# American Institute of Mathematical Sciences

January  2019, 39(1): 345-367. doi: 10.3934/dcds.2019014

## Convergence and stability of generalized gradient systems by Łojasiewicz inequality with application in continuum Kuramoto model

 1 Department of Mathematics and Institute for Advanced Study in Mathematics, Harbin Institute of Technology, Harbin 150001, China 2 Department of Mathematics, Harbin Institute of Technology, Harbin, 150001, China 3 Department of Mathematics, Suihua University, Suihua 152000, China

* Corresponding author: Xiaoping Xue

Received  January 2018 Revised  July 2018 Published  October 2018

Fund Project: This work was supported by NSF of China grants 11731010 and 11671109.

The existence and uniqueness/multiplicity of phase locked solution for continuum Kuramoto model was studied in [12,29]. However, its asymptotic behavior is still unknown. In this paper we concern the asymptotic property of classic solutions to continuum Kuramoto model. In particular, we prove the convergence towards a phase locked state and its stability, provided suitable initial data and coupling strength. The main strategy is the quasi-gradient flow approach based on Łojasiewicz inequality. For this aim, we establish a Łojasiewicz type inequality in infinite dimensions for continuum Kuramoto model which is a nonlocal integro-differential equation. General theorems for convergence and stability of (generalized) quasi-gradient system in an abstract setting are also provided based on Łojasiewicz inequality.

Citation: Zhuchun Li, Yi Liu, Xiaoping Xue. Convergence and stability of generalized gradient systems by Łojasiewicz inequality with application in continuum Kuramoto model. Discrete and Continuous Dynamical Systems, 2019, 39 (1) : 345-367. doi: 10.3934/dcds.2019014
##### References:
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B, 96 (2006), 933-957.  doi: 10.1016/j.jctb.2006.05.002. [24] G. S. Medvedev, The nonlinear heat equation on W-random graphs, Arch. Rational Mech. Anal., 212 (2014), 781-803.  doi: 10.1007/s00205-013-0706-9. [25] G. S. Medvedev, Small-world networks of Kuramoto oscillators, Physica D, 266 (2014), 13-22.  doi: 10.1016/j.physd.2013.09.008. [26] E. Ott and T. M. Antonsen, Low dimensional behavior of large systems of globally coupled oscillators, Chaos, 18 (2008), 037113, 6pp doi: 10.1063/1.2930766. [27] S. H. Strogatz, From Kuramoto to Crawford: Exploring the onset of synchronization in populations of coupled oscillators, Phsica D, 143 (2000), 1-20.  doi: 10.1016/S0167-2789(00)00094-4. [28] S. H. Strogatz and J. Mirollo, Stability of incoherence in a population of coupled oscillators, J. Statist. Phys., 63 (1991), 613-635.  doi: 10.1007/BF01029202. [29] W. C. Troy, Existence and exact multiplicity of phaselocked solutions of a Kuramoto model of mutually coupled oscillators, SIAM J. Appl. Math., 75 (2015), 1745-1760.  doi: 10.1137/15100309X. [30] J. L. van Hemmen and W. F. Wreszinski, Lyapunov function for the Kuramoto model of nonlinearly coupled oscillators, J. Stat. Phys., 72 (1993), 145-166.

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##### References:
 [1] J. J. Acebron, L. L. Bonilla, C. J. Perez-Vicente, F. Ritort and R. Spigler, The Kuramoto model: A simple paradigm for synchronization phenomena, Rev. Mod. Phys., 77 (2005), 137-185. [2] 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. [3] D. Benedetto, E. Caglioti and U. Montemagno, On the complete phase synchronization for the Kuramoto model in the mean-field limit, Comm. Math. Sci., 13 (2015), 1775-1786.  doi: 10.4310/CMS.2015.v13.n7.a6. [4] 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. [5] R. Chill, On the Lojasiewicz-Simon gradient inequality, J. Funct. Anal., 201 (2003), 572-601.  doi: 10.1016/S0022-1236(02)00102-7. [6] Y. P. 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. [7] Y.-P. Choi, Z. Li, S.-Y. Ha, X. Xue and S.-B. Yun, Complete entrainment of Kuramoto oscillators with inertia on networks via gradient-like flow, J. Diff. Eqs., 257 (2014), 2591-2621.  doi: 10.1016/j.jde.2014.05.054. [8] H. Dietert, Stability and bifurcation for the Kuramoto model, J. Math. Pures Appl., 105 (2016), 451-489.  doi: 10.1002/cpa.21741. [9] H. Dietert, B. Fernandez and D. Gérard-Varet, Landau damping to partially locked states in the Kuramoto model, Comm. Pure Appl. Math., 71 (2018), 953-993.  doi: 10.4310/CMS.2013.v11.n2.a7. [10] 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. [11] F. Dörfler and F. Bullo, On the critical coupling for Kuramoto oscillator, SIAM. J. Appl. Dyn. Syst., 10 (2011), 1070-1099.  doi: 10.1137/10081530X. [12] G. B. Ermentrout, Synchronization in a pool of mutually coupled oscillators with random frequencies, J. Math. Bio., 22 (1985), 1-9.  doi: 10.1007/BF00276542. [13] S. Y. Ha, T. Ha and J. H. Kim, On the complete synchronization of the Kuramoto phase model, Physica D, 239 (2010), 1692-1700.  doi: 10.1016/j.physd.2010.05.003. [14] S.-Y. Ha, Z. Li and X. Xue, Formation of phase-locked states in a population of locally interacting Kuramoto oscillators, J. Diff. Eqs., 255 (2013), 3053-3070.  doi: 10.1016/j.jde.2013.07.013. [15] A. Haraux and M. A. Jendoubi, The Łojasiewicz gradient inequality in the infinite-dimensional Hilbert space framework, J. Funct. Anal., 260 (2011), 2826-2842.  doi: 10.1016/j.jfa.2011.01.012. [16] S.-B. Hsu, Ordinary differential equations with applications, WorldScientific, Singapore, (2006), 31-33. [17] Y. Kuromoto, International symposium on mathematical problems in mathematical physics, Lect. Notes Theoret. Phys., 30 (1975), 420. [18] Y. Kuromoto, Chemical Oscillations, Waves and Turbulence, Springer-Verlag, Berlin, 1984. doi: 10.1007/978-3-642-69689-3. [19] Z. Li and X. Xue, Convergence of analytic gradient-type systems with periodicity and its applications in Kuramoto models, Applied Mathematics Letters, 2018. doi: 10.1016/j.aml.2018.10.015. [20] Z. Li, X. Xue and D. Yu, On the Łojasiewicz exponent of Kuramoto model, J. Math. Phys., 56 (2015), 0227041, 20pp. doi: 10.1063/1.4908104. [21] Z. Li, X. Xue and D. Yu, Synchronization and tansient stability in power grids based on Łojasiewicz inequalities, SIAM J. Control Optim., 52 (2014), 2482-2511.  doi: 10.1137/130950604. [22] S. Łojasiewicz, Une propriété topologique des sous-ensembles analytiques réels, in Les Équations aux Dérivées Partielles, Éditions du Centre National de la Recherche Scientifique, Paris, (1963), 87-89. [23] L. Lovász and B. Szegedy, Limits of dense graph sequences, J. Combin. Theory Ser. B, 96 (2006), 933-957.  doi: 10.1016/j.jctb.2006.05.002. [24] G. S. Medvedev, The nonlinear heat equation on W-random graphs, Arch. Rational Mech. Anal., 212 (2014), 781-803.  doi: 10.1007/s00205-013-0706-9. [25] G. S. Medvedev, Small-world networks of Kuramoto oscillators, Physica D, 266 (2014), 13-22.  doi: 10.1016/j.physd.2013.09.008. [26] E. Ott and T. M. Antonsen, Low dimensional behavior of large systems of globally coupled oscillators, Chaos, 18 (2008), 037113, 6pp doi: 10.1063/1.2930766. [27] S. H. Strogatz, From Kuramoto to Crawford: Exploring the onset of synchronization in populations of coupled oscillators, Phsica D, 143 (2000), 1-20.  doi: 10.1016/S0167-2789(00)00094-4. [28] S. H. Strogatz and J. Mirollo, Stability of incoherence in a population of coupled oscillators, J. Statist. Phys., 63 (1991), 613-635.  doi: 10.1007/BF01029202. [29] W. C. Troy, Existence and exact multiplicity of phaselocked solutions of a Kuramoto model of mutually coupled oscillators, SIAM J. Appl. Math., 75 (2015), 1745-1760.  doi: 10.1137/15100309X. [30] J. L. van Hemmen and W. F. Wreszinski, Lyapunov function for the Kuramoto model of nonlinearly coupled oscillators, J. Stat. Phys., 72 (1993), 145-166.
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