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高阶耦合相振子系统的同步动力学

蔡宗楷 徐灿 郑志刚

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高阶耦合相振子系统的同步动力学

蔡宗楷, 徐灿, 郑志刚

Collective dynamics of higher-order coupled phase oscillators

Cai Zong-Kai, Xu Can, Zheng Zhi-Gang
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  • 由大量耦合相振子组成的Kuramoto模型是研究各种自持续振荡系统同步相变和集体动力学的重要模型. 近些年, 高阶耦合Kuramoto模型引起了广泛的研究兴趣, 尤其高阶耦合结构在模拟编码和信息存储的动力学方面起到重要作用. 为了研究高阶耦合的影响, 本文通过考虑频率与耦合之间的关联对高阶耦合的Kuramoto模型进行了推广, 所得到的模型出现了一些新颖的动力学现象, 包括多集团态(多团簇态)、双稳态、爆炸性同步以及振荡态. 对无序态的线性稳定分析得到表征系统由无序向同步转变的临界耦合强度, 利用自洽方法分析得到系统的多团簇态, 并进一步在等效低维子空间中对多团簇态进行线性稳定性分析得到稳定的多团簇态解以及去同步相变点. 对理论分析结果的讨论总结了系统由迟滞到振荡态的转变. 此外, 本文强调结合表征系统不对称性的Kuramoto序参量和表征系统多团簇态的Daido序参量可以对系统宏观动力学给出完整的描述. 通过本文的研究可以进一步加深对高阶耦合相振子系统中耦合异质性以及爆炸性同步的理解.
    The Kuramoto model consisting of large ensembles of coupled phase oscillators serves as an illustrative paradigm for studying the synchronization transitions and collective behaviors in various self-sustained systems. In recent years, the research of the high-order coupled phase oscillators has attracted extensive interest for the high-order coupled structure playing an essential role in modeling the dynamics of code and data storage. By studying the effects of high-order coupling, we extend the Kuramoto model of high-order structure by considering the correlations between frequency and coupling, which reflects the intrinsic properties of heterogeneity of interactions between oscillators. Several novel dynamic phenomena occur in the model, including clustering, extensive multistability, explosive synchronization and oscillatory state. The universal form of the critical coupling strength characterizing the transition from disorder to order is obtained via an analysis of the stability of the incoherent state. Furthermore, we present the self-consistent approach and find the multi-cluster with their expressions of order parameters. The stability analysis of multi-cluster is performed in the subspace getting stability condition together with the stable solutions of order parameters. The discussion of all the results summarizes the mechanism of the transition from hysteresis to oscillatory states. In addition, we emphasize that the combination of the Kuramoto order parameter capturing the asymmetry of the system and the Daido order parameter representing the clustering can give a complete description of the macroscopic dynamics of the system. The research of this paper can improve the understanding of the effects of the heterogeneity among populations and the explosive synchronization occurring in higher-order coupled phase oscillators.
      通信作者: 徐灿, xucan@hqu.edu.cn ; 郑志刚, zgzheng@hqu.edu.cn
    • 基金项目: 国家自然科学基金(批准号: 11875135)、国家自然科学基金青年科学基金(批准号: 11905068)、泉州市科技计划项目(批准号: 2018C085R)和华侨大学科研经费(批准号: ZQN-810)资助的课题
      Corresponding author: Xu Can, xucan@hqu.edu.cn ; Zheng Zhi-Gang, zgzheng@hqu.edu.cn
    • Funds: Project supported by the National Natural Science Foundation of China (Grant No. 11875135), the Young Scientists Fund of the National Natural Science Foundation of China (Grant No. 11905068), the Scientific and Technological Foundation of Quanzhou, China (Grant No. 2018C085R), and the Scientific Research Funds of Huaqiao University, China (Grant No. ZQN-810)
    [1]

    Wiesenfeld K, Colet P, Strogatz S H 1998 Phys. Rev. E 57 1563

    [2]

    Rohden M, Sorge A, Timme M, Witthaut D 2012 Phys. Rev. Lett. 109 064101Google Scholar

    [3]

    Pikovsky A, Rosenblum M, Kurths J 2001 Synchronization: A Universal Concept in Nonlinear Sciences (Cambridge: Cambridge University Press) pp18–28

    [4]

    Hoppensteadt F C, Izhikevich E M 1999 Phys. Rev. Lett. 82 2983Google Scholar

    [5]

    Strogatz S H 2003 Sync: The Emerging Science of Spontaneous Order (New York: Hypernion) pp59–60

    [6]

    Arenas A, Díaz-Guilera A, Kurths J, Moreno Y, Zhou C 2008 Phys. Rep. 469 93Google Scholar

    [7]

    郑志刚 2019 复杂系统的涌现动力学: 从同步到集体输运 (北京: 科学出版社) 第95−176页

    Zheng Z G 2019 Emergence Dynamics in Complex Systems: From Synchronization to Collective Transport (Beijing: Science Press) pp95−176 (in Chinese)

    [8]

    Kuramoto Y 1975 Int. Symp. on Mathematical Problems in Theoretical Physics (Lecture Notes in Physics Vol. 30) ed Araki H (New York: Springer) pp4−20

    [9]

    Strogatz S H 2000 Physica D 143 1Google Scholar

    [10]

    Acebrón J A, Bonilla L L, Pérez Vicente C J, Ritort F, Spigler R 2005 Rev. Mod. Phys. 77 137Google Scholar

    [11]

    Pikovsky A, Rosenblum M 2015 Chaos 25 097616Google Scholar

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    Daido H 1992 Prog. Theor. Phys. 88 1213Google Scholar

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    Skardal P S, Ott E, Restrepo J G 2011 Phys. Rev. E 84 036208Google Scholar

    [14]

    Komarov M, Pikovsky A 2013 Phys. Rev. Lett. 111 204101Google Scholar

    [15]

    Xu C, Xiang H, Gao J, Zheng Z 2016 Sci. Rep. 6 31133Google Scholar

    [16]

    Wang H, Han W, Yang J 2017 Phys. Rev. E 96 022202Google Scholar

    [17]

    Gong C C, Pikovsky A 2019 Phys. Rev. E 100 062210Google Scholar

    [18]

    Czolczynski K, Perlikowski P, Stefanski A, Kapitaniak T 2013 Commun. Nonlinear Sci. Numer. Simul. 18 386Google Scholar

    [19]

    Goldobin E, Koelle D, Kleiner R, Mints R G 2011 Phys. Rev. Lett. 107 227001Google Scholar

    [20]

    Goldobin E, Kleiner R, Koelle D, Mints R G 2103 Phys. Rev. Lett. 11 057004

    [21]

    Kiss I Z, Zhai Y, Hudson J L 2005 Phys. Rev. Lett. 94 248301Google Scholar

    [22]

    Kiss I Z, Zhai Y, Hudson J L 2006 Prog. Theor. Phys. Suppl. 161 99Google Scholar

    [23]

    Ashwin P, Rodrigues A 2016 Physica D 325 14Google Scholar

    [24]

    León I, Pazó D 2019 Phys. Rev. E 100 012211Google Scholar

    [25]

    Giusti C, Ghrist R, Bassett D S 2016 J. Comput. Neurosci. 41 1Google Scholar

    [26]

    Komarov M, Pikovsky A 2015 Phys. Rev. E 92 020901Google Scholar

    [27]

    Bick C, Ashwin P, Rodrigues A 2016 Chaos 26 094814Google Scholar

    [28]

    Millán A P, Torres J J, Bianconi G 2020 Phys. Rev. Lett. 124 218301Google Scholar

    [29]

    Skardal P S, Arenas A 2020 Commun. Phys. 3 218Google Scholar

    [30]

    Skardal P S, Arenas A 2019 Phys. Rev. Lett. 122 248301Google Scholar

    [31]

    Xu C, Wang X, Skardal P S 2020 Phys. Rev. Res. 2 023281Google Scholar

    [32]

    Xu C, Skardal P S 2021 Phys. Rev. Res. 3 013013Google Scholar

    [33]

    Vlasov V, Rosenblum M, Pikovsky A 2016 J. Phys. A: Math.Theor. 49 31LT02Google Scholar

    [34]

    Chen B, Engelbrecht J R, Mirollo R 2017 J. Phys. A: Math. Theor. 50 355101Google Scholar

    [35]

    王学彬, 徐灿, 郑志刚 2020 物理学报 69 170501Google Scholar

    Wang X B, Xu C, Zheng Z G 2020 Acta Phys. Sin. 69 170501Google Scholar

    [36]

    Iatsenko D, Petkoski S, McClintock P V E, Stefanovska A 2013 Phys. Rev. Lett. 110 064101Google Scholar

    [37]

    Hong H, Strogatz S H 2011 Phys. Rev. Lett. 106 054102Google Scholar

    [38]

    Wang H, Li X 2011 Phys. Rev. E 83 066214Google Scholar

    [39]

    Zhang X, Hu X, Kurths J, Liu Z 2013 Phys. Rev. E 88 010802Google Scholar

    [40]

    Yuan D, Zhang M, Zhong J 2014 Phys. Rev. E 89 012910Google Scholar

    [41]

    Bi H, Hu X, Boccaletti S, Wang X, Zou Y, Liu Z, Guan S 2016 Phys. Rev. Lett. 117 204101Google Scholar

    [42]

    朱廷祥, 吴晔, 肖井华 2012 物理学报 62 040502Google Scholar

    Zhu T X, Wu Y, Xiao J H 2012 Acta Phys. Sin. 62 040502Google Scholar

    [43]

    Xu C, Gao J, Xiang H, Jia W, Guan S, Zheng Z 2016 Phys. Rev. E 94 062204Google Scholar

    [44]

    Xu C, Boccaletti S, Guan S, Zheng Z 2018 Phys. Rev. E 98 050202Google Scholar

    [45]

    Xu C, Boccaletti S, Zheng Z, Guan S 2019 New J. Phys. 21 113018Google Scholar

    [46]

    Xiao Y, Jia W, Xu C, Lü H, Zheng Z 2017 Europhys. Lett. 118 60005Google Scholar

    [47]

    郑志刚, 翟云 2020 中国科学: 物理学 力学 天文学 50 010505Google Scholar

    Zheng Z G, Zhai Y 2020 Sci. Sin. Phys., Mech. Astron. 50 010505Google Scholar

    [48]

    Gómez-Gardeñes J, Gómez S, Arenas A, Moreno Y 2011 Phys. Rev. Lett. 106 128701Google Scholar

    [49]

    Zou Y, Pereira T, Small M, Liu Z, Kurths J 2014 Phys. Rev. Lett. 112 114102Google Scholar

    [50]

    Xu C, Gao J, Sun Y, Huang X, Zheng Z 2015 Sci. Rep. 5 12039Google Scholar

    [51]

    Vlasov V, Zou Y, Pereira T 2015 Phys. Rev. E 92 012904Google Scholar

    [52]

    管曙光 2020 中国科学: 物理学 力学 天文学 50 010504Google Scholar

    Guan S G 2020 Sci. Sin. Phys., Mech. Astron. 50 010504Google Scholar

    [53]

    Xu C, Gao J, Boccaletti S, Zheng Z and Guan S 2019 Phys. Rev. E 100 012212Google Scholar

    [54]

    Mirollo R, Strogatz S H 2007 J. Nonlinear Sci. 17 309Google Scholar

    [55]

    Omel’chenko O E, Wolfrum M 2013 Physica D 263 74Google Scholar

    [56]

    Strogatz S H, Mirollo R E 1991 J. Stat. Phys. 63 613Google Scholar

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    Xu C, Zheng Z 2019 Nonlinear Dyn. 98 2365Google Scholar

  • 图 1  序参量$ R_2 $$ R_1 $随耦合强度$ K $的相变图. $ g(\omega) $为双峰洛伦兹分布, 且$ \varDelta = 0.10 $, $ \omega_0 $分别取(a)$ 0.08 $, (b)$ 0.12 $, (c)$ 0.30 $, (d)$ 0.40 $. 其中正三角形$ \vartriangle $表示耦合强度$ K $增大的方向, 倒三角形$ \triangledown $表示耦合强度$ K $ 减小的方向. (a2)−(d2)中$ R1 $的相变曲线自上往下$ \alpha $分别取$ 1.00 $(粉, 方形), $ 0.90 $(青, 圆形), $ 0.80 $(蓝, 菱形), $ 0.70 $(绿, 左三角), $ 0.60 $(红, 右三角)

    Fig. 1.  Phase transition diagram of order parameters $ R_ 2 $ and $ R_1 $ with the coupling strength $ K $. $ g(\omega) $ is bimodal Lorentz distribution with $ \varDelta = 0.10 $, and $ \omega_ 0 = $$ 0.08 $(a), $ 0.12 $(b), $ 0.30 $(c), $ 0.40 $(d), respectively. The regular triangle $ \vartriangle $ indicates the direction of the increase of coupling strength $ K $ and the inverted triangle $ \triangledown $ indicates the direction of the decrease of coupling strength $ K $. In (a2)−(d2), phase transition of $ R_1 $ with $ \alpha = $$ 1.00 $(pink, square), $ 0.90 $(cyan, circle), $ 0.80 $(blue, diamond), $ 0.70 $(green, left triangle) and $ 0.60 $(red, right triangle) from top to bottom, respectively.

    图 2  耦合强度$ K = 1.6 $时序参量$ R_2 $$ R_1 $随时间$ t $的演化. $ g(\omega) $为双峰洛伦兹分布, 且$\varDelta = 0.1$, $ \omega_0 $分别取(a), (b)$ 0.30 $, (c), (d)$ 0.40 $. (b), (d)中$ R_1(t) $曲线自上往下$ \alpha $分别取$ 1.0 $(粉, 实线), $ 0.9 $(青, 划线), $ 0.8 $(蓝, 点线), $ 0.7 $(绿, 点划线), $ 0.6 $(红, 双点划线)

    Fig. 2.  Evolution of the order parameters $ R_ 2 $ and $ R_1 $ with time $ t $ at coupling strength $ K = 1.60 $. $ g(\omega) $ is bimodal Lorentz distribution with $ \varDelta = 0.10 $, and $ \omega_0 = $$ 0.30 $ ((a), (b)), $ 0.40 $((c), (d)), respectively. The evolution of $ R_ 1(t) $ with $ \alpha = $$ 1.00 $(pink, solid line), $ 0.90 $(cyan, dash line), $ 0.80 $(blue, dot line), $ 0.70 $(green, dash dot line), $ 0.60 $(red, dash dots line) from top to bottom, respectively.

    图 3  $ g(\omega) $为均匀分布, $ \gamma = 1.0 $ (a), (b)序参量$ R_2 $$ R_1 $随耦合强度$ K $的相变图. 其中相变曲线自上往下$ \alpha $分别取$ 1.00 $(粉, 方形), $ 0.90 $(青, 圆形), $ 0.80 $(蓝, 菱形), $ 0.70 $(绿, 左三角), $ 0.60 $(红, 右三角). (c), (d)耦合强度$ K = 1.90 $时序参量$ R_2 $$ R_1 $随时间$ t $的演化. 图中$ R_1 $曲线自上往下$ \alpha $分别取$ 1.0 $(粉, 实线), $ 0.9 $(青, 划线), $ 0.8 $(蓝, 点线), $ 0.7 $(绿, 点划线), $ 0.6 $(红, 双点划线)

    Fig. 3.  $ g(\omega) $ is uniform distribution with $ \gamma = 1.0 $ (a), (b) Phase transition diagram of order parameters $ R_ 2 $ and $ R_1 $ with the coupling strength $ K $. Phase transition of $ R_1 $ with $ \alpha = $$ 1.00 $(pink, square), $ 0.90 $(cyan, circle), $ 0.80 $(blue, diamond), $ 0.70 $(green, left triangle) and $ 0.60 $(red, right triangle) from top to bottom, respectively. (c), (d) Evolution of the order parameters $ R_ 2 $ and $ R_1 $ with time $ t $ at coupling strength $ K = 1.90 $. The curve of $ R_ 1 $ with $ \alpha = $$ 1.00 $(pink, solid line), $ 0.90 $(cyan, dash line), $ 0.80 $(blue, dot line), $ 0.70 $(green, dash dot line), $ 0.60 $(red, dash dots line) from top to bottom, respectively.

  • [1]

    Wiesenfeld K, Colet P, Strogatz S H 1998 Phys. Rev. E 57 1563

    [2]

    Rohden M, Sorge A, Timme M, Witthaut D 2012 Phys. Rev. Lett. 109 064101Google Scholar

    [3]

    Pikovsky A, Rosenblum M, Kurths J 2001 Synchronization: A Universal Concept in Nonlinear Sciences (Cambridge: Cambridge University Press) pp18–28

    [4]

    Hoppensteadt F C, Izhikevich E M 1999 Phys. Rev. Lett. 82 2983Google Scholar

    [5]

    Strogatz S H 2003 Sync: The Emerging Science of Spontaneous Order (New York: Hypernion) pp59–60

    [6]

    Arenas A, Díaz-Guilera A, Kurths J, Moreno Y, Zhou C 2008 Phys. Rep. 469 93Google Scholar

    [7]

    郑志刚 2019 复杂系统的涌现动力学: 从同步到集体输运 (北京: 科学出版社) 第95−176页

    Zheng Z G 2019 Emergence Dynamics in Complex Systems: From Synchronization to Collective Transport (Beijing: Science Press) pp95−176 (in Chinese)

    [8]

    Kuramoto Y 1975 Int. Symp. on Mathematical Problems in Theoretical Physics (Lecture Notes in Physics Vol. 30) ed Araki H (New York: Springer) pp4−20

    [9]

    Strogatz S H 2000 Physica D 143 1Google Scholar

    [10]

    Acebrón J A, Bonilla L L, Pérez Vicente C J, Ritort F, Spigler R 2005 Rev. Mod. Phys. 77 137Google Scholar

    [11]

    Pikovsky A, Rosenblum M 2015 Chaos 25 097616Google Scholar

    [12]

    Daido H 1992 Prog. Theor. Phys. 88 1213Google Scholar

    [13]

    Skardal P S, Ott E, Restrepo J G 2011 Phys. Rev. E 84 036208Google Scholar

    [14]

    Komarov M, Pikovsky A 2013 Phys. Rev. Lett. 111 204101Google Scholar

    [15]

    Xu C, Xiang H, Gao J, Zheng Z 2016 Sci. Rep. 6 31133Google Scholar

    [16]

    Wang H, Han W, Yang J 2017 Phys. Rev. E 96 022202Google Scholar

    [17]

    Gong C C, Pikovsky A 2019 Phys. Rev. E 100 062210Google Scholar

    [18]

    Czolczynski K, Perlikowski P, Stefanski A, Kapitaniak T 2013 Commun. Nonlinear Sci. Numer. Simul. 18 386Google Scholar

    [19]

    Goldobin E, Koelle D, Kleiner R, Mints R G 2011 Phys. Rev. Lett. 107 227001Google Scholar

    [20]

    Goldobin E, Kleiner R, Koelle D, Mints R G 2103 Phys. Rev. Lett. 11 057004

    [21]

    Kiss I Z, Zhai Y, Hudson J L 2005 Phys. Rev. Lett. 94 248301Google Scholar

    [22]

    Kiss I Z, Zhai Y, Hudson J L 2006 Prog. Theor. Phys. Suppl. 161 99Google Scholar

    [23]

    Ashwin P, Rodrigues A 2016 Physica D 325 14Google Scholar

    [24]

    León I, Pazó D 2019 Phys. Rev. E 100 012211Google Scholar

    [25]

    Giusti C, Ghrist R, Bassett D S 2016 J. Comput. Neurosci. 41 1Google Scholar

    [26]

    Komarov M, Pikovsky A 2015 Phys. Rev. E 92 020901Google Scholar

    [27]

    Bick C, Ashwin P, Rodrigues A 2016 Chaos 26 094814Google Scholar

    [28]

    Millán A P, Torres J J, Bianconi G 2020 Phys. Rev. Lett. 124 218301Google Scholar

    [29]

    Skardal P S, Arenas A 2020 Commun. Phys. 3 218Google Scholar

    [30]

    Skardal P S, Arenas A 2019 Phys. Rev. Lett. 122 248301Google Scholar

    [31]

    Xu C, Wang X, Skardal P S 2020 Phys. Rev. Res. 2 023281Google Scholar

    [32]

    Xu C, Skardal P S 2021 Phys. Rev. Res. 3 013013Google Scholar

    [33]

    Vlasov V, Rosenblum M, Pikovsky A 2016 J. Phys. A: Math.Theor. 49 31LT02Google Scholar

    [34]

    Chen B, Engelbrecht J R, Mirollo R 2017 J. Phys. A: Math. Theor. 50 355101Google Scholar

    [35]

    王学彬, 徐灿, 郑志刚 2020 物理学报 69 170501Google Scholar

    Wang X B, Xu C, Zheng Z G 2020 Acta Phys. Sin. 69 170501Google Scholar

    [36]

    Iatsenko D, Petkoski S, McClintock P V E, Stefanovska A 2013 Phys. Rev. Lett. 110 064101Google Scholar

    [37]

    Hong H, Strogatz S H 2011 Phys. Rev. Lett. 106 054102Google Scholar

    [38]

    Wang H, Li X 2011 Phys. Rev. E 83 066214Google Scholar

    [39]

    Zhang X, Hu X, Kurths J, Liu Z 2013 Phys. Rev. E 88 010802Google Scholar

    [40]

    Yuan D, Zhang M, Zhong J 2014 Phys. Rev. E 89 012910Google Scholar

    [41]

    Bi H, Hu X, Boccaletti S, Wang X, Zou Y, Liu Z, Guan S 2016 Phys. Rev. Lett. 117 204101Google Scholar

    [42]

    朱廷祥, 吴晔, 肖井华 2012 物理学报 62 040502Google Scholar

    Zhu T X, Wu Y, Xiao J H 2012 Acta Phys. Sin. 62 040502Google Scholar

    [43]

    Xu C, Gao J, Xiang H, Jia W, Guan S, Zheng Z 2016 Phys. Rev. E 94 062204Google Scholar

    [44]

    Xu C, Boccaletti S, Guan S, Zheng Z 2018 Phys. Rev. E 98 050202Google Scholar

    [45]

    Xu C, Boccaletti S, Zheng Z, Guan S 2019 New J. Phys. 21 113018Google Scholar

    [46]

    Xiao Y, Jia W, Xu C, Lü H, Zheng Z 2017 Europhys. Lett. 118 60005Google Scholar

    [47]

    郑志刚, 翟云 2020 中国科学: 物理学 力学 天文学 50 010505Google Scholar

    Zheng Z G, Zhai Y 2020 Sci. Sin. Phys., Mech. Astron. 50 010505Google Scholar

    [48]

    Gómez-Gardeñes J, Gómez S, Arenas A, Moreno Y 2011 Phys. Rev. Lett. 106 128701Google Scholar

    [49]

    Zou Y, Pereira T, Small M, Liu Z, Kurths J 2014 Phys. Rev. Lett. 112 114102Google Scholar

    [50]

    Xu C, Gao J, Sun Y, Huang X, Zheng Z 2015 Sci. Rep. 5 12039Google Scholar

    [51]

    Vlasov V, Zou Y, Pereira T 2015 Phys. Rev. E 92 012904Google Scholar

    [52]

    管曙光 2020 中国科学: 物理学 力学 天文学 50 010504Google Scholar

    Guan S G 2020 Sci. Sin. Phys., Mech. Astron. 50 010504Google Scholar

    [53]

    Xu C, Gao J, Boccaletti S, Zheng Z and Guan S 2019 Phys. Rev. E 100 012212Google Scholar

    [54]

    Mirollo R, Strogatz S H 2007 J. Nonlinear Sci. 17 309Google Scholar

    [55]

    Omel’chenko O E, Wolfrum M 2013 Physica D 263 74Google Scholar

    [56]

    Strogatz S H, Mirollo R E 1991 J. Stat. Phys. 63 613Google Scholar

    [57]

    Xu C, Zheng Z 2019 Nonlinear Dyn. 98 2365Google Scholar

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出版历程
  • 收稿日期:  2021-06-29
  • 修回日期:  2021-07-29
  • 上网日期:  2021-08-15
  • 刊出日期:  2021-11-20

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