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December  2013, 8(4): 943-968. doi: 10.3934/nhm.2013.8.943

Global existence and asymptotic behavior of measure valued solutions to the kinetic Kuramoto--Daido model with inertia

1. 

Department of Mathematics, Imperial College London, London SW7 2AZ, United Kingdom

2. 

Department of Mathematical Sciences and Research Institute of Mathematics, Seoul National University, Seoul 151-747

3. 

Department of Mathematical Sciences, Seoul National University, Seoul 151-747, South Korea

Received  December 2012 Revised  June 2013 Published  November 2013

We present the global existence and long-time behavior of measure-valued solutions to the kinetic Kuramoto--Daido model with inertia. For the global existence of measure-valued solutions, we employ a Neunzert's mean-field approach for the Vlasov equation to construct approximate solutions. The approximate solutions are empirical measures generated by the solution to the Kuramoto--Daido model with inertia, and we also provide an a priori local-in-time stability estimate for measure-valued solutions in terms of a bounded Lipschitz distance. For the asymptotic frequency synchronization, we adopt two frameworks depending on the relative strength of inertia and show that the diameter of the projected frequency support of the measure-valued solutions exponentially converge to zero.
Citation: Young-Pil Choi, Seung-Yeal Ha, Seok-Bae Yun. Global existence and asymptotic behavior of measure valued solutions to the kinetic Kuramoto--Daido model with inertia. Networks and Heterogeneous Media, 2013, 8 (4) : 943-968. doi: 10.3934/nhm.2013.8.943
References:
[1]

J. A. 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. doi: 10.1103/RevModPhys.77.137.

[2]

J. A. Acebron, L. L. Bonilla and R. Spigler, Synchronization in populations of globally coupled oscillators with inertial effect, Phys. Rev. E., 62 (2000), 3437-3454. doi: 10.1103/PhysRevE.62.3437.

[3]

J. A. Acebron and R. Spigler, Adaptive frequency model for phase-frequency synchronization in large populations of globally coupled nonlinear oscillators, Phys. Rev. Lett., 81 (1998), 2229-2332. doi: 10.1103/PhysRevLett.81.2229.

[4]

J. Buck and E. Buck, Biology of synchronous flashing of fireflies, Nature, 211 (1966), 562-564. doi: 10.1038/211562a0.

[5]

N. J. Balmforth and R. Sassi, A shocking display of synchrony, Physica D., 143 (2000), 21-55. doi: 10.1016/S0167-2789(00)00095-6.

[6]

J. A. Carrillo, Y.-P. Choi, S.-Y. Ha, M.-J. Kang and Y. Kim, Contractivity of the Wasserstein metric for the kinetic Kuramoto equation, preprint, arXiv:1301.1883.

[7]

H. Chiba, Continuous limit of the moments system for the globally coupled phase oscillator, Discrete Contin. Dyn. Syst., 33 (2013), 1891-1903. doi: 10.3934/dcds.2013.33.1891.

[8]

Y.-P. Choi, S.-Y. Ha and S. E. Noh, Remarks on the nonlinear stability of the Kuramoto model with inertia, to appear in Quart. Appl. Math.

[9]

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.

[10]

Y.-P. Choi, S.-Y. Ha and S.-B. Yun, Complete synchronization of Kuramoto oscillators with finite inertia, Physica D., 240 (2011), 32-44. doi: 10.1016/j.physd.2010.08.004.

[11]

N. Chopra and M. W. Spong, On exponential synchronization of Kuramoto oscillators, IEEE Trans. Autom. Control., 54 (2009), 353-357. doi: 10.1109/TAC.2008.2007884.

[12]

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.

[13]

H. Daido, Onset of cooperative entrainment in limit-cycle oscillators with uniform all-to-all interactions: Bifurcation of the order function, Physica D., 91 (1996), 24-66. doi: 10.1016/0167-2789(95)00260-X.

[14]

B. C. Daniels, S. T. Dissanayake and B. R. Trees, Synchronization of coupled rotators: Josephson junction ladders and the locally coupled Kuramoto model, Phys. Rev. E., 67 (2003), 026216. doi: 10.1103/PhysRevE.67.026216.

[15]

F. Dorfler and F. Bullo, On the critical coupling for Kuramoto oscillators, SIAM J. Appl. Dyn. Syst., 10 (2011), 1070-1099. doi: 10.1137/10081530X.

[16]

G. B. Ermentrout, Synchronization in a pool of mutually coupled oscillators with random frequencies, J. Math. Biol., 22 (1985), 1-9. doi: 10.1007/BF00276542.

[17]

S.-Y. Ha, T. Y. Ha and J.-H. Kim, On the complete synchronization for the Kuramoto model, Physica D., 239 (2010), 1692-1700. doi: 10.1016/j.physd.2010.05.003.

[18]

S.-Y. Ha and J.-G. Liu, A simple proof of the Cucker-Smale flocking dynamics and mean-field limit, Commun. Math. Sci., 7 (2009), 297-325. doi: 10.4310/CMS.2009.v7.n2.a2.

[19]

H. Hong, M. Y. Choi, J. Yi and K.-S. Soh, Inertia effects on periodic synchronization in a system of coupled oscillators, Phys. Rev. E., 59 (1999), 353-363. doi: 10.1103/PhysRevE.59.353.

[20]

H. Hong, G. S. Jeon and M. Y. Choi, Spontaneous phase oscillation induced by inertia and time delay, Phys. Rev. E., 65 (2002), 026208. doi: 10.1103/PhysRevE.65.026208.

[21]

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

[22]

Y. Kuramoto, International symposium on mathematical problems in mathematical physics, Lecture Notes in Theoretical Physics., 39 (1975), 420-422.

[23]

C. Lancellotti, On the Vlasov limit for systems of nonlinearly coupled oscillators without noise, Transp. Theory Stat. Phys., 34 (2005), 523-535. doi: 10.1080/00411450508951152.

[24]

M. M. Lavrentiev and R. Spigler, Existence and uniqueness of solutions to the Kuramoto-Sakaguchi nonliner parabolic integrodifferential equation, Differ. Integr. Eq., 13 (2000), 649-667.

[25]

H. Neunzert, An introduction to the nonlinear Boltzmann-Vlasov equation, In Kinetic Theories and the Boltzmann Equation, Lecture Notes in Mathematics 1048, Springer, Berlin, Heidelberg. doi: 10.1007/BFb0071878.

[26]

A. Pikovsky, M. Rosenblum and J. Kurths, Synchrnization: A Universal Concept in Nonlinear Sciences, Cambridge University Press, Cambridge, 2001. doi: 10.1017/CBO9780511755743.

[27]

P.-A. Raviart, An analysis of particle methods, in Numerical Methods in Fluid Dynamics (Como, 1983), 243-324, Lecture Notes in Mathematics, 1127, Springer, Berlin, 1985. doi: 10.1007/BFb0074532.

[28]

H. Sakaguchi and Y. Kuramoto, A soluble active rotator model showing phase transitions via mutual entraintment, Prog. Theor. Phys., 76 (1986), 576-581. doi: 10.1143/PTP.76.576.

[29]

H. Sphohn, Large Scale Dynamics of Interacting Particles, Springer-Verlag, Berlin and Heidelberg, 1991. doi: 10.1007/978-3-642-84371-6.

[30]

S. H. Strogatz, From Kuramoto to Crawford: Exploring the onset of synchronization in populations of coupled oscillators, Physica D., 143 (2000), 1-20. doi: 10.1016/S0167-2789(00)00094-4.

[31]

H. A. Tanaka, A. J. Lichtenberg and S. Oishi, First order phase transition resulting from finite inertia in coupled oscillator systems, Phys. Rev. Lett., 78 (1997), 2104-2107. doi: 10.1103/PhysRevLett.78.2104.

[32]

H. A. Tanaka, A. J. Lichtenberg and S. Oishi, Self-synchronization of coupled oscillators with hysteretic responses, Physica D., 100 (1997), 279-300. doi: 10.1016/S0167-2789(96)00193-5.

[33]

S. Watanabe and J. W. Swift, Stability of periodic solutions in series arrays of Josephson junctions with internal capacitance, J. Nonlinear Sci., 7 (1997), 503-536. doi: 10.1007/s003329900038.

[34]

S. Watanabe and S. H. Strogatz, Constants of motion for superconducting Josephson arrays, Physica D., 74 (1994), 197-253. doi: 10.1016/0167-2789(94)90196-1.

[35]

K. Wiesenfeld, R. Colet and S. H. Strogatz, Synchronization transitions in a disordered Josephson series arrays, Phys. Rev. Lett., 76 (1996), 404-407. doi: 10.1103/PhysRevLett.76.404.

[36]

K. Wiesenfeld, R. Colet and S. H. Strogatz, Frequency locking in Josephson arrays: Connection with the Kuramoto model, Phys. Rev. E., 57 (1988), 1563-1569. doi: 10.1103/PhysRevE.57.1563.

[37]

K. Wiesenfeld and J. W. Swift, Averaged equations for Josephson junction series arrays, Phys. Rev. E., 51 (1995), 1020-1025. doi: 10.1103/PhysRevE.51.1020.

[38]

A. T. Winfree, Biological rhythms and the behavior of populations of coupled oscillators, J. Theor. Biol., 16 (1967), 15-42. doi: 10.1016/0022-5193(67)90051-3.

show all references

References:
[1]

J. A. 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. doi: 10.1103/RevModPhys.77.137.

[2]

J. A. Acebron, L. L. Bonilla and R. Spigler, Synchronization in populations of globally coupled oscillators with inertial effect, Phys. Rev. E., 62 (2000), 3437-3454. doi: 10.1103/PhysRevE.62.3437.

[3]

J. A. Acebron and R. Spigler, Adaptive frequency model for phase-frequency synchronization in large populations of globally coupled nonlinear oscillators, Phys. Rev. Lett., 81 (1998), 2229-2332. doi: 10.1103/PhysRevLett.81.2229.

[4]

J. Buck and E. Buck, Biology of synchronous flashing of fireflies, Nature, 211 (1966), 562-564. doi: 10.1038/211562a0.

[5]

N. J. Balmforth and R. Sassi, A shocking display of synchrony, Physica D., 143 (2000), 21-55. doi: 10.1016/S0167-2789(00)00095-6.

[6]

J. A. Carrillo, Y.-P. Choi, S.-Y. Ha, M.-J. Kang and Y. Kim, Contractivity of the Wasserstein metric for the kinetic Kuramoto equation, preprint, arXiv:1301.1883.

[7]

H. Chiba, Continuous limit of the moments system for the globally coupled phase oscillator, Discrete Contin. Dyn. Syst., 33 (2013), 1891-1903. doi: 10.3934/dcds.2013.33.1891.

[8]

Y.-P. Choi, S.-Y. Ha and S. E. Noh, Remarks on the nonlinear stability of the Kuramoto model with inertia, to appear in Quart. Appl. Math.

[9]

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.

[10]

Y.-P. Choi, S.-Y. Ha and S.-B. Yun, Complete synchronization of Kuramoto oscillators with finite inertia, Physica D., 240 (2011), 32-44. doi: 10.1016/j.physd.2010.08.004.

[11]

N. Chopra and M. W. Spong, On exponential synchronization of Kuramoto oscillators, IEEE Trans. Autom. Control., 54 (2009), 353-357. doi: 10.1109/TAC.2008.2007884.

[12]

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.

[13]

H. Daido, Onset of cooperative entrainment in limit-cycle oscillators with uniform all-to-all interactions: Bifurcation of the order function, Physica D., 91 (1996), 24-66. doi: 10.1016/0167-2789(95)00260-X.

[14]

B. C. Daniels, S. T. Dissanayake and B. R. Trees, Synchronization of coupled rotators: Josephson junction ladders and the locally coupled Kuramoto model, Phys. Rev. E., 67 (2003), 026216. doi: 10.1103/PhysRevE.67.026216.

[15]

F. Dorfler and F. Bullo, On the critical coupling for Kuramoto oscillators, SIAM J. Appl. Dyn. Syst., 10 (2011), 1070-1099. doi: 10.1137/10081530X.

[16]

G. B. Ermentrout, Synchronization in a pool of mutually coupled oscillators with random frequencies, J. Math. Biol., 22 (1985), 1-9. doi: 10.1007/BF00276542.

[17]

S.-Y. Ha, T. Y. Ha and J.-H. Kim, On the complete synchronization for the Kuramoto model, Physica D., 239 (2010), 1692-1700. doi: 10.1016/j.physd.2010.05.003.

[18]

S.-Y. Ha and J.-G. Liu, A simple proof of the Cucker-Smale flocking dynamics and mean-field limit, Commun. Math. Sci., 7 (2009), 297-325. doi: 10.4310/CMS.2009.v7.n2.a2.

[19]

H. Hong, M. Y. Choi, J. Yi and K.-S. Soh, Inertia effects on periodic synchronization in a system of coupled oscillators, Phys. Rev. E., 59 (1999), 353-363. doi: 10.1103/PhysRevE.59.353.

[20]

H. Hong, G. S. Jeon and M. Y. Choi, Spontaneous phase oscillation induced by inertia and time delay, Phys. Rev. E., 65 (2002), 026208. doi: 10.1103/PhysRevE.65.026208.

[21]

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

[22]

Y. Kuramoto, International symposium on mathematical problems in mathematical physics, Lecture Notes in Theoretical Physics., 39 (1975), 420-422.

[23]

C. Lancellotti, On the Vlasov limit for systems of nonlinearly coupled oscillators without noise, Transp. Theory Stat. Phys., 34 (2005), 523-535. doi: 10.1080/00411450508951152.

[24]

M. M. Lavrentiev and R. Spigler, Existence and uniqueness of solutions to the Kuramoto-Sakaguchi nonliner parabolic integrodifferential equation, Differ. Integr. Eq., 13 (2000), 649-667.

[25]

H. Neunzert, An introduction to the nonlinear Boltzmann-Vlasov equation, In Kinetic Theories and the Boltzmann Equation, Lecture Notes in Mathematics 1048, Springer, Berlin, Heidelberg. doi: 10.1007/BFb0071878.

[26]

A. Pikovsky, M. Rosenblum and J. Kurths, Synchrnization: A Universal Concept in Nonlinear Sciences, Cambridge University Press, Cambridge, 2001. doi: 10.1017/CBO9780511755743.

[27]

P.-A. Raviart, An analysis of particle methods, in Numerical Methods in Fluid Dynamics (Como, 1983), 243-324, Lecture Notes in Mathematics, 1127, Springer, Berlin, 1985. doi: 10.1007/BFb0074532.

[28]

H. Sakaguchi and Y. Kuramoto, A soluble active rotator model showing phase transitions via mutual entraintment, Prog. Theor. Phys., 76 (1986), 576-581. doi: 10.1143/PTP.76.576.

[29]

H. Sphohn, Large Scale Dynamics of Interacting Particles, Springer-Verlag, Berlin and Heidelberg, 1991. doi: 10.1007/978-3-642-84371-6.

[30]

S. H. Strogatz, From Kuramoto to Crawford: Exploring the onset of synchronization in populations of coupled oscillators, Physica D., 143 (2000), 1-20. doi: 10.1016/S0167-2789(00)00094-4.

[31]

H. A. Tanaka, A. J. Lichtenberg and S. Oishi, First order phase transition resulting from finite inertia in coupled oscillator systems, Phys. Rev. Lett., 78 (1997), 2104-2107. doi: 10.1103/PhysRevLett.78.2104.

[32]

H. A. Tanaka, A. J. Lichtenberg and S. Oishi, Self-synchronization of coupled oscillators with hysteretic responses, Physica D., 100 (1997), 279-300. doi: 10.1016/S0167-2789(96)00193-5.

[33]

S. Watanabe and J. W. Swift, Stability of periodic solutions in series arrays of Josephson junctions with internal capacitance, J. Nonlinear Sci., 7 (1997), 503-536. doi: 10.1007/s003329900038.

[34]

S. Watanabe and S. H. Strogatz, Constants of motion for superconducting Josephson arrays, Physica D., 74 (1994), 197-253. doi: 10.1016/0167-2789(94)90196-1.

[35]

K. Wiesenfeld, R. Colet and S. H. Strogatz, Synchronization transitions in a disordered Josephson series arrays, Phys. Rev. Lett., 76 (1996), 404-407. doi: 10.1103/PhysRevLett.76.404.

[36]

K. Wiesenfeld, R. Colet and S. H. Strogatz, Frequency locking in Josephson arrays: Connection with the Kuramoto model, Phys. Rev. E., 57 (1988), 1563-1569. doi: 10.1103/PhysRevE.57.1563.

[37]

K. Wiesenfeld and J. W. Swift, Averaged equations for Josephson junction series arrays, Phys. Rev. E., 51 (1995), 1020-1025. doi: 10.1103/PhysRevE.51.1020.

[38]

A. T. Winfree, Biological rhythms and the behavior of populations of coupled oscillators, J. Theor. Biol., 16 (1967), 15-42. doi: 10.1016/0022-5193(67)90051-3.

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