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A significant phenomenon in nature is that of collective synchronization, in which a large population of coupled oscillators spontaneously synchronizes at a common frequency. Nonlinearly coupled systems with local interactions are of special importance, in particular, the Kuramoto model in its nearest-neighbor version. In this paper the dynamics of a ring of Kuramoto phase oscillators with unidirectional couplings is investigated. We simulate numerically the bifurcation tree of average frequency observed and the multiple stable states in the synchronization region with the increase of the coupling strength for N4, which cannot be found for N3. Oscillators synchronize at a common frequency =0 when K is larger than a critical value of N=3. Multiple branches with 0 will appear besides the zero branch, and the number of branches increases with increasing oscillators for the system N3. We further present a theoretical analysis on the feature and stability of the multiple synchronous states and obtain the asymptotically stable solutions. When the system of N=2 reaches synchronization, the dynamic equation has two solutions: one is stable and the other is unstable. And there is also one stable solution for N=3 when the system is in global synchronization. For the larger system (N3), we study the identical oscillators and can find all the multiple branches on the bifurcation tree. Our results show that the phase difference between neighboring oscillators has different fixed values corresponding to the numbers of different branches. The behaviors in the synchronization region computed by numerical simulation are consistent with theoretical calculation very well. The systems in which original states belong to different stable states will evolve to the same incoherent state with an adiabatic decreasing of coupling strength. Behaviors of synchronization of all oscillators are exactly the same in non-synchronous region whenever the system evolves from an arbitrary branch according to the bifurcation trees. This result suggests that the only incoherent state can be attributed to the movement ergodicity in the phase space of coupled oscillators in an asynchronous region. When the system achieves synchronization, the phenomenon of the coexistence of multiple stable states will emerge because of the broken ergodicity. All these analyses indicate that the multiple stable states of synchronization in nonlinear coupling systems are indeed generically observable, which can have potential engineering applications.
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Keywords:
- Kuramoto model /
- synchronization /
- phase oscillator /
- multiple stable states
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[2] Strogatz S 2004 Sync: How Order Emerges From Chaos In the Universe, Nature, and Daily Life (Hyperion, New York)
[3] Kuramoto Y 1984 Chemical Oscilations, Waves and Turbulence (Springer-Verlag, Berlin)
[4] Wiesenfeld K, Colet P, Strogatz S H 1996 Phys. Rev. Lett. 76 404
[5] Cross M C, Zumdieck A, Lifshitz R, Rogers J L 2004 Phys. Rev. Lett. 93 224101
[6] Ermentrout B 1991 J. Math. Biol. 29 571
[7] Vinogradova et al T M 2006 Circ. Res. 98 505
[8] Stam C J 2005 Clin. Neurophysiol. 116 2266
[9] Javaloyes J, Perrin M, Politi A 2008 Phys. Rev. E. 78 011108
[10] Zhu T X,Wu Y,Xiao J H 2012 Acta Phys. Sin. 62 040502 (in Chinese) [朱廷祥, 吴晔, 肖井华 2012 物理学报 62 040502]
[11] Feng C, Zou Y L, Wei F Q 2013 Acta Phys. Sin. 62 070506 (in Chinese) [冯聪, 邹艳丽, 韦芳琼 2013 物理学报 62 070506]
[12] Ma X J, Wang Y, Zheng Z G 2009 Acta Phys. Sin. 58 4426 (in Chinese) [马晓娟, 王延, 郑志刚 2009 物理学报 58 4426]
[13] Park M J, Kwon O M, Park J H, Lee S M, Cha E J 2011 Chin. Phys. B 20 110504
[14] Cai G L, Jiang S Q, Cai S M, Tian L X 2013 Chinese Physics B 22 0502
[15] Kuramoto Y 1975 in International Symposium on Mathematical Problems in Theoretical Physics, Lecture Notes in Physics Vol. 39, edited by H. Araki (Springer, New York, 1975)
[16] Strogatz S H 2000 Physica D 143 1
[17] Acebron J A, Bonilla L L, Vicente J C P, Ritort F, Spigler R 2005 Rev. Mod. Phys. 77 137
[18] Zheng Z G, Hu G, Hu B 1998 Phys. Rev. Lett. 81 5318
[19] Hu B, Zheng G Z 2000 International Journal of Bifurcation and Chaos 10 2399
[20] Ochab J, Gora P F 2010 Acta Physica Polonica B Proceedings Supplement 3 453
[21] Strogatz S H, Mirollo R E 1988 Physica D 31 143
[22] Sakaguchi H 1988 Prog. Theor. Phys. 79 1069
[23] Rogers J L, Wille L T 1996 Phys. Rev. E 54 R2193
[24] El-Nashar H F, Cerdeira H A 2009 Chaos 19 033127
[25] Muruganandam P, Ferreira F F, El-Nashar H, Cerdeira H A 2008 Pramana 70 1143
[26] Maistrenko Y, Popovych O, Burylko O, Tass P A 2004 Phys. Rev. Lett. 93 084102
[27] Brede M 2007 Phys. Lett. A 372 2618
[28] Chen M Y, Shang Y, Zou Y, Kurths J 2008 Phys. Rev. E 77 027101
[29] Liu W Q, Wu Y, Xiao J H, Zhan M 2013 Europhys. Lett. 101 38002
[30] Gomez G J, Gomez S, Arenas A, Moreno Y 2011 Phys. Rev. Lett. 106 128701
[31] Ji P, Peron T, Menck P, Rodrigues F, Kurths J 2013 Phys. Rev. Lett. 110 218701
[32] Zheng Z G, Hu B, Hu G 2000 Phys. Rev. E 62 402
[33] Wu Y, Xiao J H, Hu G, Zhan M 2012 Europhys. Lett. 97 40005
[34] Huang X, Zhan M, Li F, Zheng Z G 2014 J. Phys. A: Math. Theor 47 125101
[35] Tilles P, Ferreira F, Cerdeira 2011 Phys. Rev. E 83 066206
[36] Kim S, Park S H, Ryu C S 1997 Phys. Rev. Lett. 79 2911
[37] Ochab J, Góra P F 2009 arXiv preprint arXiv:0909.0043
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[1] Pikovsky A, Rosenblum M, Kurths J 2001 Synchronization: A Universal Concept in Nonlinear Dynamics (Cambridge University Press, Cambridge, England)
[2] Strogatz S 2004 Sync: How Order Emerges From Chaos In the Universe, Nature, and Daily Life (Hyperion, New York)
[3] Kuramoto Y 1984 Chemical Oscilations, Waves and Turbulence (Springer-Verlag, Berlin)
[4] Wiesenfeld K, Colet P, Strogatz S H 1996 Phys. Rev. Lett. 76 404
[5] Cross M C, Zumdieck A, Lifshitz R, Rogers J L 2004 Phys. Rev. Lett. 93 224101
[6] Ermentrout B 1991 J. Math. Biol. 29 571
[7] Vinogradova et al T M 2006 Circ. Res. 98 505
[8] Stam C J 2005 Clin. Neurophysiol. 116 2266
[9] Javaloyes J, Perrin M, Politi A 2008 Phys. Rev. E. 78 011108
[10] Zhu T X,Wu Y,Xiao J H 2012 Acta Phys. Sin. 62 040502 (in Chinese) [朱廷祥, 吴晔, 肖井华 2012 物理学报 62 040502]
[11] Feng C, Zou Y L, Wei F Q 2013 Acta Phys. Sin. 62 070506 (in Chinese) [冯聪, 邹艳丽, 韦芳琼 2013 物理学报 62 070506]
[12] Ma X J, Wang Y, Zheng Z G 2009 Acta Phys. Sin. 58 4426 (in Chinese) [马晓娟, 王延, 郑志刚 2009 物理学报 58 4426]
[13] Park M J, Kwon O M, Park J H, Lee S M, Cha E J 2011 Chin. Phys. B 20 110504
[14] Cai G L, Jiang S Q, Cai S M, Tian L X 2013 Chinese Physics B 22 0502
[15] Kuramoto Y 1975 in International Symposium on Mathematical Problems in Theoretical Physics, Lecture Notes in Physics Vol. 39, edited by H. Araki (Springer, New York, 1975)
[16] Strogatz S H 2000 Physica D 143 1
[17] Acebron J A, Bonilla L L, Vicente J C P, Ritort F, Spigler R 2005 Rev. Mod. Phys. 77 137
[18] Zheng Z G, Hu G, Hu B 1998 Phys. Rev. Lett. 81 5318
[19] Hu B, Zheng G Z 2000 International Journal of Bifurcation and Chaos 10 2399
[20] Ochab J, Gora P F 2010 Acta Physica Polonica B Proceedings Supplement 3 453
[21] Strogatz S H, Mirollo R E 1988 Physica D 31 143
[22] Sakaguchi H 1988 Prog. Theor. Phys. 79 1069
[23] Rogers J L, Wille L T 1996 Phys. Rev. E 54 R2193
[24] El-Nashar H F, Cerdeira H A 2009 Chaos 19 033127
[25] Muruganandam P, Ferreira F F, El-Nashar H, Cerdeira H A 2008 Pramana 70 1143
[26] Maistrenko Y, Popovych O, Burylko O, Tass P A 2004 Phys. Rev. Lett. 93 084102
[27] Brede M 2007 Phys. Lett. A 372 2618
[28] Chen M Y, Shang Y, Zou Y, Kurths J 2008 Phys. Rev. E 77 027101
[29] Liu W Q, Wu Y, Xiao J H, Zhan M 2013 Europhys. Lett. 101 38002
[30] Gomez G J, Gomez S, Arenas A, Moreno Y 2011 Phys. Rev. Lett. 106 128701
[31] Ji P, Peron T, Menck P, Rodrigues F, Kurths J 2013 Phys. Rev. Lett. 110 218701
[32] Zheng Z G, Hu B, Hu G 2000 Phys. Rev. E 62 402
[33] Wu Y, Xiao J H, Hu G, Zhan M 2012 Europhys. Lett. 97 40005
[34] Huang X, Zhan M, Li F, Zheng Z G 2014 J. Phys. A: Math. Theor 47 125101
[35] Tilles P, Ferreira F, Cerdeira 2011 Phys. Rev. E 83 066206
[36] Kim S, Park S H, Ryu C S 1997 Phys. Rev. Lett. 79 2911
[37] Ochab J, Góra P F 2009 arXiv preprint arXiv:0909.0043
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