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, S. H. Strogatz, Chimera States for Coupled Oscillators, Phys. Rev. Lett., 93 (2004), 174102.

[2]

D. M. Abrams, R. Mirollo, S. H. Strogatz, D. A. Wiley, Solvable model for chimera states of coupled oscillators, Phys. Rev. Lett., 101 (2008), 084103. doi: 10.1103/PhysRevLett.101.084103.

[3]

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

[4]

L.L. Bonilla, J.C. Neu, 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.

[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.

[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.

[7]

J.D. Crawford, 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.

[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.

[9]

G. Filatrella, A.H. Nielsen, 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.

[10]

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

[11]

S.Y. Ha, 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.

[12]

S.Y. Ha, 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.

[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).

[14]

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

[15]

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

[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.

[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.

[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.

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

Y. Kuramoto, S. Shima, D. Battogtokh, 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.

[21]

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

[22]

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

[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.

[24]

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

[25]

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

[26]

Z. Li, Y. Kim, 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.

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

A. Matsumura, 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.

[29]

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

[30]

H. Risken, The Fokker-Planck Equation, Springer, Berlin, 1989.

[31]

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

[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.

[33]

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

[34]

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

[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.
[36]

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

[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.

show all references

References:
[1]

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

[2]

D. M. Abrams, R. Mirollo, S. H. Strogatz, D. A. Wiley, Solvable model for chimera states of coupled oscillators, Phys. Rev. Lett., 101 (2008), 084103. doi: 10.1103/PhysRevLett.101.084103.

[3]

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

[4]

L.L. Bonilla, J.C. Neu, 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.

[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.

[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.

[7]

J.D. Crawford, 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.

[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.

[9]

G. Filatrella, A.H. Nielsen, 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.

[10]

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

[11]

S.Y. Ha, 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.

[12]

S.Y. Ha, 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.

[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).

[14]

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

[15]

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

[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.

[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.

[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.

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

Y. Kuramoto, S. Shima, D. Battogtokh, 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.

[21]

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

[22]

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

[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.

[24]

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

[25]

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

[26]

Z. Li, Y. Kim, 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.

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

A. Matsumura, 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.

[29]

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

[30]

H. Risken, The Fokker-Planck Equation, Springer, Berlin, 1989.

[31]

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

[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.

[33]

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

[34]

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

[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.
[36]

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

[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.

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