离散混沌网络系统中共同噪声诱导同步的条件
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摘要
噪声普遍存在于各种真实系统和人造系统中,它对系统的同步动力学具有多重影响.本文探索外部噪声对高维离散复杂网络系统同步行为的积极作用.首先构建外部共同噪声驱动下两个参数相同、未耦合的离散混沌网络模型,然后利用Birkhoff遍历定理与矩阵论等相关理论,严格证明了噪声诱导两个离散混沌网络系统取得同步的充分条件,进一步借助于具体的混沌网络模型,利用数值仿真验证了理论分析的有效性.数值模拟的结果表明当网络模型参数满足理论分析的充分条件时,共同噪声可以诱导两个参数相同、未耦合的离散混沌网络在随机意义下取得同步,而且同步效果不依赖于复杂网络拓扑结构的选取.
Noise is ubiquitous in the various real and artificial systems,which plays multiple effects on synchronization of nonlinear systems. This study proposes to explore the positive influence of external common noise on the synchronous behavior of high dimensional discrete complex network. A network model of two uncoupled discrete chaotic networks with identical parameters under the excitation of external common noise is firstly constructed. By using the Birkhoff traversal theorem and the matrix theory,some sufficient conditions for synchronization of two discrete chaotic network systems are then proved. Meanwhile,by employing a specific chaotic network model,numerical simulation is used to verify the effectiveness of the theoretical analysis. The numerical results illustrate that when the parameters of network model meet the sufficient conditions for common noise induced synchronization,the two discrete chaotic networks can achieve synchronization in random sense. Moreover,the synchronization is robust against the variation of complex network topology. This paper obtains the theoretical conditions for common noise induced synchronization in the discrete chaotic networks for the first time,which not only can enrich the research of noise induced synchronization in some extent,but also can help to understand the positive effect of noise on ordered dynamics in high-dimensional nonlinear chaotic systems.
Noise is ubiquitous in the various real and artificial systems,which plays multiple effects on synchronization of nonlinear systems. This study proposes to explore the positive influence of external common noise on the synchronous behavior of high dimensional discrete complex network. A network model of two uncoupled discrete chaotic networks with identical parameters under the excitation of external common noise is firstly constructed. By using the Birkhoff traversal theorem and the matrix theory,some sufficient conditions for synchronization of two discrete chaotic network systems are then proved. Meanwhile,by employing a specific chaotic network model,numerical simulation is used to verify the effectiveness of the theoretical analysis. The numerical results illustrate that when the parameters of network model meet the sufficient conditions for common noise induced synchronization,the two discrete chaotic networks can achieve synchronization in random sense. Moreover,the synchronization is robust against the variation of complex network topology. This paper obtains the theoretical conditions for common noise induced synchronization in the discrete chaotic networks for the first time,which not only can enrich the research of noise induced synchronization in some extent,but also can help to understand the positive effect of noise on ordered dynamics in high-dimensional nonlinear chaotic systems.
引文
1 Pikovsky A,Rosenblum M,Kurths J.Synchronization:Auniversal concept in nonlinear science.Physics Today,2003,56(1):45~45
2 Boccaletti S,Kurths J,Osipov G,et al.The synchronization of chaotic systems.Physics Reports,2002,366(1-2):1~101
3 Huygens C.Horoloqium oscillatorium.Paris:FMuguet,1673
4 PecoraL M,Carroll T L.Synchronization in chaotic systems.Physical Review Letters,1990,64(8):821~825
5 Albert R,Barabási A L.Statistical mechanics of complex networks.Reviews of Modern Physics,2002,74:47~97
6 Newman M E J.The structure and function of complex network.Siam Review,2003,45(2):167~256
7 Wang X F.Complex networks:topology,dynamics and synchronization.International Journal of Bifurcation&Chaos,2002,12(5):885~916
8 Dorogovtsev S N,Mendes J F F.Evolution of networks.Advances in Physics,2002,51(4):1079~1187
9 Barabási A L,Crandall R E.The new science of networks.Physics Today,2003,6(5):243~270
10 Wu X J,Lu H T.Projective lag synchronization of the general complex dynamical.Communications in Nonlinear Science&Numerical Simulation,2012,17(11):4417~4429
11 Liu H,Chen J,Lu J A,et al.Generalized synchronization in complex dynamical networks via adaptive couplings.Physica A,2010,389(8):1759~1770
12敬晓丹,吕翎.非线性耦合完全网络的时空混沌同步.物理学报,2009,58(11):7539~7543(Jing X D,Lv L.The synchronization of spatiotemporal chaos of all-to-all network using nonlinear coupling.Acta Physsica Sinica,2009,58(11):7539~7543(in Chinese))
13李岩,吕翎,栾玲.环形加权网络的时空混沌延迟同步.物理学报,2009,58(7):4463~4468(Li Y,Lv L,Luan L.Lag synchronization of spatiotemporal chaos in a weighted network with ring connection.Acta Physsica Sinica,2009,58(7):4463~4468(in Chinese))
14 Li C P,Sun W G,Kurths J.Synchronization between two coupled complex networks.Physical Review E,2007,76(4pt 2):046204
15 Dai H,Jia L X,Zhang Y B.Adaptive generalized matrix projective lag synchronization between two different complex networks with non-identical nodes and different dimensions.Chinese Physics B,2012,21(12):141~152
16 Chen J R,Jiao L C,Wu J S,et al.Adaptive synchronization between two different complex networks with time-varing delay coupling.Chinese Physics Letters,2009,26(6):060505
17张檬,吕翎,吕娜等.结构与参量不确定的网络与网络之间的混沌同步.物理学报,2012,61(22):220508(Zhang M,Lv L,Lv N,et al.Chaos synchronization between complex networks with uncertain structures and unknown parameters.Acta Physica Sinica,2012,61(22):220508(in Chinese))
18方同.工程随机振动.北京:国防工业出版社,1995(Fang T.Engineering random vibration.Beijing:National Defense Industry Press,1995(in Chinese))
19朱位秋.非线性随机动力学与控制.北京:科学出版社,2003(Zhu W Q.Nonlinear stochastic dynamics and control.Beijing:Science Press,2003(in Chinese))
20 Arnold L.Random Dynamical Systems.Berlin:Springer,1994
21 Stratonovich R.Topics in the theory of random noise.New York:Gordonand Breach Science Publishers,1963
22 Lin W,He Y B.Complete synchronization of the noiseperturbed Chua's circuits.Chaos,2005,15(2):23705
23 Xiao Y Z,Xu W,Li X C,et al.The effect of noise on the complete synchronization of two bidirectionally coupled.Chaos,2009,19(1):013131
24关新平,范正平,彭海朋等.对一类具有随机扰动的混沌系统同步的新方法.电子学报,2001,29(10):1427~1429(Guan X P,Fan Z P,Peng H P,et al.A new approach of synchronization to a class of chaotic system with random perturbation.Acta Electronica Sinica,2001,29(10):1427~1429(in Chinese))
25 Sun Z K,Yang X L.Generating and enhancing lag synchronization of chaotic systems by white noise.Chaos,2011,21(3):033114
26 Yang X L,Xu W.Study on phase synchronization of stochastic chaotic system.Chinese Physics B,2004,17(6):2004~2009
27 Guan S G,Lai Y C,Lai C H.Effect of noise on generalized chaotic synchronization.Physical Review E,2006,73(4):046210
28 Sun Y Z,Zhao D H.Effects of noise on the outer synchronization of two unidirectionally coupled complex dynamical networks.Chaos,2012,22(2):023131
29 Cao L,Ma Y.Linear generalized outer synchronization between two different complex dynamical networks with noise perturbation.International Journal of Nonlinear Science,2012,13(3):373~379
30 Sun Y Z,Li W,Ruan J.Generalized outer synchronization between complex dynamical networks with time delay and noise perturbation.Communications in Nonlinear Science,2013,18(4):989~998
31张丽,杨晓丽,孙中奎.噪声环境下时滞耦合网络的广义投影滞后同步.物理学报,2013,62(24):240502(Zhang L,Yang X L,Sun Z K.Generalized projective lag synchronization between delay-coupled networks under circumstance noise.Acta Physsica Sinica,2013,62(24):240502(in Chinese))
32 Yang X L,Wei T.Revealing network topology and dynamical parameters in delay-coupled complex network subjected to random noise.Nonlinear Dynamics,2015,82(1-2):319~332
33 Maritan A,Banavar J R.Chaos,noise,and synchronization.Physical Review Letters,1994,72(10):1451~1454
34 Shuai J W,Wong K W.Noise and synchronization in chaotic neural networks.Physical Review E,1998,57(6):7002~7007
35 Toral R,Mirasso C R,Hernandez-Garcia E,et al.Analytical and numerical studies of noise-induced synchronization of chaotic systems.Chaos,2001,11(3):665~673
36 Zhou C S,Kurths J.Noise-induced phase synchronization and synchronization transitions in chaotic oscillators.Physical Review Letters,2002,88(23):230602
37 Longa L,Curado E M F,Oliveira F A.Roundoff-induced coalescence of chaotic trajectories.Physical Review E,1996,54(3):2201~2204
38 Herzel H,Freund J.Chaos,noise,and synchronization reconsidered.Physical Review E,1995,52(2):3328~3241
39 Yoshimura K,Valiusaityte I,Davis P.Synchronization induced by common colored noise in limit cycle and chaotic systems.Physical Review E,2007,75(2):026208
40 Yang X L,Xu W,Sun Z K.Synchronization of a chaotic particle with potential.Physics Letters A,2006,353(2):179~184
41 Uchida A,McAllister R,Roy R.Consistency of nonlinear system response to complex drive signals.Physical Review Letters,2004,93(24):244102
42 Mainen Z F,Sejnowski T J.Reliability of spike timing in neocortical neurons.Science,1995,268:1503~1505
43 Royama T.Analytical population dynamics.London:Chapman&Hall,1992:695~702
44 Yoshimura K,Muramatsu J,Davis P.Conditions for common-noise-induced synchronization in time-delay systems.Physica D,2008,237(23):3146~3152
45刘茂省.复杂网络上的动力学模型分析及随机影响[博士学位论文].上海:复旦大学,2009(Liu M S.Dynamic model analysis and stochastic effects in complex networks[Ph.D Thesis].Shanghai:Fudan University,2009(in Chinese))
46 Atay F M,Jost J,Wende A.Delays,connection topology,and synchronization of coupled chaotic maps.Physical Review Letters,2004,92(14):144101
47 Newman M E J,Watts D J.Renormalization group analysis of the small-world network model.Physics Letters A,1999,263(4-6):341~346
2 Boccaletti S,Kurths J,Osipov G,et al.The synchronization of chaotic systems.Physics Reports,2002,366(1-2):1~101
3 Huygens C.Horoloqium oscillatorium.Paris:FMuguet,1673
4 PecoraL M,Carroll T L.Synchronization in chaotic systems.Physical Review Letters,1990,64(8):821~825
5 Albert R,Barabási A L.Statistical mechanics of complex networks.Reviews of Modern Physics,2002,74:47~97
6 Newman M E J.The structure and function of complex network.Siam Review,2003,45(2):167~256
7 Wang X F.Complex networks:topology,dynamics and synchronization.International Journal of Bifurcation&Chaos,2002,12(5):885~916
8 Dorogovtsev S N,Mendes J F F.Evolution of networks.Advances in Physics,2002,51(4):1079~1187
9 Barabási A L,Crandall R E.The new science of networks.Physics Today,2003,6(5):243~270
10 Wu X J,Lu H T.Projective lag synchronization of the general complex dynamical.Communications in Nonlinear Science&Numerical Simulation,2012,17(11):4417~4429
11 Liu H,Chen J,Lu J A,et al.Generalized synchronization in complex dynamical networks via adaptive couplings.Physica A,2010,389(8):1759~1770
12敬晓丹,吕翎.非线性耦合完全网络的时空混沌同步.物理学报,2009,58(11):7539~7543(Jing X D,Lv L.The synchronization of spatiotemporal chaos of all-to-all network using nonlinear coupling.Acta Physsica Sinica,2009,58(11):7539~7543(in Chinese))
13李岩,吕翎,栾玲.环形加权网络的时空混沌延迟同步.物理学报,2009,58(7):4463~4468(Li Y,Lv L,Luan L.Lag synchronization of spatiotemporal chaos in a weighted network with ring connection.Acta Physsica Sinica,2009,58(7):4463~4468(in Chinese))
14 Li C P,Sun W G,Kurths J.Synchronization between two coupled complex networks.Physical Review E,2007,76(4pt 2):046204
15 Dai H,Jia L X,Zhang Y B.Adaptive generalized matrix projective lag synchronization between two different complex networks with non-identical nodes and different dimensions.Chinese Physics B,2012,21(12):141~152
16 Chen J R,Jiao L C,Wu J S,et al.Adaptive synchronization between two different complex networks with time-varing delay coupling.Chinese Physics Letters,2009,26(6):060505
17张檬,吕翎,吕娜等.结构与参量不确定的网络与网络之间的混沌同步.物理学报,2012,61(22):220508(Zhang M,Lv L,Lv N,et al.Chaos synchronization between complex networks with uncertain structures and unknown parameters.Acta Physica Sinica,2012,61(22):220508(in Chinese))
18方同.工程随机振动.北京:国防工业出版社,1995(Fang T.Engineering random vibration.Beijing:National Defense Industry Press,1995(in Chinese))
19朱位秋.非线性随机动力学与控制.北京:科学出版社,2003(Zhu W Q.Nonlinear stochastic dynamics and control.Beijing:Science Press,2003(in Chinese))
20 Arnold L.Random Dynamical Systems.Berlin:Springer,1994
21 Stratonovich R.Topics in the theory of random noise.New York:Gordonand Breach Science Publishers,1963
22 Lin W,He Y B.Complete synchronization of the noiseperturbed Chua's circuits.Chaos,2005,15(2):23705
23 Xiao Y Z,Xu W,Li X C,et al.The effect of noise on the complete synchronization of two bidirectionally coupled.Chaos,2009,19(1):013131
24关新平,范正平,彭海朋等.对一类具有随机扰动的混沌系统同步的新方法.电子学报,2001,29(10):1427~1429(Guan X P,Fan Z P,Peng H P,et al.A new approach of synchronization to a class of chaotic system with random perturbation.Acta Electronica Sinica,2001,29(10):1427~1429(in Chinese))
25 Sun Z K,Yang X L.Generating and enhancing lag synchronization of chaotic systems by white noise.Chaos,2011,21(3):033114
26 Yang X L,Xu W.Study on phase synchronization of stochastic chaotic system.Chinese Physics B,2004,17(6):2004~2009
27 Guan S G,Lai Y C,Lai C H.Effect of noise on generalized chaotic synchronization.Physical Review E,2006,73(4):046210
28 Sun Y Z,Zhao D H.Effects of noise on the outer synchronization of two unidirectionally coupled complex dynamical networks.Chaos,2012,22(2):023131
29 Cao L,Ma Y.Linear generalized outer synchronization between two different complex dynamical networks with noise perturbation.International Journal of Nonlinear Science,2012,13(3):373~379
30 Sun Y Z,Li W,Ruan J.Generalized outer synchronization between complex dynamical networks with time delay and noise perturbation.Communications in Nonlinear Science,2013,18(4):989~998
31张丽,杨晓丽,孙中奎.噪声环境下时滞耦合网络的广义投影滞后同步.物理学报,2013,62(24):240502(Zhang L,Yang X L,Sun Z K.Generalized projective lag synchronization between delay-coupled networks under circumstance noise.Acta Physsica Sinica,2013,62(24):240502(in Chinese))
32 Yang X L,Wei T.Revealing network topology and dynamical parameters in delay-coupled complex network subjected to random noise.Nonlinear Dynamics,2015,82(1-2):319~332
33 Maritan A,Banavar J R.Chaos,noise,and synchronization.Physical Review Letters,1994,72(10):1451~1454
34 Shuai J W,Wong K W.Noise and synchronization in chaotic neural networks.Physical Review E,1998,57(6):7002~7007
35 Toral R,Mirasso C R,Hernandez-Garcia E,et al.Analytical and numerical studies of noise-induced synchronization of chaotic systems.Chaos,2001,11(3):665~673
36 Zhou C S,Kurths J.Noise-induced phase synchronization and synchronization transitions in chaotic oscillators.Physical Review Letters,2002,88(23):230602
37 Longa L,Curado E M F,Oliveira F A.Roundoff-induced coalescence of chaotic trajectories.Physical Review E,1996,54(3):2201~2204
38 Herzel H,Freund J.Chaos,noise,and synchronization reconsidered.Physical Review E,1995,52(2):3328~3241
39 Yoshimura K,Valiusaityte I,Davis P.Synchronization induced by common colored noise in limit cycle and chaotic systems.Physical Review E,2007,75(2):026208
40 Yang X L,Xu W,Sun Z K.Synchronization of a chaotic particle with potential.Physics Letters A,2006,353(2):179~184
41 Uchida A,McAllister R,Roy R.Consistency of nonlinear system response to complex drive signals.Physical Review Letters,2004,93(24):244102
42 Mainen Z F,Sejnowski T J.Reliability of spike timing in neocortical neurons.Science,1995,268:1503~1505
43 Royama T.Analytical population dynamics.London:Chapman&Hall,1992:695~702
44 Yoshimura K,Muramatsu J,Davis P.Conditions for common-noise-induced synchronization in time-delay systems.Physica D,2008,237(23):3146~3152
45刘茂省.复杂网络上的动力学模型分析及随机影响[博士学位论文].上海:复旦大学,2009(Liu M S.Dynamic model analysis and stochastic effects in complex networks[Ph.D Thesis].Shanghai:Fudan University,2009(in Chinese))
46 Atay F M,Jost J,Wende A.Delays,connection topology,and synchronization of coupled chaotic maps.Physical Review Letters,2004,92(14):144101
47 Newman M E J,Watts D J.Renormalization group analysis of the small-world network model.Physics Letters A,1999,263(4-6):341~346