March  2017, 12(1): 25-57. doi: 10.3934/nhm.2017002

Global-in-time solution and stability of Kuramoto-Sakaguchi equation under non-local Coupling

Midori-Cho 3-9-11, Musashino-Shi, Tokyo 180-8585, Japan

Received  May 2016 Revised  September 2016 Published  February 2017

Fund Project: We thank the anonymous referees for valuable comments to improve the quality of this paper.

In this paper, we focus on the global-in-time solvability of the Kuramoto-Sakaguchi equation under non-local coupling. We further study the nonlinear stability of the trivial stationary solution in the presence of sufficiently large diffusivity, and the existence of the solution under vanishing diffusion.

Citation: Hirotada Honda. Global-in-time solution and stability of Kuramoto-Sakaguchi equation under non-local Coupling. Networks & Heterogeneous Media, 2017, 12 (1) : 25-57. doi: 10.3934/nhm.2017002
References:
[1]

D. M. Abrams and S. H. Strogatz, Chimera States for Coupled Oscillators, Phys. Rev. Lett., 93 (2004), 174102.   Google Scholar

[2]

D. M. AbramsR. MirolloS. H. Strogatz and D. A. Wiley, Solvable model for chimera states of coupled oscillators, Phys. Rev. Lett., 101 (2008), 084103.  doi: 10.1103/PhysRevLett.101.084103.  Google Scholar

[3]

J.A. Acebrón, Kuramoto model: A simple paradigm for synchronization phenomena, Rev. Mod. Phys., 77 (2005), 137-185.   Google Scholar

[4]

L.L. BonillaJ.C. Neu and R. Spigler, Nonlinear stability of incoherence and collective synchronization in a population of coupled oscillators, J. Stat. Phys., 1992 (67), 313-330.  doi: 10.1007/BF01049037.  Google Scholar

[5]

H. Chiba, A proof of the Kuramoto conjecture for a bifurcation structure of the infinite dimensional Kuramoto model, Ergod. Theory Dyn. Syst., 35 (2015), 762-834.  doi: 10.1017/etds.2013.68.  Google Scholar

[6]

J.D. Crawford, Amplitude expansions for instabilities in populations of globally-coupled oscillators, J. Stat. Phys., 74 (1994), 1047-1084.  doi: 10.1007/BF02188217.  Google Scholar

[7]

J.D. Crawford and K. T. R. Davies, Synchronization of globally coupled phase oscillators: singularities and scaling for general couplings, Physica D, 125 (1999), 1-46.  doi: 10.1016/S0167-2789(98)00235-8.  Google Scholar

[8]

H. Daido, Population dynamics of randomly interacting self-oscillators. Ⅰ. Tractable models without frustration, Prog. Theo. Phys., 77 (1987), 622-634.  doi: 10.1143/PTP.77.622.  Google Scholar

[9]

G. FilatrellaA.H. Nielsen and N. F. Pedersen, Analysis of a power grid using a Kuramoto-like model, Eur. Phys. J. B, 61 (2008), 485-491.  doi: 10.1140/epjb/e2008-00098-8.  Google Scholar

[10]

C. W. Gardiner, Handbook of Stochastic Methods, Springer, Berlin, 2009.  Google Scholar

[11]

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

[12]

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.  Google Scholar

[13]

H. Honda and A. Tani, Mathematical analysis of synchronization from the perspective of network science, to appear in Mathematical Analysis of Continuum Mechanics and Industrial Applications (Proceedings of the international conference CoMFoS15) (eds. H. Itou et al.), Springer Singapore, (2017). Google Scholar

[14]

T. Ichinomiya, Frequency synchronization in a random oscillator network, Phys. Rev. E, 70 (2004), 026116.  doi: 10.1103/PhysRevE.70.026116.  Google Scholar

[15]

Y. KawamuraH. Nakao and Y. Kuramoto, Noise-induced turbulence in nonlocally coupled oscillators, Phys. Rev. E, 75 (2007), 036209, 17pp..  doi: 10.1103/PhysRevE.75.036209.  Google Scholar

[16]

Y. Kawamura, From the Kuramoto-Sakaguchi model to the Kuramoto-Sivashinsky equation, Phys. Rev. E, 89 (2014), 010901.  doi: 10.1103/PhysRevE.89.010901.  Google Scholar

[17]

Y. Kuramoto, Self-entrainment of a population of coupled non-linear oscillators, in Int. Symp. on Mathematical problems in theoretical physics (eds. H. Araki), Springer, New York, 39 (1975), 420-422.  Google Scholar

[18]

Y. Kuramoto, Rhythms and turbulence in population of chemical oscillations, Physica A, 106 (1981), 128-143.  doi: 10.1016/0378-4371(81)90214-4.  Google Scholar

[19] Y. Kuramoto, Chemical Oscillations, Waves, and Turbulence, Springer-Verlag, Berlin, 1984.  doi: 10.1007/978-3-642-69689-3.  Google Scholar
[20]

Y. KuramotoS. ShimaD. Battogtokh and Y. Shiogai, Mean-field theory revives in self-oscillatory fields with non-local coupling, Prog. Theor. Phys. Suppl., 161 (2006), 127-143.  doi: 10.1143/PTPS.161.127.  Google Scholar

[21]

Y. Kuramoto, Departmental Bulletin Paper, (Japanese), Kyoto University, 2007. Google Scholar

[22]

Y. Kuramoto and D. Battogtokh, Coexistence of coherence and incoherence in nonlocally coupled phase oscillators, Nonlinear Phenom. Complex Syst., 5 (2002), 380-385.   Google Scholar

[23]

O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasi-Linear Equations of Parabolic Type, Translations of Mathematical Monographs, 23, American Mathematical Society, 1968.  Google Scholar

[24]

M. Lavrentiev and R. S. Spigler, Existence and uniqueness of solutions to the KuramotoSakaguchi nonlinear parabolic integrodifferential equatio, Differential and Integral Equations, 13 (2000), 649-667.   Google Scholar

[25]

M. LavrentievR.S. Spigler and A. Tani, Existence, uniqueness, and regularity for the Kuramoto-Sakaguchi equation with unboundedly supported frequency distribution, Differential and Integral Equations, 27 (2014), 879-8992.   Google Scholar

[26]

Z. LiY. Kim and S. H. Ha, Asymptotic synchronous behavior of Kuramoto type models with frustrations, Networks and Heterogeneous Media, 9 (2014), 33-64.  doi: 10.3934/nhm.2014.9.33.  Google Scholar

[27] J. L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications, Vol.1, Springer-Verlag, Berlin, 1972.   Google Scholar
[28]

A. Matsumura and T. Nishida, The initial value problem for the equations of motion of viscous and heat conductive gases, J. Math. Kyoto Univ., 20 (1980), 67-104.   Google Scholar

[29]

H. Nakao and A. S. Mikhailov, Diffusion-induced instability and chaos in random oscillator networks, Phys. Rev. E, 79 (2009), 036214.   Google Scholar

[30]

H. Risken, The Fokker-Planck Equation, Springer, Berlin, 1989. doi: 10.1007/978-3-642-61544-3.  Google Scholar

[31]

Y. Shiogai and Y. Kuramoto, Wave propagation in nonlocally coupled oscillators with noise, Prog. Thoer. Phys. Suppl., 150 (2003), 435-438.   Google Scholar

[32]

A. Sjöberg, On the Korteweg-de Vries equation, J. Math. Anal. Appl., 29 (1970), 569-579.  doi: 10.1016/0022-247X(70)90068-5.  Google Scholar

[33]

R. L. Stratonovich, Topics in the Theory of Random Noise, Gordon and Breach, New York, 1967. Google Scholar

[34]

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

[35] R. Temam, nfinite-Dimensional Dynamical Systems in Mechanics and Physics, 2nd edition, Springer-Verlag, New York, 1997.  doi: 10.1007/978-1-4612-0645-3.  Google Scholar
[36]

M. Tsutsumi and T. Mukasa, Parabolic regularizations for the generalized Kortewegde Vries equation, Funkcialaj Ekvacioj, 14 (1971), 89-110.   Google Scholar

[37]

A.T. Winfree, Biological rhythms and the behavior of populations of coupled oscillators, J. Theoret. Biol, 16 (1967), 15-43.  doi: 10.1016/0022-5193(67)90051-3.  Google Scholar

show all references

References:
[1]

D. M. Abrams and S. H. Strogatz, Chimera States for Coupled Oscillators, Phys. Rev. Lett., 93 (2004), 174102.   Google Scholar

[2]

D. M. AbramsR. MirolloS. H. Strogatz and D. A. Wiley, Solvable model for chimera states of coupled oscillators, Phys. Rev. Lett., 101 (2008), 084103.  doi: 10.1103/PhysRevLett.101.084103.  Google Scholar

[3]

J.A. Acebrón, Kuramoto model: A simple paradigm for synchronization phenomena, Rev. Mod. Phys., 77 (2005), 137-185.   Google Scholar

[4]

L.L. BonillaJ.C. Neu and R. Spigler, Nonlinear stability of incoherence and collective synchronization in a population of coupled oscillators, J. Stat. Phys., 1992 (67), 313-330.  doi: 10.1007/BF01049037.  Google Scholar

[5]

H. Chiba, A proof of the Kuramoto conjecture for a bifurcation structure of the infinite dimensional Kuramoto model, Ergod. Theory Dyn. Syst., 35 (2015), 762-834.  doi: 10.1017/etds.2013.68.  Google Scholar

[6]

J.D. Crawford, Amplitude expansions for instabilities in populations of globally-coupled oscillators, J. Stat. Phys., 74 (1994), 1047-1084.  doi: 10.1007/BF02188217.  Google Scholar

[7]

J.D. Crawford and K. T. R. Davies, Synchronization of globally coupled phase oscillators: singularities and scaling for general couplings, Physica D, 125 (1999), 1-46.  doi: 10.1016/S0167-2789(98)00235-8.  Google Scholar

[8]

H. Daido, Population dynamics of randomly interacting self-oscillators. Ⅰ. Tractable models without frustration, Prog. Theo. Phys., 77 (1987), 622-634.  doi: 10.1143/PTP.77.622.  Google Scholar

[9]

G. FilatrellaA.H. Nielsen and N. F. Pedersen, Analysis of a power grid using a Kuramoto-like model, Eur. Phys. J. B, 61 (2008), 485-491.  doi: 10.1140/epjb/e2008-00098-8.  Google Scholar

[10]

C. W. Gardiner, Handbook of Stochastic Methods, Springer, Berlin, 2009.  Google Scholar

[11]

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

[12]

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.  Google Scholar

[13]

H. Honda and A. Tani, Mathematical analysis of synchronization from the perspective of network science, to appear in Mathematical Analysis of Continuum Mechanics and Industrial Applications (Proceedings of the international conference CoMFoS15) (eds. H. Itou et al.), Springer Singapore, (2017). Google Scholar

[14]

T. Ichinomiya, Frequency synchronization in a random oscillator network, Phys. Rev. E, 70 (2004), 026116.  doi: 10.1103/PhysRevE.70.026116.  Google Scholar

[15]

Y. KawamuraH. Nakao and Y. Kuramoto, Noise-induced turbulence in nonlocally coupled oscillators, Phys. Rev. E, 75 (2007), 036209, 17pp..  doi: 10.1103/PhysRevE.75.036209.  Google Scholar

[16]

Y. Kawamura, From the Kuramoto-Sakaguchi model to the Kuramoto-Sivashinsky equation, Phys. Rev. E, 89 (2014), 010901.  doi: 10.1103/PhysRevE.89.010901.  Google Scholar

[17]

Y. Kuramoto, Self-entrainment of a population of coupled non-linear oscillators, in Int. Symp. on Mathematical problems in theoretical physics (eds. H. Araki), Springer, New York, 39 (1975), 420-422.  Google Scholar

[18]

Y. Kuramoto, Rhythms and turbulence in population of chemical oscillations, Physica A, 106 (1981), 128-143.  doi: 10.1016/0378-4371(81)90214-4.  Google Scholar

[19] Y. Kuramoto, Chemical Oscillations, Waves, and Turbulence, Springer-Verlag, Berlin, 1984.  doi: 10.1007/978-3-642-69689-3.  Google Scholar
[20]

Y. KuramotoS. ShimaD. Battogtokh and Y. Shiogai, Mean-field theory revives in self-oscillatory fields with non-local coupling, Prog. Theor. Phys. Suppl., 161 (2006), 127-143.  doi: 10.1143/PTPS.161.127.  Google Scholar

[21]

Y. Kuramoto, Departmental Bulletin Paper, (Japanese), Kyoto University, 2007. Google Scholar

[22]

Y. Kuramoto and D. Battogtokh, Coexistence of coherence and incoherence in nonlocally coupled phase oscillators, Nonlinear Phenom. Complex Syst., 5 (2002), 380-385.   Google Scholar

[23]

O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasi-Linear Equations of Parabolic Type, Translations of Mathematical Monographs, 23, American Mathematical Society, 1968.  Google Scholar

[24]

M. Lavrentiev and R. S. Spigler, Existence and uniqueness of solutions to the KuramotoSakaguchi nonlinear parabolic integrodifferential equatio, Differential and Integral Equations, 13 (2000), 649-667.   Google Scholar

[25]

M. LavrentievR.S. Spigler and A. Tani, Existence, uniqueness, and regularity for the Kuramoto-Sakaguchi equation with unboundedly supported frequency distribution, Differential and Integral Equations, 27 (2014), 879-8992.   Google Scholar

[26]

Z. LiY. Kim and S. H. Ha, Asymptotic synchronous behavior of Kuramoto type models with frustrations, Networks and Heterogeneous Media, 9 (2014), 33-64.  doi: 10.3934/nhm.2014.9.33.  Google Scholar

[27] J. L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications, Vol.1, Springer-Verlag, Berlin, 1972.   Google Scholar
[28]

A. Matsumura and T. Nishida, The initial value problem for the equations of motion of viscous and heat conductive gases, J. Math. Kyoto Univ., 20 (1980), 67-104.   Google Scholar

[29]

H. Nakao and A. S. Mikhailov, Diffusion-induced instability and chaos in random oscillator networks, Phys. Rev. E, 79 (2009), 036214.   Google Scholar

[30]

H. Risken, The Fokker-Planck Equation, Springer, Berlin, 1989. doi: 10.1007/978-3-642-61544-3.  Google Scholar

[31]

Y. Shiogai and Y. Kuramoto, Wave propagation in nonlocally coupled oscillators with noise, Prog. Thoer. Phys. Suppl., 150 (2003), 435-438.   Google Scholar

[32]

A. Sjöberg, On the Korteweg-de Vries equation, J. Math. Anal. Appl., 29 (1970), 569-579.  doi: 10.1016/0022-247X(70)90068-5.  Google Scholar

[33]

R. L. Stratonovich, Topics in the Theory of Random Noise, Gordon and Breach, New York, 1967. Google Scholar

[34]

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

[35] R. Temam, nfinite-Dimensional Dynamical Systems in Mechanics and Physics, 2nd edition, Springer-Verlag, New York, 1997.  doi: 10.1007/978-1-4612-0645-3.  Google Scholar
[36]

M. Tsutsumi and T. Mukasa, Parabolic regularizations for the generalized Kortewegde Vries equation, Funkcialaj Ekvacioj, 14 (1971), 89-110.   Google Scholar

[37]

A.T. Winfree, Biological rhythms and the behavior of populations of coupled oscillators, J. Theoret. Biol, 16 (1967), 15-43.  doi: 10.1016/0022-5193(67)90051-3.  Google Scholar

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