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 and 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.

[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.

[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.

[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.

[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.

[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.

[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.

[8]

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

[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.

[10]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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. 

[20]

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

[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.

[22]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

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

[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.

[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.

[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.

[38]

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

[39]

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

[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.

[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.

[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.

[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.

[44]

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

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.

[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.

[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.

[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.

[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.

[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.

[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.

[8]

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

[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.

[10]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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. 

[20]

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

[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.

[22]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

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

[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.

[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.

[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.

[38]

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

[39]

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

[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.

[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.

[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.

[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.

[44]

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

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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