Robustness in the instability of the incoherent state for the Kuramoto-Sakaguchi-Fokker-Planck equation with frustration
Authors:
Seung-Yeal Ha, Jaeseung Lee and Yinglong Zhang
Journal:
Quart. Appl. Math. 77 (2019), 631-654
MSC (2010):
Primary 35B35, 35B40
DOI:
https://doi.org/10.1090/qam/1533
Published electronically:
January 25, 2019
MathSciNet review:
3962586
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Abstract: We study the robustness in the nonlinear instability of the incoherent state for the Kuramoto-Sakaguchi-Fokker-Planck (KS-FP for short) equation in the presence of frustrations. For this, we construct a new unstable mode for the corresponding linear part of the perturbation around the incoherent state, and we show that the nonlinear perturbation stays close to the unstable mode in some small time interval which depends on the initial size of the perturbations. Our instability results improve the previous results on the KS-FP with zero frustration [J. Stat. Phys. 160 (2015), pp. 477–496] by providing a new linear unstable mode and detailed energy estimates.
References
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- 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.
- Arkady Pikovsky, Michael Rosenblum, and Jürgen Kurths, Synchronization, Cambridge Nonlinear Science Series, vol. 12, Cambridge University Press, Cambridge, 2001. A universal concept in nonlinear sciences. MR 1869044
- Hidetsugu Sakaguchi, Cooperative phenomena in coupled oscillator systems under external fields, Progr. Theoret. Phys. 79 (1988), no. 1, 39–46. MR 937229, DOI https://doi.org/10.1143/PTP.79.39
- Hidetsugu Sakaguchi and Yoshiki Kuramoto, A soluble active rotator model showing phase transitions via mutual entrainment, Progr. Theoret. Phys. 76 (1986), no. 3, 576–581. MR 869973, DOI https://doi.org/10.1143/PTP.76.576
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- Steven H. Strogatz, From Kuramoto to Crawford: exploring the onset of synchronization in populations of coupled oscillators, Phys. D 143 (2000), no. 1-4, 1–20. Bifurcations, patterns and symmetry. MR 1783382, DOI https://doi.org/10.1016/S0167-2789%2800%2900094-4
- A. T. Winfree, Biological rhythms and the behavior of populations of coupled oscillators, J. Theoret. Biol. 16 (1967), 15–42.
- Z. G. Zheng, Frustration effect on synchronization and chaos in coupled oscillators, Chin. Phys. Soc. 10 (2001), 703–707.
References
- 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. Mod. Phys. 77 (2005), 137–185.
- Debora Amadori, Seung-Yeal Ha, and Jinyeong Park, On the global well-posedness of BV weak solutions to the Kuramoto-Sakaguchi equation, J. Differential Equations 262 (2017), no. 2, 978–1022. MR 3569413, DOI https://doi.org/10.1016/j.jde.2016.10.004
- François Bolley, José A. Cañizo, and José A. Carrillo, Mean-field limit for the stochastic Vicsek model, Appl. Math. Lett. 25 (2012), no. 3, 339–343. MR 2855983, DOI https://doi.org/10.1016/j.aml.2011.09.011
- Dario Benedetto, Emanuele Caglioti, and Umberto Montemagno, Exponential dephasing of oscillators in the kinetic Kuramoto model, J. Stat. Phys. 162 (2016), no. 4, 813–823. MR 3456977, DOI https://doi.org/10.1007/s10955-015-1426-3
- José A. Carrillo, Young-Pil Choi, Seung-Yeal Ha, Moon-Jin Kang, and Yongduck Kim, Contractivity of transport distances for the kinetic Kuramoto equation, J. Stat. Phys. 156 (2014), no. 2, 395–415. MR 3215629, DOI https://doi.org/10.1007/s10955-014-1005-z
- H. Daido, Quasientrainment and slow relaxation in a population of oscillators with random and frustrated interactions, Phys. Rev. Lett. 68 (1992), 1073–1076.
- Filip De Smet and Dirk Aeyels, Partial entrainment in the finite Kuramoto-Sakaguchi model, Phys. D 234 (2007), no. 2, 81–89. MR 2371859, DOI https://doi.org/10.1016/j.physd.2007.06.025
- Helge Dietert, Stability and bifurcation for the Kuramoto model, J. Math. Pures Appl. (9) 105 (2016), no. 4, 451–489 (English, with English and French summaries). MR 3471147, DOI https://doi.org/10.1016/j.matpur.2015.11.001
- Florian Dörfler and Francesco Bullo, Synchronization in complex networks of phase oscillators: a survey, Automatica J. IFAC 50 (2014), no. 6, 1539–1564. MR 3214901, DOI https://doi.org/10.1016/j.automatica.2014.04.012
- S.-Y. Ha, D. Kim, J. Lee and S. E., Noh, Particle and kinetic models for swarming particles on a sphere and stability properties, J. Stat. Phys., (2018) https://doi.org/10.1007/s10955-018-2169-8
- Seung-Yeal Ha, Doheon Kim, Jaeseung Lee, and Yinglong Zhang, Remarks on the stability properties of the Kuramoto-Sakaguchi-Fokker-Planck equation with frustration, Z. Angew. Math. Phys. 69 (2018), no. 4, Art. 94, 25. MR 3816984, DOI https://doi.org/10.1007/s00033-018-0984-z
- Seung-Yeal Ha, Yongduck Kim, and Zhuchun Li, Asymptotic synchronous behavior of Kuramoto type models with frustrations, Netw. Heterog. Media 9 (2014), no. 1, 33–64. MR 3195344, DOI https://doi.org/10.3934/nhm.2014.9.33
- Seung-Yeal Ha, Dongnam Ko, and Yinglong Zhang, Emergence of phase-locking in the Kuramoto model for identical oscillators with frustration, SIAM J. Appl. Dyn. Syst. 17 (2018), no. 1, 581–625. MR 3765900, DOI https://doi.org/10.1137/17M1112959
- Seung-Yeal Ha, Dongnam Ko, Jinyeong Park, and Xiongtao Zhang, Collective synchronization of classical and quantum oscillators, EMS Surv. Math. Sci. 3 (2016), no. 2, 209–267. MR 3576533, DOI https://doi.org/10.4171/EMSS/17
- Seung-Yeal Ha and Qinghua Xiao, Remarks on the nonlinear stability of the Kuramoto-Sakaguchi equation, J. Differential Equations 259 (2015), no. 6, 2430–2457. MR 3353651, DOI https://doi.org/10.1016/j.jde.2015.03.038
- Seung-Yeal Ha and Qinghua Xiao, Nonlinear instability of the incoherent state for the Kuramoto-Sakaguchi-Fokker-Plank equation, J. Stat. Phys. 160 (2015), no. 2, 477–496. MR 3360470, DOI https://doi.org/10.1007/s10955-015-1270-5
- Y. Kuramoto, International symposium on mathematical problems in mathematical physics, Lecture Notes in Physics, 39 (1975), 420.
- Carlo Lancellotti, On the Vlasov limit for systems of nonlinearly coupled oscillators without noise, Transport Theory Statist. Phys. 34 (2005), no. 7, 523–535. MR 2265477, DOI https://doi.org/10.1080/00411450508951152
- Z. Levnajić, Emergent multistability and frustration in phase-repulsive networks of oscillators, Phys. Rev. E 84 (2011), 016231.
- E. Oh, C. Choi, B. Kahng, and D. Kim, Modular synchronization in complex networks with a gauge Kuramoto model, EPL 83 (2008), 68003.
- 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.
- Arkady Pikovsky, Michael Rosenblum, and Jürgen Kurths, Synchronization, A universal concept in nonlinear sciences, Cambridge Nonlinear Science Series, vol. 12, Cambridge University Press, Cambridge, 2001. MR 1869044
- Hidetsugu Sakaguchi, Cooperative phenomena in coupled oscillator systems under external fields, Progr. Theoret. Phys. 79 (1988), no. 1, 39–46. MR 937229, DOI https://doi.org/10.1143/PTP.79.39
- Hidetsugu Sakaguchi and Yoshiki Kuramoto, A soluble active rotator model showing phase transitions via mutual entrainment, Progr. Theoret. Phys. 76 (1986), no. 3, 576–581. MR 869973, DOI https://doi.org/10.1143/PTP.76.576
- Steven H. Strogatz and Renato E. Mirollo, Stability of incoherence in a population of coupled oscillators, J. Statist. Phys. 63 (1991), no. 3-4, 613–635. MR 1115806, DOI https://doi.org/10.1007/BF01029202
- Steven H. Strogatz, From Kuramoto to Crawford: exploring the onset of synchronization in populations of coupled oscillators, Phys. D 143 (2000), no. 1-4, 1–20. Bifurcations, patterns and symmetry. MR 1783382, DOI https://doi.org/10.1016/S0167-2789%2800%2900094-4
- A. T. Winfree, Biological rhythms and the behavior of populations of coupled oscillators, J. Theoret. Biol. 16 (1967), 15–42.
- Z. G. Zheng, Frustration effect on synchronization and chaos in coupled oscillators, Chin. Phys. Soc. 10 (2001), 703–707.
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Additional Information
Seung-Yeal Ha
Affiliation:
Department of Mathematical Sciences and Research Institute of Mathematics, Seoul National University, Seoul 08826, Korea; and Korea Institute for Advanced Study, Hoegiro 87, Seoul 02455, Korea
MR Author ID:
684438
Email:
syha@snu.ac.kr
Jaeseung Lee
Affiliation:
The Research Institute of Basic Sciences, Seoul National University, Seoul 08826, Korea
MR Author ID:
1201866
Email:
jaeseunglee@snu.ac.kr
Yinglong Zhang
Affiliation:
Department of Mathematical Sciences, Industrial and Mathematical Data Analytics Research Center, Seoul National University, Seoul 08826, Korea
MR Author ID:
1122209
Email:
yinglongzhang@amss.ac.cn
Keywords:
Frustration,
Kuramoto-Sakaguchi-Fokker-Planck equation,
mean-field equation,
nonlinear instability
Received by editor(s):
July 25, 2018
Published electronically:
January 25, 2019
Additional Notes:
The work of the first and third authors was supported by the National Research Foundation of Korea(NRF) Grant funded by the Korean Government(MSIT)(No.2017R1A5A1015626)
Article copyright:
© Copyright 2019
Brown University
