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耦合Stuart-Landau极限环系统为研究含振幅自由度的自持续振荡系统同步相变和集体动力学提供了一个重要的范式模型. 本文研究了平均场反馈下全局耦合Stuart-Landau极限环系统中3种典型的可解集体动力学: 非相干态、振幅死亡及锁频态. 在热力学极限
$N\rightarrow\infty$ 情形下, 利用非相干态线性稳定性分析揭示了耦合系统中同步发生的临界条件, 发现了增强平均场反馈强度可使得耦合系统在更小的扩散耦合强度下出现同步相变行为; 通过对振幅死亡态的线性稳定性分析得到了参数空间中振幅死亡的稳定区, 发现了平均场反馈强度可有效地消除耦合系统中的振幅死亡现象; 从理论上分析了锁频态的存在性条件, 并从锁频态序参量的自洽关系中推导出了振幅死亡区的边界线. 本文的研究揭示了平均场反馈对耦合非线性系统中集体行为的动力学控制作用, 加深了平均场反馈技术对耦合诱导的集体行为影响的理解, 进一步阐释了复杂耦合系统中自组织行为的涌现规律与机制.Coupled Stuart-Landau limit-cycle system serves as an important paradigmatic model for studying synchronization transitions and collective dynamics in self-sustained nonlinear systems with amplitude degree of freedom. In this paper, we extensively investigate three typical solvable collective behaviors in globally coupled Stuart-Landau limit-cycle systems under mean-field feedback: incoherence, amplitude death, and locked states. In the thermodynamic limit of$N\rightarrow\infty$ , the critical condition characterizing the transition from incoherence to synchronization is explicitly obtained via performing the linear stability of the incoherent states. It is found that the synchronization transition occurs at a smaller coupling strength when the strength of mean-field feedback is gradually enhanced. The stable regions of amplitude death are theoretically obtained via an analysis of the linear stability of coupled systems around the origin. The results indicate that the existence of mean-field feedback can effectively eliminate the amplitude death phenomenon in the coupled systems; furthermore, the existence of locked states is analyzed theoretically, and in particular, the boundary of stable amplitude death region is re-derived from the self-consistent relation of the order parameter for the locked states. This work reveals the key role of mean-field feedback in controlling the collective dynamics of coupled nonlinear systems, deepens the understanding of the influence of mean-field feedback technology on the coupling-induced collective behaviors, and is conductive to our further understanding of the emerging rules and the underlying mechanisms of self-organized behavior in complex coupled systems.[1] Kuramoto Y 1984 Chemical Oscillations, Waves and Turbulence (Berlin: Springer-Verlag
[2] Pikovsky A, Rosenblum M, Kurths J 2001 Synchronization: A Universal Concept in Nonlinear Sciences (Cambridge: Cambridge University Press
[3] Strogatz S 2003 Sync: The Emerging Science of Spontaneous Order (London: Pengiun Press Science
[4] 郑志刚 2004 耦合非 线性系统的时空动力学与合作行为 (北京: 高等教育出版社)
Zheng Z G 2004 Spatiotemporal Dynamics and Cooperative Behaviors in Coupled Nonlinear Systems (Beijing: Higher Education Press
[5] 丁大为, 卢小齐, 胡永兵, 杨宗立, 王威, 张红伟 2022 物理学报 71 230501Google Scholar
Ding D W, Lu X Q, Hu Y B, Yang Z L, Wang W, Zhang H W 2022 Acta Phys. Sin. 71 230501Google Scholar
[6] 蒋宏帆, 林机, 胡贝贝, 张肖 2023 物理学报 72 104205Google Scholar
Jiang H F, Lin J, Hu B B, Zhang X 2023 Acta Phys. Sin. 72 104205Google Scholar
[7] 沈力峰, 王建波, 杜占玮, 许小可 2023 物理学报 72 068701Google Scholar
Shen L F, Wang J B, Du Z W, Xu X K 2023 Acta Phys. Sin. 72 068701Google Scholar
[8] Kuramoto Y 1975 Self-entrainment of a Population of Coupled Nonlinear Oscillators, in: International Symposium on Mathematical Problems in Theoretical Physics (Berlin Heidelberg: Springer-Verlag) pp420–428
[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] 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
[12] 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
[13] 郑志刚, 翟云, 王学彬, 陈宏斌, 徐灿 2020 物理学报 69 080502Google Scholar
Zheng Z G, Zhai Y, Wang X B, Chen H B, Xu C 2020 Acta Phys. Sin. 69 080502Google Scholar
[14] 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
[15] 蔡宗楷, 徐灿, 郑志刚 2021 物理学报 70 220501Google Scholar
Cai Z K, Xu C, Zheng Z G 2021 Acta Phys. Sin. 70 220501Google Scholar
[16] Katz Y, Tunstrøm K, Ioannou C C, Huepe C, Couzin I D 2011 Proc. Natl. Acad. Sci. U.S.A. 108 18720Google Scholar
[17] Vicsek T, Zafeiris A 2012 Phys. Rep. 517 71Google Scholar
[18] O'Keeffe K P, Hong H, Strogatz S H 2017 Nat. Commun. 8 1504Google Scholar
[19] Cross M C, Hohenberg P C 1993 Rev. Mod. Phys. 65 851Google Scholar
[20] Nakagawa N, Kuramoto Y 1993 Prog. Theor. Phys. 89 313Google Scholar
[21] 刘若琪, 贾萌萌, 范伟丽, 贺亚峰, 刘富成 2022 物理学报 71 248201Google Scholar
Liu R Q, Jia M M, Fan W L, He Y F, Liu F C 2022 Acta Phys. Sin. 71 248201Google Scholar
[22] 杨佳奇, 刘文军 2023 物理学报 72 100504Google Scholar
Yang J Q, Liu W J 2023 Acta Phys. Sin. 72 100504Google Scholar
[23] Saxena G, Prasad A, Ramaswamy R 2012 Phys. Rep. 521 205Google Scholar
[24] Zou W, Senthilkumar D V, Zhan M, Kurths J 2021 Phys. Rep. 931 1Google Scholar
[25] Mirollo R E, Strogatz S H 1989 J. Stat. Phys. 60 245
[26] Aronson D G, Ermentrout G B, Kopell N 1990 Physica D 41 403Google Scholar
[27] Zou W, Senthilkumar D V, Nagao R, Kiss I Z, Tang Y, Koseska A, Duan J Q, Kurths J 2015 Nat. Commun. 6 7709Google Scholar
[28] Ramana Reddy D V, Sen A, Johnston G L 1998 Phys. Rev. Lett. 80 5109Google Scholar
[29] Atay F M 2003 Phys. Rev. Lett. 91 094101Google Scholar
[30] Zou W, Senthilkumar D V, Zhan M, Kurths J 2013 Phys. Rev. Lett. 111 014101Google Scholar
[31] Konishi K 2003 Phys. Rev. E 68 067202Google Scholar
[32] Karnatak R, Ramaswamy R, Prasad A 2007 Phys. Rev. E 76 035201
[33] Zou W, He S J, Yao C G 2022 Appl. Math. Lett. 131 108052Google Scholar
[34] Resmi V, Ambika G, Amritkar R E 2011 Phys. Rev. E 84 046212Google Scholar
[35] Matthews P C, Strogatz S H 1990 Phys. Rev. Lett. 65 1701Google Scholar
[36] Matthews P C, Mirollo R E, Strogatz S H 1991 Physica D 52 293Google Scholar
[37] Zou W, He S J, Senthilkumar D V, Kurths J 2023 Phys. Rev. Lett. 130 107202Google Scholar
[38] Schwab D J, Baetica A, Mehta P 2012 Physica D 241 1782Google Scholar
[39] Lee W S, Ott E, Antonsen T M 2013 Chaos 23 033116Google Scholar
[40] Wang C Q, Garnier N B 2016 Chaos 26 113119Google Scholar
[41] Kemeth F P, Haugland S W, Krischer K 2018 Phys. Rev. Lett. 120 214101Google Scholar
[42] Röhm A, Lüdge K, Schneider I 2018 Chaos 28 063114Google Scholar
[43] Kemeth F P, Haugland S W, Krischer K 2019 Chaos 29 023107Google Scholar
[44] León I, Pazó D 2019 Phys. Rev. E 100 012211Google Scholar
[45] León I, Pazó D 2020 Phys. Rev. E 102 042203
[46] Kak S C 1993 Circuits, Systems and Signal Processing 12 263Google Scholar
[47] Becskei A, Séraphin B, Serrano L 2001 Embo. J. 20 2528Google Scholar
[48] Draghici S 1997 Int. J. Neural Syst. 8 113Google Scholar
[49] Gough J E, Gohm R, Yanagisawa M 2008 Phys. Rev. A 78 062104Google Scholar
[50] Lloyd S 2000 Phys. Rev. A 62 022108Google Scholar
[51] Little S, Brown P 2012 Ann. N. Y. Acad. Sci. 1265 9Google Scholar
[52] Chandrasekar V K, Karthiga S, Lakshmanan M 2015 Phys. Rev. E 92 012903Google Scholar
[53] Zhao N N, Sun Z K 2020 Int. J. Bifurcation and Chaos 30 2050094Google Scholar
[54] 史东鑫, 李兴瑞, 单美华 2023 应用数学进展 12 1940Google Scholar
Shi D X, Li X R, Shan M H 2023 Adv. Appl. Math. 12 1940Google Scholar
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图 1 平均场反馈对全局扩散耦合系统(1)中非相干态的影响 (a) 不同平均场反馈强度
$ K_2 $ 下, 序参量R随扩散耦合强度$ K_1 $ 的相变图, 其中频率标准差固定为$ \varDelta=0.2 $ ; (b) 不同平均场反馈强度$ K_2 $ 下, 使得非相干态失稳的临界扩散耦合强度$ K_{1 {\rm{c}}} $ 与频率标准差Δ的关系. 实线为(21)式给出的理论预测, 蓝色的方形实点为数值结果Fig. 1. Effect of mean-field feedback on the incoherence in coupled system (1): (a) Phase transition diagrams of order parameter R with the strengths of diffusive coupling
$ K_1 $ under different strengths of the mean-field feedback$ K_2 $ , where the standard deviation of natural frequencies is fixed at$ \varDelta=0.2 $ ; (b) dependence of the critical diffusive coupling strength$ K_{1 {\rm{c}}} $ , beyond which the incoherence becomes destabilized, on the standard deviation Δ of natural frequencies under different strengths of the mean-field feedback$ K_2 $ . The solid lines are the plots of the theoretical prediction given in Eq. (21), whereas the blue squares denote the numerical results图 2 平均场反馈对全局扩散耦合系统(1)中振幅死亡的影响 (a) 稳定的振幅死亡扩散耦合强度
$ K_{1} $ 区间随平均场反馈强度$ K_2 $ 的变化, 其中频率标准差固定为$ \varDelta=1.5 $ ; (b) 不同平均场反馈强度$ K_2 $ 下, 参数空间$ (K_{1}, \varDelta) $ 中振幅死亡的稳定区. (a)与(b)中, 蓝色的方形实点表示数值模拟结果, 实斜线为(27)式中的理论预测结果Fig. 2. Effect of mean-field feedback on amplitude death (AD) in coupled system (1): (a) Stable AD interval of diffusive coupling
$ K_{1} $ versus the strength of the mean-field feedback$ K_2 $ , where$ \varDelta=1.5 $ is fixed; (b) stable regions of AD in the parameter space of$ (K_{1}, \varDelta) $ under different strengths of the mean-field feedback$ K_2 $ . In both panels, the blue squares represent the numerical results, whereas the solid lines denote the theoretical prediction by Eq. (27) -
[1] Kuramoto Y 1984 Chemical Oscillations, Waves and Turbulence (Berlin: Springer-Verlag
[2] Pikovsky A, Rosenblum M, Kurths J 2001 Synchronization: A Universal Concept in Nonlinear Sciences (Cambridge: Cambridge University Press
[3] Strogatz S 2003 Sync: The Emerging Science of Spontaneous Order (London: Pengiun Press Science
[4] 郑志刚 2004 耦合非 线性系统的时空动力学与合作行为 (北京: 高等教育出版社)
Zheng Z G 2004 Spatiotemporal Dynamics and Cooperative Behaviors in Coupled Nonlinear Systems (Beijing: Higher Education Press
[5] 丁大为, 卢小齐, 胡永兵, 杨宗立, 王威, 张红伟 2022 物理学报 71 230501Google Scholar
Ding D W, Lu X Q, Hu Y B, Yang Z L, Wang W, Zhang H W 2022 Acta Phys. Sin. 71 230501Google Scholar
[6] 蒋宏帆, 林机, 胡贝贝, 张肖 2023 物理学报 72 104205Google Scholar
Jiang H F, Lin J, Hu B B, Zhang X 2023 Acta Phys. Sin. 72 104205Google Scholar
[7] 沈力峰, 王建波, 杜占玮, 许小可 2023 物理学报 72 068701Google Scholar
Shen L F, Wang J B, Du Z W, Xu X K 2023 Acta Phys. Sin. 72 068701Google Scholar
[8] Kuramoto Y 1975 Self-entrainment of a Population of Coupled Nonlinear Oscillators, in: International Symposium on Mathematical Problems in Theoretical Physics (Berlin Heidelberg: Springer-Verlag) pp420–428
[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] 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
[12] 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
[13] 郑志刚, 翟云, 王学彬, 陈宏斌, 徐灿 2020 物理学报 69 080502Google Scholar
Zheng Z G, Zhai Y, Wang X B, Chen H B, Xu C 2020 Acta Phys. Sin. 69 080502Google Scholar
[14] 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
[15] 蔡宗楷, 徐灿, 郑志刚 2021 物理学报 70 220501Google Scholar
Cai Z K, Xu C, Zheng Z G 2021 Acta Phys. Sin. 70 220501Google Scholar
[16] Katz Y, Tunstrøm K, Ioannou C C, Huepe C, Couzin I D 2011 Proc. Natl. Acad. Sci. U.S.A. 108 18720Google Scholar
[17] Vicsek T, Zafeiris A 2012 Phys. Rep. 517 71Google Scholar
[18] O'Keeffe K P, Hong H, Strogatz S H 2017 Nat. Commun. 8 1504Google Scholar
[19] Cross M C, Hohenberg P C 1993 Rev. Mod. Phys. 65 851Google Scholar
[20] Nakagawa N, Kuramoto Y 1993 Prog. Theor. Phys. 89 313Google Scholar
[21] 刘若琪, 贾萌萌, 范伟丽, 贺亚峰, 刘富成 2022 物理学报 71 248201Google Scholar
Liu R Q, Jia M M, Fan W L, He Y F, Liu F C 2022 Acta Phys. Sin. 71 248201Google Scholar
[22] 杨佳奇, 刘文军 2023 物理学报 72 100504Google Scholar
Yang J Q, Liu W J 2023 Acta Phys. Sin. 72 100504Google Scholar
[23] Saxena G, Prasad A, Ramaswamy R 2012 Phys. Rep. 521 205Google Scholar
[24] Zou W, Senthilkumar D V, Zhan M, Kurths J 2021 Phys. Rep. 931 1Google Scholar
[25] Mirollo R E, Strogatz S H 1989 J. Stat. Phys. 60 245
[26] Aronson D G, Ermentrout G B, Kopell N 1990 Physica D 41 403Google Scholar
[27] Zou W, Senthilkumar D V, Nagao R, Kiss I Z, Tang Y, Koseska A, Duan J Q, Kurths J 2015 Nat. Commun. 6 7709Google Scholar
[28] Ramana Reddy D V, Sen A, Johnston G L 1998 Phys. Rev. Lett. 80 5109Google Scholar
[29] Atay F M 2003 Phys. Rev. Lett. 91 094101Google Scholar
[30] Zou W, Senthilkumar D V, Zhan M, Kurths J 2013 Phys. Rev. Lett. 111 014101Google Scholar
[31] Konishi K 2003 Phys. Rev. E 68 067202Google Scholar
[32] Karnatak R, Ramaswamy R, Prasad A 2007 Phys. Rev. E 76 035201
[33] Zou W, He S J, Yao C G 2022 Appl. Math. Lett. 131 108052Google Scholar
[34] Resmi V, Ambika G, Amritkar R E 2011 Phys. Rev. E 84 046212Google Scholar
[35] Matthews P C, Strogatz S H 1990 Phys. Rev. Lett. 65 1701Google Scholar
[36] Matthews P C, Mirollo R E, Strogatz S H 1991 Physica D 52 293Google Scholar
[37] Zou W, He S J, Senthilkumar D V, Kurths J 2023 Phys. Rev. Lett. 130 107202Google Scholar
[38] Schwab D J, Baetica A, Mehta P 2012 Physica D 241 1782Google Scholar
[39] Lee W S, Ott E, Antonsen T M 2013 Chaos 23 033116Google Scholar
[40] Wang C Q, Garnier N B 2016 Chaos 26 113119Google Scholar
[41] Kemeth F P, Haugland S W, Krischer K 2018 Phys. Rev. Lett. 120 214101Google Scholar
[42] Röhm A, Lüdge K, Schneider I 2018 Chaos 28 063114Google Scholar
[43] Kemeth F P, Haugland S W, Krischer K 2019 Chaos 29 023107Google Scholar
[44] León I, Pazó D 2019 Phys. Rev. E 100 012211Google Scholar
[45] León I, Pazó D 2020 Phys. Rev. E 102 042203
[46] Kak S C 1993 Circuits, Systems and Signal Processing 12 263Google Scholar
[47] Becskei A, Séraphin B, Serrano L 2001 Embo. J. 20 2528Google Scholar
[48] Draghici S 1997 Int. J. Neural Syst. 8 113Google Scholar
[49] Gough J E, Gohm R, Yanagisawa M 2008 Phys. Rev. A 78 062104Google Scholar
[50] Lloyd S 2000 Phys. Rev. A 62 022108Google Scholar
[51] Little S, Brown P 2012 Ann. N. Y. Acad. Sci. 1265 9Google Scholar
[52] Chandrasekar V K, Karthiga S, Lakshmanan M 2015 Phys. Rev. E 92 012903Google Scholar
[53] Zhao N N, Sun Z K 2020 Int. J. Bifurcation and Chaos 30 2050094Google Scholar
[54] 史东鑫, 李兴瑞, 单美华 2023 应用数学进展 12 1940Google Scholar
Shi D X, Li X R, Shan M H 2023 Adv. Appl. Math. 12 1940Google Scholar
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