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:
[1]

J. J. AcebronL. L. BonillaC. J. Perez-VicenteF. Ritort and R. Spigler, The Kuramoto model: A simple paradigm for synchronization phenomena, Rev. Mod. Phys., 77 (2005), 137-185. 

[2]

D. BenedettoE. 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. BenedettoE. 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. 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.

[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. ChoiS. -Y. HaS. 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. ChoiZ. LiS.-Y. HaX. 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. DietertB. 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. HaT. 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. HaZ. 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. LiX. 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. 

show all references

References:
[1]

J. J. AcebronL. L. BonillaC. J. Perez-VicenteF. Ritort and R. Spigler, The Kuramoto model: A simple paradigm for synchronization phenomena, Rev. Mod. Phys., 77 (2005), 137-185. 

[2]

D. BenedettoE. 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. BenedettoE. 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. 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.

[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. ChoiS. -Y. HaS. 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. ChoiZ. LiS.-Y. HaX. 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. DietertB. 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. HaT. 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. HaZ. 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. LiX. 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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