March  2020, 15(1): 111-124. doi: 10.3934/nhm.2020005

The General Non-Abelian Kuramoto Model on the 3-sphere

1. 

Faculty of Natural Sciences and Mathematics, University of Montenegro, 81000 Podgorica, Montenegro

2. 

Faculty of Technical Engineering, University of Bihać, 77000 Bihać, Bosnia and Herzegovina

* Corresponding author: Aladin Crnkić

Received  March 2019 Revised  July 2019 Published  December 2019

We introduce non-Abelian Kuramoto model on $ S^3 $ in the most general form. Following an analogy with the classical Kuramoto model (on the circle $ S^1 $), we study some interesting variations of the model on $ S^3 $ that are obtained for particular coupling functions. As a partial case, by choosing "standard" coupling function one obtains a previously known model, that is referred to as Kuramoto-Lohe model on $ S^3 $.

We briefly address two particular models: Kuramoto models on $ S^3 $ with frustration and with external forcing. These models on higher dimensional manifolds have not been studied so far. By choosing suitable values of parameters we observe different nontrivial dynamical regimes even in the simplest setup of globally coupled homogeneous population.

Although non-Abelian Kuramoto models can be introduced on various symmetric spaces, we restrict our analysis to the case when underlying manifold is the 3-sphere. Due to geometric and algebraic properties of this specific manifold, variations of this model are meaningful and geometrically well justified.

Citation: Vladimir Jaćimović, Aladin Crnkić. The General Non-Abelian Kuramoto Model on the 3-sphere. Networks & Heterogeneous Media, 2020, 15 (1) : 111-124. doi: 10.3934/nhm.2020005
References:
[1]

J. A. Acebrón, L. L. Bonilla, C. J. P. Vicente, F. Ritort and R. Spigler, The Kuramoto model: A simple paradigm for synchronization phenomena, Reviews of Modern Physics, 77 (2005), 137. Google Scholar

[2]

T. M. Antonsen, R. T. Faghih, M. Girvan, E. Ott and J. Platig, External periodic driving of large systems of globally coupled phase oscillators, Chaos: An Interdisciplinary Journal of Nonlinear Science, 18 (2008), 037112, 10pp. doi: 10.1063/1.2952447.  Google Scholar

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A. ArenasA. Díaz-GuileraJ. KurthsY. Moreno and C. Zhou, Synchronization in complex networks, Physics Reports, 469 (2008), 93-153.  doi: 10.1016/j.physrep.2008.09.002.  Google Scholar

[4]

M. CaponigroA. C. Lai and B. Piccoli, A nonlinear model of opinion formation on the sphere, Discrete and Continuous Dynamical Systems - A, 35 (2015), 4241-4268.  doi: 10.3934/dcds.2015.35.4241.  Google Scholar

[5]

S. Chandra, M. Girvan and E. Ott, Continuous versus discontinuous transitions in the d-dimensional generalized Kuramoto model: Odd d is different, Physical Review X, 9 (2019), 011002. Google Scholar

[6]

S. Chandra, M. Girvan and E. Ott, Complexity reduction ansatz for systems of interacting orientable agents: Beyond the Kuramoto model, Chaos: An Interdisciplinary Journal of Nonlinear Science, 29 (2019), 053107, 8pp. doi: 10.1063/1.5093038.  Google Scholar

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L. M. Childs and S. H. Strogatz, Stability diagram for the forced Kuramoto model, Chaos: An Interdisciplinary Journal of Nonlinear Science, 18 (2008), 043128, 9pp. doi: 10.1063/1.3049136.  Google Scholar

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L. DeVille, Synchronization and stability for quantum Kuramoto, Journal of Statistical Physics, 174 (2019), 160-187.  doi: 10.1007/s10955-018-2168-9.  Google Scholar

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Z.-M. GuM. ZhaoT. ZhouC.-P. Zhu and B.-H. Wang, Phase synchronization of non-Abelian oscillators on small-world networks, Physics Letters A, 362 (2007), 115-119.  doi: 10.1016/j.physleta.2006.10.010.  Google Scholar

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W. R. Hamilton, On quaternions or a new system of imaginaries in algebra, Philosophical Magazine, 25 (1844), 489-495. Google Scholar

[11]

S. Y. HaY. Kim and Z. Li, Asymptotic synchronous behavior of Kuramoto type models with frustrations, Networks and Heterogeneous Media, 9 (2014), 33-64.  doi: 10.3934/nhm.2014.9.33.  Google Scholar

[12]

S. Y. HaY. Kim and Z. Li, Large-time dynamics of Kuramoto oscillators under the effects of inertia and frustration, SIAM Journal on Applied Dynamical Systems, 13 (2014), 466-492.  doi: 10.1137/130926559.  Google Scholar

[13]

S.-Y. HaD. KoJ. Park and X. Zhang, Collective synchronization of classical and quantum oscillators, EMS Surveys in Mathematical Sciences, 3 (2016), 209-267.  doi: 10.4171/EMSS/17.  Google Scholar

[14]

S.-Y. HaD. Ko and S. W. Ryoo, Emergent dynamics of a generalized Lohe model on some class of Lie groups, Journal of Statistical Physics, 168 (2017), 171-207.  doi: 10.1007/s10955-017-1797-8.  Google Scholar

[15]

S. Y. HaD. Ko and Y. Zhang, Emergence of phase-locking in the Kuramoto model for identical oscillators with frustration, SIAM Journal on Applied Dynamical Systems, 17 (2018), 581-625.  doi: 10.1137/17M1112959.  Google Scholar

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S.-Y. Ha and S. W. Ryoo, On the emergence and orbital stability of phase-locked states for the Lohe model, Journal of Statistical Physics, 163 (2016), 411-439.  doi: 10.1007/s10955-016-1481-4.  Google Scholar

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V. Jaćimović and A. Crnkić, Low-dimensional dynamics in non-Abelian Kuramoto model on the 3-sphere, Chaos: An Interdisciplinary Journal of Nonlinear Science, 28 (2018), 083105, 8pp. doi: 10.1063/1.5029485.  Google Scholar

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D. J. Jörg, L. G. Morelli, S. Ares and F. Jülicher, Synchronization dynamics in the presence of coupling delays and phase shifts, Physical Review Letters, 112 (2014), 174101. Google Scholar

[19]

Y. Kuramoto, Self-entrainment of a population of coupled nonlinear oscillators, Proc. International Symposium on Mathematical Problems in Theoretical Physics, 39 (1975), 420-422.   Google Scholar

[20]

Y. Kuramoto, Cooperative dynamics of oscillator communitya study based on lattice of rings, Progress of Theoretical Physics Supplement, 79 (1984), 223-240.   Google Scholar

[21]

W. S. Lee, E. Ott and T. M. Antonsen, Large coupled oscillator systems with heterogeneous interaction delays, Physical Review Letters, 103 (2009), 044101. doi: 10.1103/PhysRevLett.103.044101.  Google Scholar

[22]

M. Lipton, R. Mirollo and S. H. Strogatz, On Higher Dimensional Generalized Kuramoto Oscillator Systems, arXiv: 1907.07150. Google Scholar

[23]

M. A. Lohe, Higher-dimensional generalizations of the Watanabe-Strogatz transform for vector models of synchronization, Journal of Physics A: Mathematical and Theoretical, 51 (2018), 225101, 24pp. doi: 10.1088/1751-8121/aac030.  Google Scholar

[24]

M. A. Lohe, Non-abelian Kuramoto models and synchronization, Journal of Physics A: Mathematical and Theoretical, 42 (2009), 395101, 25pp. doi: 10.1088/1751-8113/42/39/395101.  Google Scholar

[25]

M. A. Lohe, Quantum synchronization over quantum networks, Journal of Physics A: Mathematical and Theoretical, 43 (2010), 465301, 20pp. doi: 10.1088/1751-8113/43/46/465301.  Google Scholar

[26]

J. MarkdahlJ. Thunberg and J. Gonçalves, Almost global consensus on the $n$-sphere, IEEE Transactions on Automatic Control, 63 (2018), 1664-1675.  doi: 10.1109/TAC.2017.2752799.  Google Scholar

[27]

D. Métivier and S. Gupta, Bifurcations in the time-delayed Kuramoto model of coupled oscillators: Exact results, Journal of Statistical Physics, 176 (2019), 279-298.  doi: 10.1007/s10955-019-02299-z.  Google Scholar

[28]

B. Niu and Y. Guo, Bifurcation analysis on the globally coupled Kuramoto oscillators with distributed time delays, Physica D: Nonlinear Phenomena, 266 (2014), 23-33.  doi: 10.1016/j.physd.2013.10.003.  Google Scholar

[29]

B. NiuY. Guo and W. Jiang, An approach to normal forms of Kuramoto model with distributed delays and the effect of minimal delay, Physics Letters A, 379 (2015), 2018-2024.  doi: 10.1016/j.physleta.2015.06.028.  Google Scholar

[30]

B. Niu, J. Zhang and J. Wei, Multiple-parameter bifurcation analysis in a Kuramoto model with time delay and distributed shear, AIP Advances, 8 (2018), 055111. doi: 10.1063/1.5029512.  Google Scholar

[31]

R. Olfati-Saber, Swarms on sphere: A programmable swarm with synchronous behaviors like oscillator networks, in Proc. 45th IEEE Conf. Decision and Control, (2006), 5060–5066. doi: 10.1109/CDC.2006.376811.  Google Scholar

[32]

E. Omel'chenko and M. Wolfrum, Bifurcations in the Sakaguchi–Kuramoto model, Physica D: Nonlinear Phenomena, 263 (2013), 74-85.  doi: 10.1016/j.physd.2013.08.004.  Google Scholar

[33] A. PikovskyM. Rosenblum and J. Kurths, Synchronization: A Universal Concept in Nonlinear Sciences, Cambridge University Press, Cambridge, 2001.  doi: 10.1017/CBO9780511755743.  Google Scholar
[34]

F. A. RodriguesT. K. D. PeronP. Ji and J. Kurths, The Kuramoto model in complex networks, Physics Reports, 610 (2016), 1-98.  doi: 10.1016/j.physrep.2015.10.008.  Google Scholar

[35]

M. Rosenblum and A. Pikovsky, Delayed feedback control of collective synchrony: An approach to suppression of pathological brain rhythms, Physical Review E, 70 (2004), 041904, 11pp. doi: 10.1103/PhysRevE.70.041904.  Google Scholar

[36]

H. Sakaguchi, Cooperative phenomena in coupled oscillator systems under external fields, Progress of Theoretical Physics, 79 (1988), 39-46.  doi: 10.1143/PTP.79.39.  Google Scholar

[37]

H. Sakaguchi and Y. Kuramoto, A soluble active rotater model showing phase transitions via mutual entertainment, Progress of Theoretical Physics, 76 (1986), 576-581.  doi: 10.1143/PTP.76.576.  Google Scholar

[38]

A. Sarlette and R. Sepulchre, Consensus optimization on manifolds, SIAM Journal on Control and Optimization, 48 (2009), 56-76.  doi: 10.1137/060673400.  Google Scholar

[39]

A. Sarlette, Geometry and Symmetries in Coordination Control, Ph.D. thesis, Université de Liège, 2009. Google Scholar

[40]

H. G. Schuster and P. Wagner, Mutual entrainment of two limit cycle oscillators with time delayed coupling, Progress of Theoretical Physics, 81 (1989), 939-945.  doi: 10.1143/PTP.81.939.  Google Scholar

[41]

S. Shinomoto and Y. Kuramoto, Phase transitions in active rotator systems, Progress of Theoretical Physics, 75 (1986), 1105-1110.  doi: 10.1143/PTP.75.1105.  Google Scholar

[42]

S. H. Strogatz, From Kuramoto to Crawford: Exploring the onset of synchronization in populations of coupled oscillators, Physica D: Nonlinear Phenomena, 143 (2000), 1-20.  doi: 10.1016/S0167-2789(00)00094-4.  Google Scholar

[43]

T. Tanaka, Solvable model of the collective motion of heterogeneous particles interacting on a sphere, New Journal of Physics, 16 (2014), 023016. doi: 10.1088/1367-2630/16/2/023016.  Google Scholar

[44]

M. K. S. Yeung and S. H. Strogatz, Time delay in the Kuramoto model of coupled oscillators, Physical Review Letters, 82 (1999), 648. Google Scholar

show all references

References:
[1]

J. A. Acebrón, L. L. Bonilla, C. J. P. Vicente, F. Ritort and R. Spigler, The Kuramoto model: A simple paradigm for synchronization phenomena, Reviews of Modern Physics, 77 (2005), 137. Google Scholar

[2]

T. M. Antonsen, R. T. Faghih, M. Girvan, E. Ott and J. Platig, External periodic driving of large systems of globally coupled phase oscillators, Chaos: An Interdisciplinary Journal of Nonlinear Science, 18 (2008), 037112, 10pp. doi: 10.1063/1.2952447.  Google Scholar

[3]

A. ArenasA. Díaz-GuileraJ. KurthsY. Moreno and C. Zhou, Synchronization in complex networks, Physics Reports, 469 (2008), 93-153.  doi: 10.1016/j.physrep.2008.09.002.  Google Scholar

[4]

M. CaponigroA. C. Lai and B. Piccoli, A nonlinear model of opinion formation on the sphere, Discrete and Continuous Dynamical Systems - A, 35 (2015), 4241-4268.  doi: 10.3934/dcds.2015.35.4241.  Google Scholar

[5]

S. Chandra, M. Girvan and E. Ott, Continuous versus discontinuous transitions in the d-dimensional generalized Kuramoto model: Odd d is different, Physical Review X, 9 (2019), 011002. Google Scholar

[6]

S. Chandra, M. Girvan and E. Ott, Complexity reduction ansatz for systems of interacting orientable agents: Beyond the Kuramoto model, Chaos: An Interdisciplinary Journal of Nonlinear Science, 29 (2019), 053107, 8pp. doi: 10.1063/1.5093038.  Google Scholar

[7]

L. M. Childs and S. H. Strogatz, Stability diagram for the forced Kuramoto model, Chaos: An Interdisciplinary Journal of Nonlinear Science, 18 (2008), 043128, 9pp. doi: 10.1063/1.3049136.  Google Scholar

[8]

L. DeVille, Synchronization and stability for quantum Kuramoto, Journal of Statistical Physics, 174 (2019), 160-187.  doi: 10.1007/s10955-018-2168-9.  Google Scholar

[9]

Z.-M. GuM. ZhaoT. ZhouC.-P. Zhu and B.-H. Wang, Phase synchronization of non-Abelian oscillators on small-world networks, Physics Letters A, 362 (2007), 115-119.  doi: 10.1016/j.physleta.2006.10.010.  Google Scholar

[10]

W. R. Hamilton, On quaternions or a new system of imaginaries in algebra, Philosophical Magazine, 25 (1844), 489-495. Google Scholar

[11]

S. Y. HaY. Kim and Z. Li, Asymptotic synchronous behavior of Kuramoto type models with frustrations, Networks and Heterogeneous Media, 9 (2014), 33-64.  doi: 10.3934/nhm.2014.9.33.  Google Scholar

[12]

S. Y. HaY. Kim and Z. Li, Large-time dynamics of Kuramoto oscillators under the effects of inertia and frustration, SIAM Journal on Applied Dynamical Systems, 13 (2014), 466-492.  doi: 10.1137/130926559.  Google Scholar

[13]

S.-Y. HaD. KoJ. Park and X. Zhang, Collective synchronization of classical and quantum oscillators, EMS Surveys in Mathematical Sciences, 3 (2016), 209-267.  doi: 10.4171/EMSS/17.  Google Scholar

[14]

S.-Y. HaD. Ko and S. W. Ryoo, Emergent dynamics of a generalized Lohe model on some class of Lie groups, Journal of Statistical Physics, 168 (2017), 171-207.  doi: 10.1007/s10955-017-1797-8.  Google Scholar

[15]

S. Y. HaD. Ko and Y. Zhang, Emergence of phase-locking in the Kuramoto model for identical oscillators with frustration, SIAM Journal on Applied Dynamical Systems, 17 (2018), 581-625.  doi: 10.1137/17M1112959.  Google Scholar

[16]

S.-Y. Ha and S. W. Ryoo, On the emergence and orbital stability of phase-locked states for the Lohe model, Journal of Statistical Physics, 163 (2016), 411-439.  doi: 10.1007/s10955-016-1481-4.  Google Scholar

[17]

V. Jaćimović and A. Crnkić, Low-dimensional dynamics in non-Abelian Kuramoto model on the 3-sphere, Chaos: An Interdisciplinary Journal of Nonlinear Science, 28 (2018), 083105, 8pp. doi: 10.1063/1.5029485.  Google Scholar

[18]

D. J. Jörg, L. G. Morelli, S. Ares and F. Jülicher, Synchronization dynamics in the presence of coupling delays and phase shifts, Physical Review Letters, 112 (2014), 174101. Google Scholar

[19]

Y. Kuramoto, Self-entrainment of a population of coupled nonlinear oscillators, Proc. International Symposium on Mathematical Problems in Theoretical Physics, 39 (1975), 420-422.   Google Scholar

[20]

Y. Kuramoto, Cooperative dynamics of oscillator communitya study based on lattice of rings, Progress of Theoretical Physics Supplement, 79 (1984), 223-240.   Google Scholar

[21]

W. S. Lee, E. Ott and T. M. Antonsen, Large coupled oscillator systems with heterogeneous interaction delays, Physical Review Letters, 103 (2009), 044101. doi: 10.1103/PhysRevLett.103.044101.  Google Scholar

[22]

M. Lipton, R. Mirollo and S. H. Strogatz, On Higher Dimensional Generalized Kuramoto Oscillator Systems, arXiv: 1907.07150. Google Scholar

[23]

M. A. Lohe, Higher-dimensional generalizations of the Watanabe-Strogatz transform for vector models of synchronization, Journal of Physics A: Mathematical and Theoretical, 51 (2018), 225101, 24pp. doi: 10.1088/1751-8121/aac030.  Google Scholar

[24]

M. A. Lohe, Non-abelian Kuramoto models and synchronization, Journal of Physics A: Mathematical and Theoretical, 42 (2009), 395101, 25pp. doi: 10.1088/1751-8113/42/39/395101.  Google Scholar

[25]

M. A. Lohe, Quantum synchronization over quantum networks, Journal of Physics A: Mathematical and Theoretical, 43 (2010), 465301, 20pp. doi: 10.1088/1751-8113/43/46/465301.  Google Scholar

[26]

J. MarkdahlJ. Thunberg and J. Gonçalves, Almost global consensus on the $n$-sphere, IEEE Transactions on Automatic Control, 63 (2018), 1664-1675.  doi: 10.1109/TAC.2017.2752799.  Google Scholar

[27]

D. Métivier and S. Gupta, Bifurcations in the time-delayed Kuramoto model of coupled oscillators: Exact results, Journal of Statistical Physics, 176 (2019), 279-298.  doi: 10.1007/s10955-019-02299-z.  Google Scholar

[28]

B. Niu and Y. Guo, Bifurcation analysis on the globally coupled Kuramoto oscillators with distributed time delays, Physica D: Nonlinear Phenomena, 266 (2014), 23-33.  doi: 10.1016/j.physd.2013.10.003.  Google Scholar

[29]

B. NiuY. Guo and W. Jiang, An approach to normal forms of Kuramoto model with distributed delays and the effect of minimal delay, Physics Letters A, 379 (2015), 2018-2024.  doi: 10.1016/j.physleta.2015.06.028.  Google Scholar

[30]

B. Niu, J. Zhang and J. Wei, Multiple-parameter bifurcation analysis in a Kuramoto model with time delay and distributed shear, AIP Advances, 8 (2018), 055111. doi: 10.1063/1.5029512.  Google Scholar

[31]

R. Olfati-Saber, Swarms on sphere: A programmable swarm with synchronous behaviors like oscillator networks, in Proc. 45th IEEE Conf. Decision and Control, (2006), 5060–5066. doi: 10.1109/CDC.2006.376811.  Google Scholar

[32]

E. Omel'chenko and M. Wolfrum, Bifurcations in the Sakaguchi–Kuramoto model, Physica D: Nonlinear Phenomena, 263 (2013), 74-85.  doi: 10.1016/j.physd.2013.08.004.  Google Scholar

[33] A. PikovskyM. Rosenblum and J. Kurths, Synchronization: A Universal Concept in Nonlinear Sciences, Cambridge University Press, Cambridge, 2001.  doi: 10.1017/CBO9780511755743.  Google Scholar
[34]

F. A. RodriguesT. K. D. PeronP. Ji and J. Kurths, The Kuramoto model in complex networks, Physics Reports, 610 (2016), 1-98.  doi: 10.1016/j.physrep.2015.10.008.  Google Scholar

[35]

M. Rosenblum and A. Pikovsky, Delayed feedback control of collective synchrony: An approach to suppression of pathological brain rhythms, Physical Review E, 70 (2004), 041904, 11pp. doi: 10.1103/PhysRevE.70.041904.  Google Scholar

[36]

H. Sakaguchi, Cooperative phenomena in coupled oscillator systems under external fields, Progress of Theoretical Physics, 79 (1988), 39-46.  doi: 10.1143/PTP.79.39.  Google Scholar

[37]

H. Sakaguchi and Y. Kuramoto, A soluble active rotater model showing phase transitions via mutual entertainment, Progress of Theoretical Physics, 76 (1986), 576-581.  doi: 10.1143/PTP.76.576.  Google Scholar

[38]

A. Sarlette and R. Sepulchre, Consensus optimization on manifolds, SIAM Journal on Control and Optimization, 48 (2009), 56-76.  doi: 10.1137/060673400.  Google Scholar

[39]

A. Sarlette, Geometry and Symmetries in Coordination Control, Ph.D. thesis, Université de Liège, 2009. Google Scholar

[40]

H. G. Schuster and P. Wagner, Mutual entrainment of two limit cycle oscillators with time delayed coupling, Progress of Theoretical Physics, 81 (1989), 939-945.  doi: 10.1143/PTP.81.939.  Google Scholar

[41]

S. Shinomoto and Y. Kuramoto, Phase transitions in active rotator systems, Progress of Theoretical Physics, 75 (1986), 1105-1110.  doi: 10.1143/PTP.75.1105.  Google Scholar

[42]

S. H. Strogatz, From Kuramoto to Crawford: Exploring the onset of synchronization in populations of coupled oscillators, Physica D: Nonlinear Phenomena, 143 (2000), 1-20.  doi: 10.1016/S0167-2789(00)00094-4.  Google Scholar

[43]

T. Tanaka, Solvable model of the collective motion of heterogeneous particles interacting on a sphere, New Journal of Physics, 16 (2014), 023016. doi: 10.1088/1367-2630/16/2/023016.  Google Scholar

[44]

M. K. S. Yeung and S. H. Strogatz, Time delay in the Kuramoto model of coupled oscillators, Physical Review Letters, 82 (1999), 648. Google Scholar

Figure 1.  The population of $ N = 50 $ KL oscillators with intrinsic frequencies $ w_l = (0.3,1.3,0.9) $, $ u_l = (1.7,0.5,1.4) $ and phase shifts $ \alpha = \frac{\pi}{3}, \, \beta = \frac{\pi}{4}, \, \gamma = \frac{\pi}{3} $ achieves coherent state: (a) evolution of the order parameters (thick line for the global order parameter $ r $ and dashed and dotted lines for $ r_\varphi $ and $ r_\psi $ respectively) and (b) cosines of the angles between some pairs of KL oscillators. Initial conditions are sampled from the von Mises-Fisher on $ S^{3} $ with mean direction $ \mu = (0.5, 0.5, 0.5, 0.5) $ and the concentration $ \kappa = 2.5 $
Figure 2.  The population of $ N = 50 $ KL oscillators with intrinsic frequencies $ w_l = (0.3,1.3,0.9) $, $ u_l = (1.7,0.5,1.4) $ and phase shifts $ \alpha = \frac{2\pi}{3}, \, \beta = \frac{\pi}{4}, \, \gamma = \frac{\pi}{3} $ achieves a fully incoherent state: (a) evolution of the order parameters (thick line for the global order parameter $ r $ and dashed and dotted lines for $ r_\varphi $ and $ r_\psi $ respectively) and (b) cosines of the angles between some pairs of KL oscillators. Initial conditions are sampled from the von Mises-Fisher on $ S^{3} $ with mean direction $ \mu = (0.5, 0.5, 0.5, 0.5) $ and the concentration $ \kappa = 2.5 $
Figure 3.  Oscillations of the system with $ N = 50 $ KL oscillators with intrinsic frequencies $ w_l = (0.3,1.3,0.9) $, $ u_l = (1.7,0.5,1.4) $ and phase shifts $ \alpha = \frac{\pi}{2}, \, \beta = \frac{\pi}{4}, \, \gamma = \frac{\pi}{3} $: (a) evolution of the order parameters (thick line for the global order parameter $ r $ and dashed and dotted lines for $ r_\varphi $ and $ r_\psi $ respectively) and (b) cosines of the angles between some pairs of KL oscillators. Initial conditions are sampled from the von Mises-Fisher on $ S^{3} $ with mean direction $ \mu = (0.5, 0.5, 0.5, 0.5) $ and the concentration $ \kappa = 2.5 $
Figure 4.  Evolution of the global and angular order parameters (thick line for the global order parameter $ r $ and dashed and dotted lines for $ r_\varphi $ and $ r_\psi $ respectively) in the population of $ N = 50 $ nonidentical oscillators with the external forcing. The global coupling strength is set at $ K = 1 $, the frequencies of the external signal are $ p = 0 $, $ v = (1.2,2.3,3.7) $ and the intensity of external forcing is (a) $ D = 0.5+0.5\mathrm{\textbf{j}} $; (b) $ D = 0.9+0.5\mathrm{\textbf{j}} $ and (c) $ D = 1.2+0.5\mathrm{\textbf{j}} $. The right intrinsic frequencies $ u_l $ are zero, and the left intrinsic frequencies are sampled from the 3-dimensional Gaussian distribution with the expectation vector $ \left(1,2,3\right) $ and covariance matrix $ \left(\left(2,-1/4,1/3\right),\left(-1/4,2/3,1/5\right),\left(1/3,1/5,1/2\right)\right) $
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