We study synchronization in a system of Stuart–Landau oscillators with frequency-weighted coupling. For three typical unimodal frequency distributions, namely, the Lorentzian, the triangle, and the uniform, we found that the first-order transition occurs when the frequency distribution is relatively compact, while the synchronization transition is continuous when it is relatively wide. In both cases, there is a regime of Bellerophon state between the incoherent state and the synchronized state. Remarkably, we revealed novel transition behavior for such coupled oscillators with amplitudes, i.e., the regime of Bellerophon state actually contains two stages. In the first stage, the oscillators achieve chaotic phase synchronization; while in the second stage, oscillators form periodical phase synchronization. Our results suggest that Bellerophon state also exists in coupled oscillators with amplitude dynamics.
A. Pikvosky, M. Rosenblum, and J. Kurths, Synchronization: A Universal Concept in Nonlinear Sciences, Cambridge: Cambridge University Press, 2003 https://doi.org/10.1007/978-94-010-0217-2
2
J. Buck, Synchronous rhythmic flashing of fireflies (II), Q. Rev. Biol. 63(3), 265 (1988) https://doi.org/10.1086/415929
M. Rohden, A. Sorge, M. Timme, and D. Witthaut, Selforganized synchronization in decentralized power grids, Phys. Rev. Lett. 109(6), 064101 (2012) https://doi.org/10.1103/PhysRevLett.109.064101
6
K. Wiesenfeld, P. Colet, and S. H. Strogatz, Synchronization transitions in a disordered Josephson series array, Phys. Rev. Lett. 76(3), 404 (1996) https://doi.org/10.1103/PhysRevLett.76.404
7
B. Eckhardt, E. Ott, S. H. Strogatz, D. M. Abrams, and A. McRobie, Modeling walker synchronization on the Millennium Bridge, Phys. Rev. E 75(2), 021110 (2007) https://doi.org/10.1103/PhysRevE.75.021110
8
H. Chen, Y. Sun, J. Gao, C. Xu, and Z. Zheng, Order parameter analysis of synchronization transitions on star networks, Front. Phys. 12(6), 120504 (2017) https://doi.org/10.1007/s11467-017-0651-4
9
D. Yuan, J. L. Tian, F. Lin, D. W. Ma, J. Zhang, H. T. Cui, and Y. Xiao, Periodic synchronization in a system of coupled phase oscillators with attractive and repulsive interactions, Front. Phys. 13(3), 130504 (2018) https://doi.org/10.1007/s11467-018-0748-4
10
X. Huang, J. Dong, W. Jia, Z. Zheng, and C. Xu, Dynamics of clustering patterns in the Kuramoto model with unidirectional coupling, Front. Phys. 13(5), 130506 (2018) https://doi.org/10.1007/s11467-018-0783-1
S. H. Strogatz, From Kuramoto to Crawford: Exploring the onset of synchronization in populations of coupled oscillators, Physica D 143(1–4), 1 (2000) https://doi.org/10.1016/S0167-2789(00)00094-4
H. Hong and S. H. Strogatz, Kuramoto model of coupled oscillators with positive and negative coupling parameters: An example of conformist and contrarian oscillators, Phys. Rev. Lett. 106(5), 054102 (2011) https://doi.org/10.1103/PhysRevLett.106.054102
15
D. Iatsenko, S. Petkoski, P. V. E. McClintock, and A. Stefanovska, Stationary and traveling wave states of the Kuramoto model with an arbitrary distribution of frequencies and coupling strengths, Phys. Rev. Lett. 110(6), 064101 (2013) https://doi.org/10.1103/PhysRevLett.110.064101
16
J. D. Crawford, Amplitude expansions for instabilities in populations of globally-coupled oscillators, J. Stat. Phys. 74(5), 1047 (1994) https://doi.org/10.1007/BF02188217
17
Y. Kuramoto and D. Battogtokh, Coexistence of coherence and incoherence in nonlocally coupled phase oscillators, arXiv: cond-mat/0210694 (2002)
18
I. Omelchenko, O. E. Omel’chenko, P. Hövel, and E. Schöll, When nonlocal coupling between oscillators becomes stronger: Patched synchrony or multichimera states, Phys. Rev. Lett. 110(22), 224101 (2013) https://doi.org/10.1103/PhysRevLett.110.224101
19
D. M. Abrams, R. Mirollo, S. H. Strogatz, and D. A. Wiley, Solvable model for chimera states of coupled oscillators, Phys. Rev. Lett. 101(8), 084103 (2008) https://doi.org/10.1103/PhysRevLett.101.084103
20
H. Wang and X. Li, Synchronization and chimera states of frequency-weighted Kuramoto-oscillator networks, Phys. Rev. E 83(6), 066214 (2011) https://doi.org/10.1103/PhysRevE.83.066214
21
S. Boccaletti, J. A. Almendral, S. Guan, I. Leyva, Z. Liu, I. Sendiña-Nadal, Z. Wang, and Y. Zou, Explosive transitions in complex networks’ structure and dynamics: Percolation and synchronization, Phys. Rep. 660, 1 (2016) https://doi.org/10.1016/j.physrep.2016.10.004
22
H. Bi, X. Hu, S. Boccaletti, X. Wang, Y. Zou, Z. Liu, and S. Guan, Coexistence of quantized, time dependent, clusters in globally coupled oscillators, Phys. Rev. Lett. 117(20), 204101 (2016) https://doi.org/10.1103/PhysRevLett.117.204101
23
W. Zhou, Y. Zou, J. Zhou, Z. Liu, and S. Guan, Intermittent Bellerophon state in frequency-weighted Kuramoto model, Chaos 26(12), 123117 (2016) https://doi.org/10.1063/1.4972117
24
H. Bi, Y. Li, L. Zhou, and S. Guan, Nontrivial standing wave state in frequency-weighted Kuramoto model, Front. Phys. 12(3), 126801 (2017) https://doi.org/10.1007/s11467-017-0672-z
25
Y. Xiao, W. Jia, C. Xu, H. Lü, and Z. Zheng, Synchronization of phase oscillators in the generalized Sakaguchi-Kuramoto model, Europhys. Lett. 118(6), 60005 (2017) https://doi.org/10.1209/0295-5075/118/60005
26
C. Xu, S. Boccaletti, S. Guan, and Z. Zheng, Origin of Bellerophon states in globally coupled phase oscillators, Phys. Rev. E 98, 050202(R) (2018)
27
D. Yuan, M. Zhang, and J. Yang, Dynamics of the Kuramoto model in the presence of correlation between distributions of frequencies and coupling strengths, Phys. Rev. E 89(1), 012910 (2014) https://doi.org/10.1103/PhysRevE.89.012910
28
T. Qiu, S. Boccaletti, I. Bonamassa, Y. Zou, J. Zhou, Z. Liu, and S. Guan, Synchronization and Bellerophon states in conformist and contrarian oscillators, Sci. Rep. 6(1), 36713 (2016) https://doi.org/10.1038/srep36713
29
T. Qiu, I. Bonamassa, S. Boccaletti, Z. Liu, and S. Guan, Rhythmic synchronization and hybrid collective states of globally coupled oscillators, Sci. Rep. 8(1), 12950 (2018) https://doi.org/10.1038/s41598-018-31278-9
30
X. Li, T. Qiu, S. Boccaletti, I. Sendiña-Nadal, Z. Liu, and S. Guan, Synchronization clusters with quantum traits emerge as the result of a global coupling among classical phase oscillators (submitted)
