# American Institute of Mathematical Sciences

July & August  2021, 20(7&8): 2579-2612. doi: 10.3934/cpaa.2021013

## Kuramoto order parameters and phase concentration for the Kuramoto-Sakaguchi equation with frustration

 1 Department of Mathematical Sciences and Research Institute of Mathematics, Seoul National University, Seoul 08826 and Korea Institute for Advanced Study, Hoegiro 87, Seoul, 130-722, Republic of Korea 2 Center for Scientific Computation and Mathematical Modeling, University of Maryland, College Park, MD 20742, USA 3 Stochastic Analysis and Application Research Center, Korea Advanced Institute of Science and Technology, Daejeon 34141, Republic of Korea

* Corresponding author

Dedicated to the celebration of the 80th birthday of Prof. Shuxing Chen

Received  June 2020 Revised  December 2020 Published  July & August 2021 Early access  February 2021

Fund Project: The work of S. Y. Ha was supported by National Research Foundation of Korea(NRF-2020R1A2C3A01003881), and the work of Y. Zhang was supported by the National Research Foundation of Korea(NRF) grant funded by the Korea government(MSIT)(NRF-2019R1A5A1028324)

We study phase concentration for the Kuramoto-Sakaguchi(K-S) equation with frustration via detailed estimates on the dynamics of order parameters. The Kuramoto order parameters measure the overall degree of phase concentrations. When the coupling strength is sufficiently large and the size of frustration parameter is sufficiently small, we show that the amplitude order parameter has a positive lower bound uniformly in time, and we also show that the total mass concentrates on the translated phase order parameter by a frustration parameter asymptotically, whereas the mass in the region around the antipodal point decays to zero exponentially fast.

Citation: Seung-Yeal Ha, Javier Morales, Yinglong Zhang. Kuramoto order parameters and phase concentration for the Kuramoto-Sakaguchi equation with frustration. Communications on Pure and Applied Analysis, 2021, 20 (7&8) : 2579-2612. doi: 10.3934/cpaa.2021013
##### References:
 [1] J. A. Acebrón, L. L. Bonilla, C. J. P. Pérez Vicente, F. Ritort and R. Spigler, The Kuramoto model: A simple paradigm for synchronization phenomena, Rev. Modern. Phys., 77 (2005), 137-185.  doi: 10.1103/RevModPhys.77.137. [2] D. Amadori, S. Y. Ha and J. Park, On the global well-posedness of BV weak solutions for the Kuramoto-akaguchi equation, J. Differ. Equ., 262 (2017), 978-1022.  doi: 10.1016/j.jde.2016.10.004. [3] D. Benedetto, E. Caglioti and U. Montemagno, On the complete phase synchronization for the Kuramoto model in the mean-field limit, Commun. Math. Sci., 13 (2015), 1775-1786.  doi: 10.4310/CMS.2015.v13.n7.a6. [4] D. Benedetto, E. Caglioti and U. Montemagno, Exponential dephasing of oscillators in the kinetic Kuramoto model, J. Stat. Phys., 162 (2016), 813-823.  doi: 10.1007/s10955-015-1426-3. [5] M. Brede and A. C. Kalloniatis, Frustration tuning and perfect phase synchronization in the Kuramoto-Sakaguchi model, Phys. Rev. E, 93 (2916), 062315, 13 pp. doi: 10.1103/PhysRevE.93.062315. [6] J. A. Carrillo, Y. P. Choi, S. Y. Ha, M. 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. [7] Y. Choi, S. Y. Ha, S. Jung and Y. Kim, Asymptotic formation and orbital stability of phase-locked states for the Kuramoto model, Physica D, 241 (2012), 735-754.  doi: 10.1016/j.physd.2011.11.011. [8] H. Chiba, Continuous limit of the moments system for the globally coupled phase oscillators, Discrete Contin. Dyn. Syst., 33 (2013), 1891-1903.  doi: 10.3934/dcds.2013.33.1891. [9] N. Chopra and M. W. Spong, On exponential synchronization of Kuramoto oscillators, IEEE Trans. Autom. Control, 54 (2009), 353-357.  doi: 10.1109/TAC.2008.2007884. [10] H. Daido, Quasientrainment and slow relaxation in a population of oscillators with random and frustrated interactions, Phys. Rev. Lett., 68 (1992), 1073-1076. [11] F. De Smet and D. Aeyels, Partial entrainment in the finite Kuramoto-Sakaguchi model, Physica D, 234 (2007), 81-89.  doi: 10.1016/j.physd.2007.06.025. [12] H. Dietert, B. Fernandez and D. Gérard-Varet, Landau damping to partially locked states in the Kuramoto model, Commun. Pure Appl. Math., 71 (2018), 953-993.  doi: 10.1002/cpa.21741. [13] 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. [14] 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. [15] F. Dorfler and F. Bullo, On the critical coupling for Kuramoto oscillators, SIAM J. Appl. Dyn. Syst., 10 (2011), 1070-1099.  doi: 10.1137/10081530X. [16] B. Fernandez, D. Grard-Varet and G. Giacomin, Landau damping in the Kuramoto model, Ann. Henri Poincaré, 17 (2016), 1793-1823.  doi: 10.1007/s00023-015-0450-9. [17] 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), 25 pp. doi: 10.1007/s00033-018-0984-z. [18] S. Y. Ha, H. K. Kim and J. Park, Remarks on the complete synchronization for the Kuramoto model with frustrations, Anal. Appl., 16 (2018), 525-563.  doi: 10.1142/S0219530517500130. [19] S. Y. Ha, H. Kim and S. 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. [20] S. Y. Ha, Y. 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. [21] S. Y. Ha, Y. 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. [22] S. Y. Ha, Y. H. Kim, J. Morales and J. Park, Emergence of phase concentration for the Kuramoto-Sakaguchi equation, Phys. D, 401 (2020), 24 pp. doi: 10.1016/j.physd.2019.132154. [23] S. Y. Ha, D. 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. [24] S. Y. Ha, J. Lee 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. [25] S. Y. Ha, H. Park and Y. Zhang, Nonlinear stability of stationary solutions to the Kuramoto-Sakaguchi equation with frustration, Netw. Heterog. Media, 15 (2020), 427-461.  doi: 10.3934/nhm.2020026. [26] 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. [27] S. Y. Ha and Q. Xiao, Nonlinear instability of the incoherent state for the Kuramoto-Sakaguchi-Fokker-Plank equation, J. Stat. Phys., 160 (2015), 477-496.  doi: 10.1007/s10955-015-1270-5. [28] A. Jadbabaie, N. Motee and M. Barahona, On the stability of the Kuramoto model of coupled nonlinear oscillators, Proc. American Control Conf., 5 (2004), 4296-4301. [29] Y. Kuramoto, Chemical Oscillations, Waves and Turbulence, Springer-Verlag, Berlin, 1984. doi: 10.1007/978-3-642-69689-3. [30] Y. Kuramoto, International symposium on mathematical problems in mathematical physics, Lecture notes in theoretical physics, 30 (1975), 420. [31] C. Lancellotti, On the Vlasov limit for systems of nonlinearly coupled oscillators without noise, Transport Theory Statist. Phys., 34 (2005), 523-535.  doi: 10.1080/00411450508951152. [32] Z. Levnajić, Emergent multistability and frustration in phase-repulsive networks of oscillators, Phys. Rev. E, 84 (2011), 016231. [33] 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. [34] R. E. Mirollo and S. H. Strogatz, The spectrum of the partially locked state for the Kuramoto model, J. Nonlinear Sci., 17 (2007), 309-347.  doi: 10.1007/s00332-006-0806-x. [35] J. Morales and D. Poyato, On the trend to global equilibrium for Kuramoto Oscillators, arXiv: 1908.07657v1 [36] E. Oh, C. Choi, B. Kahng and D. Kim, Modular synchronization in complex networks with a gauge Kuramoto model, EPL, 83 (2008), 68003. [37] K. Park, S. W. Rhee and M. Y. Choi, Glass synchronization in the network of oscillators with random phase shift, Phys. Rev. E, 57 (1998), 5030-5035. [38] A. Pikovsky, M. Rosenblum and J. Kurths, Synchronization: A universal concept in nonlinear sciences, Cambridge University Press, Cambridge, 2001.  doi: 10.1017/CBO9780511755743. [39] H. Sakaguchi and Y. Kuramoto, A soluble active rotator model showing phase transitions via mutual entrainment, Progr. Theoret. Phys., 76 (1986), 576-581.  doi: 10.1143/PTP.76.576. [40] S. H. Strogatz, From Kuramoto to Crawford: exploring the onset of synchronization in populations of coupled oscillators, Physica D, 143 (2000), 1-20.  doi: 10.1016/S0167-2789(00)00094-4. [41] A. T. Winfree, Biological rhythms and the behavior of populations of coupled oscillators, J. Theor. Biol., 16 (1967), 15-42. [42] A. T. Winfree, The Geometry of Biological Time, Springer, New York, 1980. [43] Z. G. Zheng, Frustration effect on synchronization and chaos in coupled oscillators, Chin. Phys. Soc., 10 (2001), 703-707.

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##### References:
 [1] J. A. Acebrón, L. L. Bonilla, C. J. P. Pérez Vicente, F. Ritort and R. Spigler, The Kuramoto model: A simple paradigm for synchronization phenomena, Rev. Modern. Phys., 77 (2005), 137-185.  doi: 10.1103/RevModPhys.77.137. [2] D. Amadori, S. Y. Ha and J. Park, On the global well-posedness of BV weak solutions for the Kuramoto-akaguchi equation, J. Differ. Equ., 262 (2017), 978-1022.  doi: 10.1016/j.jde.2016.10.004. [3] D. Benedetto, E. Caglioti and U. Montemagno, On the complete phase synchronization for the Kuramoto model in the mean-field limit, Commun. Math. Sci., 13 (2015), 1775-1786.  doi: 10.4310/CMS.2015.v13.n7.a6. [4] D. Benedetto, E. Caglioti and U. Montemagno, Exponential dephasing of oscillators in the kinetic Kuramoto model, J. Stat. Phys., 162 (2016), 813-823.  doi: 10.1007/s10955-015-1426-3. [5] M. Brede and A. C. Kalloniatis, Frustration tuning and perfect phase synchronization in the Kuramoto-Sakaguchi model, Phys. Rev. E, 93 (2916), 062315, 13 pp. doi: 10.1103/PhysRevE.93.062315. [6] J. A. Carrillo, Y. P. Choi, S. Y. Ha, M. 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. [7] Y. Choi, S. Y. Ha, S. Jung and Y. Kim, Asymptotic formation and orbital stability of phase-locked states for the Kuramoto model, Physica D, 241 (2012), 735-754.  doi: 10.1016/j.physd.2011.11.011. [8] H. Chiba, Continuous limit of the moments system for the globally coupled phase oscillators, Discrete Contin. Dyn. Syst., 33 (2013), 1891-1903.  doi: 10.3934/dcds.2013.33.1891. [9] N. Chopra and M. W. Spong, On exponential synchronization of Kuramoto oscillators, IEEE Trans. Autom. Control, 54 (2009), 353-357.  doi: 10.1109/TAC.2008.2007884. [10] H. Daido, Quasientrainment and slow relaxation in a population of oscillators with random and frustrated interactions, Phys. Rev. Lett., 68 (1992), 1073-1076. [11] F. De Smet and D. Aeyels, Partial entrainment in the finite Kuramoto-Sakaguchi model, Physica D, 234 (2007), 81-89.  doi: 10.1016/j.physd.2007.06.025. [12] H. Dietert, B. Fernandez and D. Gérard-Varet, Landau damping to partially locked states in the Kuramoto model, Commun. Pure Appl. Math., 71 (2018), 953-993.  doi: 10.1002/cpa.21741. [13] 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. [14] 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. [15] F. Dorfler and F. Bullo, On the critical coupling for Kuramoto oscillators, SIAM J. Appl. Dyn. Syst., 10 (2011), 1070-1099.  doi: 10.1137/10081530X. [16] B. Fernandez, D. Grard-Varet and G. Giacomin, Landau damping in the Kuramoto model, Ann. Henri Poincaré, 17 (2016), 1793-1823.  doi: 10.1007/s00023-015-0450-9. [17] 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), 25 pp. doi: 10.1007/s00033-018-0984-z. [18] S. Y. Ha, H. K. Kim and J. Park, Remarks on the complete synchronization for the Kuramoto model with frustrations, Anal. Appl., 16 (2018), 525-563.  doi: 10.1142/S0219530517500130. [19] S. Y. Ha, H. Kim and S. 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. [20] S. Y. Ha, Y. 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. [21] S. Y. Ha, Y. 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. [22] S. Y. Ha, Y. H. Kim, J. Morales and J. Park, Emergence of phase concentration for the Kuramoto-Sakaguchi equation, Phys. D, 401 (2020), 24 pp. doi: 10.1016/j.physd.2019.132154. [23] S. Y. Ha, D. 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. [24] S. Y. Ha, J. Lee 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. [25] S. Y. Ha, H. Park and Y. Zhang, Nonlinear stability of stationary solutions to the Kuramoto-Sakaguchi equation with frustration, Netw. Heterog. Media, 15 (2020), 427-461.  doi: 10.3934/nhm.2020026. [26] 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. [27] S. Y. Ha and Q. Xiao, Nonlinear instability of the incoherent state for the Kuramoto-Sakaguchi-Fokker-Plank equation, J. Stat. Phys., 160 (2015), 477-496.  doi: 10.1007/s10955-015-1270-5. [28] A. Jadbabaie, N. Motee and M. Barahona, On the stability of the Kuramoto model of coupled nonlinear oscillators, Proc. American Control Conf., 5 (2004), 4296-4301. [29] Y. Kuramoto, Chemical Oscillations, Waves and Turbulence, Springer-Verlag, Berlin, 1984. doi: 10.1007/978-3-642-69689-3. [30] Y. Kuramoto, International symposium on mathematical problems in mathematical physics, Lecture notes in theoretical physics, 30 (1975), 420. [31] C. Lancellotti, On the Vlasov limit for systems of nonlinearly coupled oscillators without noise, Transport Theory Statist. Phys., 34 (2005), 523-535.  doi: 10.1080/00411450508951152. [32] Z. Levnajić, Emergent multistability and frustration in phase-repulsive networks of oscillators, Phys. Rev. E, 84 (2011), 016231. [33] 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. [34] R. E. Mirollo and S. H. Strogatz, The spectrum of the partially locked state for the Kuramoto model, J. Nonlinear Sci., 17 (2007), 309-347.  doi: 10.1007/s00332-006-0806-x. [35] J. Morales and D. Poyato, On the trend to global equilibrium for Kuramoto Oscillators, arXiv: 1908.07657v1 [36] E. Oh, C. Choi, B. Kahng and D. Kim, Modular synchronization in complex networks with a gauge Kuramoto model, EPL, 83 (2008), 68003. [37] K. Park, S. W. Rhee and M. Y. Choi, Glass synchronization in the network of oscillators with random phase shift, Phys. Rev. E, 57 (1998), 5030-5035. [38] A. Pikovsky, M. Rosenblum and J. Kurths, Synchronization: A universal concept in nonlinear sciences, Cambridge University Press, Cambridge, 2001.  doi: 10.1017/CBO9780511755743. [39] H. Sakaguchi and Y. Kuramoto, A soluble active rotator model showing phase transitions via mutual entrainment, Progr. Theoret. Phys., 76 (1986), 576-581.  doi: 10.1143/PTP.76.576. [40] S. H. Strogatz, From Kuramoto to Crawford: exploring the onset of synchronization in populations of coupled oscillators, Physica D, 143 (2000), 1-20.  doi: 10.1016/S0167-2789(00)00094-4. [41] A. T. Winfree, Biological rhythms and the behavior of populations of coupled oscillators, J. Theor. Biol., 16 (1967), 15-42. [42] A. T. Winfree, The Geometry of Biological Time, Springer, New York, 1980. [43] Z. G. Zheng, Frustration effect on synchronization and chaos in coupled oscillators, Chin. Phys. Soc., 10 (2001), 703-707.
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