September  2020, 15(3): 427-461. doi: 10.3934/nhm.2020026

Nonlinear stability of stationary solutions to the Kuramoto-Sakaguchi equation with frustration

1. 

Department of Mathematical Sciences and Research Institute of Mathematics, Seoul National University, Seoul 08826, Republic of Korea

2. 

Korea Institute for Advanced Study, Hoegiro 85, Seoul, 02455, Republic of Korea

3. 

Department of Mathematical Sciences, Seoul National University, Seoul 08826, Republic of Korea

4. 

Stochastic Analysis and Application Research Center, Korea Advanced Institute of Science and Technology, Daejeon 34141, Republic of Korea

Received  August 2019 Revised  May 2020 Published  September 2020 Early access  September 2020

Fund Project: The work of S.-Y. Ha is supported by the National Research Foundation of Korea(NRF) Grant (No. NRF-2020R1A2C3A01003881). The work of Y. Zhang is supported by the National Research Foundation of Korea(NRF) grant funded by the Korea government(MSIT) (No. NRF-2017R1A5A1015626 and NRF-2019R1A5A1028324)

We study measurable stationary solutions for the kinetic Kuramoto-Sakaguchi (in short K-S) equation with frustration and their stability analysis. In the presence of frustration, the total phase is not a conserved quantity anymore, but it is time-varying. Thus, we can not expect the genuinely stationary solutions for the K-S equation. To overcome this lack of conserved quantity, we introduce new variables whose total phase is conserved. In the transformed K-S equation in new variables, we derive all measurable stationary solution representing the incoherent state, complete and partial phase-locked states. We also provide several frameworks in which the complete phase-locked state is stable, whereas partial phase-locked state is semi-stable in the space of Radon measures. In particular, we show that the incoherent state is nonlinearly stable in a large frustration regime, whereas it can exhibit stable behavior or concentration phenomenon in a small frustration regime.

Citation: Seung-Yeal Ha, Hansol Park, Yinglong Zhang. Nonlinear stability of stationary solutions to the Kuramoto-Sakaguchi equation with frustration. Networks and Heterogeneous Media, 2020, 15 (3) : 427-461. doi: 10.3934/nhm.2020026
References:
[1]

J. A. AcebrónL. L. BonillaC. J. Pérez VicenteF. Ritort and R. Spigler, The Kuramoto model: A simple paradigm for synchronization phenomena, Rev. Mod. Phys., 77 (2005), 137-185.  doi: 10.1103/RevModPhys.77.137.

[2]

M. Brede and A. C. Kalloniatis, Frustration tuning and perfect phase synchronization in the Kuramoto-Sakaguchi model, Phys. Rev. E, 93 (2016), 13pp. doi: 10.1103/physreve.93.062315.

[3]

J. A. CarrilloY.-P. ChoiS.-Y. HaM.-J. Kang and Y. Kim, Contractivity of transport distances for the kinetic Kuramoto equation, J. Stat. Phys., 156 (2014), 395-415.  doi: 10.1007/s10955-014-1005-z.

[4]

H. Chiba, A proof of the Kuramoto conjecture for a bifurcation structure of the infinite-dimensional Kuramoto model, Ergodic Theory Dynam. Systems, 35 (2015), 762-834.  doi: 10.1017/etds.2013.68.

[5]

N. Chopra and M. W. Spong, On exponential synchronization of Kuramoto oscillators, IEEE Trans. Automat. Control, 54 (2009), 353-357.  doi: 10.1109/TAC.2008.2007884.

[6]

H. Daido, Quasientrainment and slow relaxation in a population of oscillators with random and frustrated interactions, Phys. Rev. Lett., 68 (1992), 1073-1076.  doi: 10.1103/PhysRevLett.68.1073.

[7]

F. De Smet and D. Aeyels, Partial entrainment in the finite Kuramoto-Sakaguchi model, Phys. D, 234 (2007), 81-89.  doi: 10.1016/j.physd.2007.06.025.

[8]

J.-G. Dong and X. Xue, Synchronization analysis of Kuramoto oscillators, Commun. Math. Sci., 11 (2013), 465-480.  doi: 10.4310/CMS.2013.v11.n2.a7.

[9]

F. Dörfler and F. Bullo, Synchronization and transient stability in power networks and nonuniform Kuramoto oscillators, SIAM J. Control Optim., 50 (2012), 1616-1642.  doi: 10.1137/110851584.

[10]

F. Dörfler and F. Bullo, Synchronization in complex networks of phase oscillators: A survey, Automatica J. IFAC, 50 (2014), 1539-1564.  doi: 10.1016/j.automatica.2014.04.012.

[11]

S.-Y. Ha, D. Kim, J. Lee and Y. Zhang, Remarks on the stability properties of the Kuramoto-Sakaguchi-Fokker-Planck equation with frustration, Z. Angew. Math. Phys., 69 (2018), 25pp. doi: 10.1007/s00033-018-0984-z.

[12]

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

[13]

S.-Y. HaY. Kim and Z. Li, Large-time dynamics of Kuramoto oscillators under the effects of inertia and frustration, SIAM J. Appl. Dyn. Syst., 13 (2014), 466-492.  doi: 10.1137/130926559.

[14]

S.-Y. HaH. K. Kim and J. Park, Remarks on the complete synchronization of Kuramoto oscillators, Nonlinearity, 28 (2015), 1441-1462.  doi: 10.1088/0951-7715/28/5/1441.

[15]

S.-Y. HaH. K. Kim and S. W. Ryoo, Emergence of phase-locked states for the Kuramoto model in a large coupling regime, Commun. Math. Sci., 14 (2016), 1073-1091.  doi: 10.4310/CMS.2016.v14.n4.a10.

[16]

S.-Y. HaD. Ko and Y. Zhang, Emergence of phase-locking in the Kuramoto model for identical oscillators with frustration, SIAM J. Appl. Dyn. Syst., 17 (2018), 581-625.  doi: 10.1137/17M1112959.

[17]

S.-Y. HaZ. Li and X. Xue, Formation of phase-locked states in a population of locally interacting Kuramoto oscillators, J. Differential Equations, 255 (2013), 3053-3070.  doi: 10.1016/j.jde.2013.07.013.

[18]

S.-Y. HaJ. Li and Y. Zhang, Robustness in the instability of the incoherent state for the Kuramoto-Sakaguchi-Fokker-Planck equation with frustration, Quart. Appl. Math., 77 (2019), 631-654.  doi: 10.1090/qam/1533.

[19]

S.-Y. Ha and Q. Xiao, Remarks on the nonlinear stability of the Kuramoto-Sakaguchi equation, J. Differential Equations, 259 (2015), 2430-2457.  doi: 10.1016/j.jde.2015.03.038.

[20]

Y. Kuramoto, Chemical Oscillations, Waves and Turbulence, Springer Series in Synergetics, 19, Springer-Verlag, Berlin, 1984. doi: 10.1007/978-3-642-69689-3.

[21]

Y. Kuramoto, International Symposium on Mathematical Problems in Mathematical Physics, Lecture Notes in Physics, 39, Springer-Verlag, Berlin, 1975.

[22]

Z. Levnajić, Emergent multistability and frustration in phase-repulsive networks of oscillators, Phys. Rev. E, 84 (2011). doi: 10.1103/PhysRevE.84.016231.

[23]

Z. Li and S.-Y. Ha, Uniqueness and well-ordering of emergent phase-locked states for the Kuramoto model with frustration and inertia, Math. Models Methods Appl. Sci., 26 (2016), 357-382.  doi: 10.1142/S0218202516400054.

[24]

R. Mirollo and S. H. Strogatz, The spectrum of the partially locked state for the Kuramoto model, J. Nonlinear Sci., 17 (2017), 309-347.  doi: 10.1007/s00332-006-0806-x.

[25]

E. Oh, C. Choi, B. Kahng and D. Kim, Modular synchronization in complex networks with a gauge Kuramoto model, EPL, 83 (2008). doi: 10.1209/0295-5075/83/68003.

[26]

K. ParkS. W. Rhee and M. Y. Choi, Glass synchronization in the network of oscillators with random phase shifts, Phys. Rev. E, 57 (1998), 5030-5035.  doi: 10.1103/PhysRevE.57.5030.

[27] A. PikovskyM. Rosenblum and J. Kurths, Synchronization. A Universal Concept in Nonlinear Sciences, Cambridge Nonlinear Science Series, 12, Cambridge University Press, Cambridge, 2001.  doi: 10.1017/CBO9780511755743.
[28]

H. Sakaguchi, Cooperative phenomena in coupled oscillator systems sunder external fields, Prog. Theoret. Phys., 79 (1988), 39-46.  doi: 10.1143/PTP.79.39.

[29]

H. Sakaguchi and Y. Kuramoto, A soluble active rotator model showing phase transitions via mutual entraintment, Progr. Theoret. Phys., 76 (1986), 576-581.  doi: 10.1143/PTP.76.576.

[30]

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

[31]

S. H. Strogatz and R. E. Mirollo, Stability of incoherence in a population of coupled oscillators, J. Statist. Phys., 63 (1991), 613-635.  doi: 10.1007/BF01029202.

[32]

A. T. Winfree, Biological rhythms and the behavior of populations of coupled oscillators, J. Theor. Biol., 16 (1967), 15-42.  doi: 10.1016/0022-5193(67)90051-3.

[33]

Z.-G. Zheng, Frustration effect on synchronization and chaos in coupled oscillators, Chin. Phys. Soc., 10 (2001), 703-707.  doi: 10.1088/1009-1963/10/8/306.

show all references

References:
[1]

J. A. AcebrónL. L. BonillaC. J. Pérez VicenteF. Ritort and R. Spigler, The Kuramoto model: A simple paradigm for synchronization phenomena, Rev. Mod. Phys., 77 (2005), 137-185.  doi: 10.1103/RevModPhys.77.137.

[2]

M. Brede and A. C. Kalloniatis, Frustration tuning and perfect phase synchronization in the Kuramoto-Sakaguchi model, Phys. Rev. E, 93 (2016), 13pp. doi: 10.1103/physreve.93.062315.

[3]

J. A. CarrilloY.-P. ChoiS.-Y. HaM.-J. Kang and Y. Kim, Contractivity of transport distances for the kinetic Kuramoto equation, J. Stat. Phys., 156 (2014), 395-415.  doi: 10.1007/s10955-014-1005-z.

[4]

H. Chiba, A proof of the Kuramoto conjecture for a bifurcation structure of the infinite-dimensional Kuramoto model, Ergodic Theory Dynam. Systems, 35 (2015), 762-834.  doi: 10.1017/etds.2013.68.

[5]

N. Chopra and M. W. Spong, On exponential synchronization of Kuramoto oscillators, IEEE Trans. Automat. Control, 54 (2009), 353-357.  doi: 10.1109/TAC.2008.2007884.

[6]

H. Daido, Quasientrainment and slow relaxation in a population of oscillators with random and frustrated interactions, Phys. Rev. Lett., 68 (1992), 1073-1076.  doi: 10.1103/PhysRevLett.68.1073.

[7]

F. De Smet and D. Aeyels, Partial entrainment in the finite Kuramoto-Sakaguchi model, Phys. D, 234 (2007), 81-89.  doi: 10.1016/j.physd.2007.06.025.

[8]

J.-G. Dong and X. Xue, Synchronization analysis of Kuramoto oscillators, Commun. Math. Sci., 11 (2013), 465-480.  doi: 10.4310/CMS.2013.v11.n2.a7.

[9]

F. Dörfler and F. Bullo, Synchronization and transient stability in power networks and nonuniform Kuramoto oscillators, SIAM J. Control Optim., 50 (2012), 1616-1642.  doi: 10.1137/110851584.

[10]

F. Dörfler and F. Bullo, Synchronization in complex networks of phase oscillators: A survey, Automatica J. IFAC, 50 (2014), 1539-1564.  doi: 10.1016/j.automatica.2014.04.012.

[11]

S.-Y. Ha, D. Kim, J. Lee and Y. Zhang, Remarks on the stability properties of the Kuramoto-Sakaguchi-Fokker-Planck equation with frustration, Z. Angew. Math. Phys., 69 (2018), 25pp. doi: 10.1007/s00033-018-0984-z.

[12]

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

[13]

S.-Y. HaY. Kim and Z. Li, Large-time dynamics of Kuramoto oscillators under the effects of inertia and frustration, SIAM J. Appl. Dyn. Syst., 13 (2014), 466-492.  doi: 10.1137/130926559.

[14]

S.-Y. HaH. K. Kim and J. Park, Remarks on the complete synchronization of Kuramoto oscillators, Nonlinearity, 28 (2015), 1441-1462.  doi: 10.1088/0951-7715/28/5/1441.

[15]

S.-Y. HaH. K. Kim and S. W. Ryoo, Emergence of phase-locked states for the Kuramoto model in a large coupling regime, Commun. Math. Sci., 14 (2016), 1073-1091.  doi: 10.4310/CMS.2016.v14.n4.a10.

[16]

S.-Y. HaD. Ko and Y. Zhang, Emergence of phase-locking in the Kuramoto model for identical oscillators with frustration, SIAM J. Appl. Dyn. Syst., 17 (2018), 581-625.  doi: 10.1137/17M1112959.

[17]

S.-Y. HaZ. Li and X. Xue, Formation of phase-locked states in a population of locally interacting Kuramoto oscillators, J. Differential Equations, 255 (2013), 3053-3070.  doi: 10.1016/j.jde.2013.07.013.

[18]

S.-Y. HaJ. Li and Y. Zhang, Robustness in the instability of the incoherent state for the Kuramoto-Sakaguchi-Fokker-Planck equation with frustration, Quart. Appl. Math., 77 (2019), 631-654.  doi: 10.1090/qam/1533.

[19]

S.-Y. Ha and Q. Xiao, Remarks on the nonlinear stability of the Kuramoto-Sakaguchi equation, J. Differential Equations, 259 (2015), 2430-2457.  doi: 10.1016/j.jde.2015.03.038.

[20]

Y. Kuramoto, Chemical Oscillations, Waves and Turbulence, Springer Series in Synergetics, 19, Springer-Verlag, Berlin, 1984. doi: 10.1007/978-3-642-69689-3.

[21]

Y. Kuramoto, International Symposium on Mathematical Problems in Mathematical Physics, Lecture Notes in Physics, 39, Springer-Verlag, Berlin, 1975.

[22]

Z. Levnajić, Emergent multistability and frustration in phase-repulsive networks of oscillators, Phys. Rev. E, 84 (2011). doi: 10.1103/PhysRevE.84.016231.

[23]

Z. Li and S.-Y. Ha, Uniqueness and well-ordering of emergent phase-locked states for the Kuramoto model with frustration and inertia, Math. Models Methods Appl. Sci., 26 (2016), 357-382.  doi: 10.1142/S0218202516400054.

[24]

R. Mirollo and S. H. Strogatz, The spectrum of the partially locked state for the Kuramoto model, J. Nonlinear Sci., 17 (2017), 309-347.  doi: 10.1007/s00332-006-0806-x.

[25]

E. Oh, C. Choi, B. Kahng and D. Kim, Modular synchronization in complex networks with a gauge Kuramoto model, EPL, 83 (2008). doi: 10.1209/0295-5075/83/68003.

[26]

K. ParkS. W. Rhee and M. Y. Choi, Glass synchronization in the network of oscillators with random phase shifts, Phys. Rev. E, 57 (1998), 5030-5035.  doi: 10.1103/PhysRevE.57.5030.

[27] A. PikovskyM. Rosenblum and J. Kurths, Synchronization. A Universal Concept in Nonlinear Sciences, Cambridge Nonlinear Science Series, 12, Cambridge University Press, Cambridge, 2001.  doi: 10.1017/CBO9780511755743.
[28]

H. Sakaguchi, Cooperative phenomena in coupled oscillator systems sunder external fields, Prog. Theoret. Phys., 79 (1988), 39-46.  doi: 10.1143/PTP.79.39.

[29]

H. Sakaguchi and Y. Kuramoto, A soluble active rotator model showing phase transitions via mutual entraintment, Progr. Theoret. Phys., 76 (1986), 576-581.  doi: 10.1143/PTP.76.576.

[30]

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

[31]

S. H. Strogatz and R. E. Mirollo, Stability of incoherence in a population of coupled oscillators, J. Statist. Phys., 63 (1991), 613-635.  doi: 10.1007/BF01029202.

[32]

A. T. Winfree, Biological rhythms and the behavior of populations of coupled oscillators, J. Theor. Biol., 16 (1967), 15-42.  doi: 10.1016/0022-5193(67)90051-3.

[33]

Z.-G. Zheng, Frustration effect on synchronization and chaos in coupled oscillators, Chin. Phys. Soc., 10 (2001), 703-707.  doi: 10.1088/1009-1963/10/8/306.

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