101-144104
[40]Ichinomiya T. Frequency Synchronization in random oscillator network. Phys. Rev. E, 2004,70:026116-026120
[41]Belykh I V, Lange E N, Hasler M. Synchronization of Bursting Neurons:what matters in the network topology. Phys. Rev. Lett,2005,94:188101-188104
[42]Wiener N. Nonlinear Problems in Random Theory. Cambridge. MA. MIT Prsss,1998
[43]Winfree A T. Biological rhythms and the behavior of population ofcoupled oscillators. Journal of Theoretical Biology,1967
[44]Kuramoto Y.Chemical Oscillator. Waves and Turbulence. Springer-Verlag,1984
[45]Heagy J F, Carroll T L, Pecora L M. Synchronous chaos in coupled oscillator systems. Phys. Rev. E,1994,50:1874-1885
[46]Heagy J F, Pecora L M, Carroll T L. Short wavelength bifurcations and size instabilities in coupled oscillator systems. Phys. Rev. Lett,1995,74:4185-4187
[47]Pecora L M. Synchronization conditions and desynchronizing patterns in coupled limit-cycle and chaotic systems. Phys. Rev. E,1998,58:347-360
[48]Fink K S, Carroll T L. Three coupled oscillators as a universal probe of synchronization stability in coupled oscillator arrays. Phys. Rev. E,2000,61:5080-5090
[49]吕翎非线性动力学与混沌(大连：大连出版社)2000
[50]Pecora L M, Carroll T L. Synchronization in chaotic systems. Phys Rev Lett,1990,64: 821-824
[51]Kocarev L, Parlitz U. General approach for chaotic synchronization with application to comunication.Phys. Rev. Lett,1995,74:5028-5031
[52]Winful H G, Rahman L. Synchronized chaos and spatiotemporal chaos in arrays of coupled lasers. Phys. Rev. Lett,1990,65:1575-1578
[53]Huberman B A, Lumer E. Dynamics of adaptive systems. IEEE. Trans. Circuits. Syst,1990, 37:547-550
[54]Xie W X, Wen C Y, Li Z G. Impulsive control for the stabilization and synchronization of Lorenz systems. Phys. Lett. A,2000,275:67-72
[55]Wu X Q,Lu J H.Parameter identification and backstepping control of uncertain Lu system, Chaos. Solitons and Fractals,2003,15:897-901
[56]Wang C, Ge S S. Adaptive synchronization of uncertain chaotic systems via backstepping design. Chao,2001,12:1199-1206
[57]Li G H. Projective synchronization of chaotic system using backstepping control. Chao, 2006,29:490-494
[58]LU J H, Zhang S C. Controlling Chen's chaotic attractor using backstepping design based on parameters identification. Physica A,1999,286:148-152
[59]Yu Y G, Zhang S C. Controlling uncertain Lu system using backstepping design. Chao, 2003,15:897-902
[60]Huang L L, Wang M. Synchronization of generalized Henon map via backstepping design. Chao,2005,23:617-620
[61]Tan X H, Zhang J, Hasler M. Synchronizing chaotic systems using backstepping design. Chao,2003,16:37-45
[62]Timme M, Wolf F, Geisel T. Topological speed limits to network synchronization. Phys. Rev. Lett,2004,92:074101-074104
[63]Pecora L M, Carroll T L. Master stability functions for synchronization coupled systems. Phys. Rev. Lett,1998,80:2109-2113
[64]Hong H, Choi M Y, Kim B J. Synchronization on small-world networks. Phys. Rev. E,2002, 65:026139-0216143
[65]Park K, Lai Y C, Gupte S, Kim J W. Synchronization in complex networks with a modular structure. Chao,2006,16:015105-015115
[66]吕翎、李钢、柴元.单向耦合映象格子时空混沌同步.物理学报,2008,57：7517-7521
[67]张旭、沈柯.时空混沌的单向耦合同步.物理学报,2002,51：2702-2705
[68]吕翎、夏晓岚.非线性耦合时空混沌系统的反同步研究.物理学报,2009,58：814-818
[69]敬晓丹、吕翎.非线性耦合完全网络的时空混沌系统同步.物理学报,2009,58：7539-7543
[70]Pearson J E. Comple patters in a simple system. Science,1993,261:777-786
[71]敬晓丹、吕翎.相空间压缩实现时空混沌系统的广义同步.物理学报,2008,57：4766～4770
[72]Hildebrand M, Eiswirth M, Carroll M. Statistics of topological defects and spatiotemporal chaos in a Reaction-Diffusion system. Phys. Rev. Lett,1995,75: 1503-1506
[73]Malchow H, Radtke B. Spatio-temporal pattern formation in coupled models of plankton dynamics and fish school motion. Nonlinear. Analysis:Real world Applications,2000, 1:53-67