Order parameter analysis of synchronization transitions on star networks

Abstract

The collective behaviors of populations of coupled oscillators have attracted significant attention in recent years. In this paper, an order parameter approach is proposed to study the low-dimensional dynamical mechanism of collective synchronizations, by adopting the star-topology of coupled oscillators as a prototype system. The order parameter equation of star-linked phase oscillators can be obtained in terms of the Watanabe–Strogatz transformation, Ott–Antonsen ansatz, and the ensemble order parameter approach. Different solutions of the order parameter equation correspond to the diverse collective states, and different bifurcations reveal various transitions among these collective states. The properties of various transitions in the star-network model are revealed by using tools of nonlinear dynamics such as time reversibility analysis and linear stability analysis.

Keywords

Kuramoto model / synchronization / order parameter / Ott–Antonsen ansatz / star network

References

[1] Y. Kuramoto, Chemical Oscillations, Waves and Turbulence, Springer Science and Business Media, 2012

[2] J. A. Acebrón, L. L. Bonilla, C. J. Pérez Vicente, F. Ritort, and R. Spigler, The Kuramoto model: A simple paradigm for synchronization phenomena, Rev. Mod. Phys. 77(1), 137 (2005)

[3] S. H. Strogatz, From Kuramoto to Crawford: Exploring the onset of synchronization in populations of coupled oscillators, Physica D 143(1–4), 1 (2000)

[4] A. Pikovsky, M. Rosenblum, and J. Kurths, Synchronization: A Universal Concept in Nonlinear Sciences, Cambridge: Cambridge University Press, 2001

[5] S. N. Dorogovtsev, A. V. Goltsev, and J. F. F. Mendes, Critical phenomena in complex networks, Rev. Mod. Phys. 80(4), 1275 (2008)

[6] A. Arenas, A. Diaz-Guilera, J. Kurths, Y. Moreno, and C. Zhou, Synchronization in complex networks, Phys. Rep. 469(3), 93 (2008)

[7] Z. Zheng, G. Hu, and B. Hu, Phase slips and phase synchronization of coupled oscillators, Phys. Rev. Lett. 81(24), 5318 (1998)

[8] Z. Zheng, G. Hu, and B. Hu, Collective phase slips and phase synchronizations in coupled oscillator systems, Phys. Rev. E 62(1), 402 (2000)

[9] D. A. Paley, N. E. Leonard, and R. Sepulchre, Oscillator models and collective motion: Splay state stabilization of self-propelled particles, in: Proc. 51st IEEE Conf. Decision Control, pp 3935–3940 (2005)

[10] M. Silber, L. Fabiny, and K. Wiesenfeld, Stability results for in-phase and splay-phase states of solid-state laser arrays, J. Opt. Soc. Am. B 10(6), 1121 (1993)

[11] S. H. Strogatz and R. E. Mirollo, Splay states in globally coupled Josephson arrays: Analytical prediction of Floquet multipliers, Phys. Rev. E 47(1), 220 (1993)

[12] L. Lü, C. Li, W. Wang, Y. Sun, Y. Wang, and A. Sun, Study on spatiotemporal chaos synchronization among complex networks with diverse structures, Nonlinear Dyn. 77(1–2), 145 (2014)

[13] J. Gómez-Gardeñes, S. Gomez, A. Arenas, and Y. Moreno, Explosive synchronization transitions in scalefree networks, Phys. Rev. Lett. 106(12), 128701 (2011)

[14] O. E. Omel’chenko and M. Wolfrum, Nonuniversal transitions to synchrony in the Sakaguchi–Kuramoto model, Phys. Rev. Lett. 109(16), 164101 (2012)

[15] D. Topaj and A. Pikovsky, Reversibility vs. synchronization in oscillator lattices, Physica D 170(2), 118 (2002)

[16] L. Zhou, C. Wang, Y. Lin, and H. He, Combinatorial synchronization of complex multiple networks with unknown parameters, Nonlinear Dyn. 79(1), 307 (2015)

[17] X. Zhang, X. Hu, J. Kurths, and Z. Liu, Explosive synchronization in a general complex network, Phys. Rev. E 88, 010802(R) (2013)

[18] Z. Zheng, Spatiotemporal Dynamics and Collective Behaviors in Coupled Nonlinear Systems, Beijing: Higher Education Press, 2004 (in Chinese)

[19] N. Yao and Z. Zheng, Chimera states in spatiotemporal systems: Theory and applications, Int. J. Mod. Phys. B 30(1), 163002 (2016)

[20] E. Ott and T. M. Antonsen, Low dimensional behavior of large systems of globally coupled oscillators, Chaos 18(3), 037113 (2008)

[21] S. Watanabe and S. H. Strogatz, Integrability of a globally coupled oscillator array, Phys. Rev. Lett. 70(16), 2391 (1993)

[22] S. Watanabe and S. H. Strogatz, Constants of motion for superconducting Josephson arrays, Physica D 74(3–4), 197 (1994)

[23] J. Gao, C. Xu, Y. Sun, and Z. Zheng, Order parameter analysis for low-dimensional behaviors of coupled phase oscillators, Sci. Rep. 6, 30184 (2016)

[24] X. Hu, S. Boccaletti, W. Huang, X. Zhang, Z. Liu, S. Guan, and C. H. Lai, Exact solution for first-order synchronization transition in a generalized Kuramoto model, Sci. Rep. 4, 7262 (2014)

[25] I. Leyva, R. Sevilla-Escoboza, J. M. Buldu, I. Sendina-Nadal, J. Gomez-Gardenes, A. Arenas, Y. Moreno, S. Gomez, R. Jaimes-Reategui, and S. Boccaletti, Explosive first-order transition to synchrony in networked chaotic oscillators, Phys. Rev. Lett. 108(16), 168702 (2012)

[26] C. Xu, J. Gao, Y. Sun, X. Huang, and Z. Zheng, Explosive or continuous: Incoherent state determines the route to synchronization, Sci. Rep. 5, 12039 (2015)

[27] P. Li, K. Zhang, X. Xu, J. Zhang, and M. Small, Reexamination of explosive synchronization in scale-free networks: The effect of disassortativity, Phys. Rev. E 87(4), 042803 (2013)

[28] T. K. Peron and F. A. Rodrigues, Explosive synchronization enhanced by time-delayed coupling, Phys. Rev. E 86(1), 016102 (2012)

[29] P. Ji, T. K. Peron, P. J. Menck, F. A. Rodrigues, and J. Kurths, Cluster explosive synchronization in complex networks, Phys. Rev. Lett. 110(21), 218701 (2013)

[30] L. Zhang, J. Chen, B. Sun, Y. Tang, M. Wang, Y. Li, and S. Xue, Nonlinear dynamic evolution and control in a new scale-free networks modeling, Nonlinear Dyn. 76(2), 1569 (2014)

[31] I. Leyva, A. Navas, I. Sendina-Nadal, J. A. Almendral, J. M. Buldu, M. Zanin, D. Papo, and S. Boccaletti, Explosive transitions to synchronization in networks of phase oscillators, Sci. Rep. 3, 1281 (2013)

[32] C. Wang, A. Pumir, N. B. Garnier, and Z. Liu, Explosive synchronization enhances selectivity: Example of the cochlea, Front. Phys. 12(5), 128901 (2017)

[33] A. Bergner, M. Frasca, G. Sciuto, A. Buscarino, E. J. Ngamga, L. Fortuna, and J. Kurths, Remote synchronization in star networks, Phys. Rev. E 85(2), 026208 (2012)

[34] O. Burylko, Y. Kazanovich, and R. Borisyuk, Bifurcations in phase oscillator networks with a central element, Physica D 241(12), 1072 (2012)

[35] S. J. S. Theesar, M. R. K. Ariffin, and S. Banerjee, Synchronization and a secure communication scheme using optical star network, Opt. Laser Technol. 54, 15 (2013)

[36] V. Vlasov, A. Pikovsky, and E. E. N. Macau, Star-type oscillatory networks with generic Kuramoto-type coupling: A model for Japanese drums synchrony, Chaos 25(12), 123120 (2015)

[37] C. Xu, Y. Sun, J. Gao, T. Qiu, Z. Zheng, and S. Guan, Synchronization of phase oscillators with frequencyweighted coupling, Sci. Rep. 6, 21926 (2016)

[38] C. Xu, H. Xiang, J. Gao, and Z. Zheng, Collective dynamics of identical phase oscillators with high-order coupling, Sci. Rep. 6, 31133 (2016)

[39] X. Huang, J. Gao, Y. Sun, Z. Zheng, and C. Xu, Effects of frustration on explosive synchronization, Front. Phys. 11(6), 110504 (2016)

[40] C. J. Goebel, Comment on “Constants of motion for superconductor arrays”, Physica D 80(1–2), 18 (1995)

[41] S. A. Marvel, R. E. Mirollo, and S. H. Strogatz, Identical phase oscillators with global sinusoidal coupling evolve by Möbius group action, Chaos 19(4), 043104 (2009)

[42] H. Sakaguchi and Y. Kuramoto, A soluble active rotater model showing phase transitions via mutual entertainment, Prog. Theor. Phys. 76(3), 576 (1986)

[43] S. A. Marvel and S. H. Strogatz, Invariant submain-fold for series arrays of Josephson junctions, Chaos 19(1), 013132 (2009)

[44] F. Dorfler and F. Bullo, Exploring synchronization in complex oscillator networks, in: Proc. 51st IEEE Conf. Decision Control, pp 7157–7170 (2012)

RIGHTS & PERMISSIONS

2017 Higher Education Press and Springer-Verlag Berlin Heidelberg