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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 & 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. doi: 10.1103/RevModPhys.77.137. Google Scholar

[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. doi: 10.1103/PhysRevE.62.3437. Google Scholar

[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. doi: 10.1103/PhysRevLett.81.2229. Google Scholar

[4]

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

[5]

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

[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, (). Google Scholar

[7]

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

[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., (). Google Scholar

[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. doi: 10.1016/j.physd.2011.11.011. Google Scholar

[10]

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

[11]

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

[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. doi: 10.1016/S0167-2789(98)00235-8. Google Scholar

[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. doi: 10.1016/0167-2789(95)00260-X. Google Scholar

[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). doi: 10.1103/PhysRevE.67.026216. Google Scholar

[15]

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

[16]

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

[17]

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

[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. doi: 10.4310/CMS.2009.v7.n2.a2. Google Scholar

[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. doi: 10.1103/PhysRevE.59.353. Google Scholar

[20]

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

[21]

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

[22]

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

[23]

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

[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. Google Scholar

[25]

H. Neunzert, An introduction to the nonlinear Boltzmann-Vlasov equation,, In Kinetic Theories and the Boltzmann Equation, (1048). doi: 10.1007/BFb0071878. Google Scholar

[26]

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

[27]

P.-A. Raviart, An analysis of particle methods,, in Numerical Methods in Fluid Dynamics (Como, 1127 (1983), 243. doi: 10.1007/BFb0074532. Google Scholar

[28]

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

[29]

H. Sphohn, Large Scale Dynamics of Interacting Particles,, Springer-Verlag, (1991). doi: 10.1007/978-3-642-84371-6. Google Scholar

[30]

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

[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. doi: 10.1103/PhysRevLett.78.2104. Google Scholar

[32]

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

[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. doi: 10.1007/s003329900038. Google Scholar

[34]

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

[35]

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

[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. doi: 10.1103/PhysRevE.57.1563. Google Scholar

[37]

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

[38]

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

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. doi: 10.1103/RevModPhys.77.137. Google Scholar

[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. doi: 10.1103/PhysRevE.62.3437. Google Scholar

[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. doi: 10.1103/PhysRevLett.81.2229. Google Scholar

[4]

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

[5]

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

[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, (). Google Scholar

[7]

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

[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., (). Google Scholar

[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. doi: 10.1016/j.physd.2011.11.011. Google Scholar

[10]

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

[11]

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

[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. doi: 10.1016/S0167-2789(98)00235-8. Google Scholar

[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. doi: 10.1016/0167-2789(95)00260-X. Google Scholar

[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). doi: 10.1103/PhysRevE.67.026216. Google Scholar

[15]

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

[16]

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

[17]

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

[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. doi: 10.4310/CMS.2009.v7.n2.a2. Google Scholar

[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. doi: 10.1103/PhysRevE.59.353. Google Scholar

[20]

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

[21]

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

[22]

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

[23]

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

[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. Google Scholar

[25]

H. Neunzert, An introduction to the nonlinear Boltzmann-Vlasov equation,, In Kinetic Theories and the Boltzmann Equation, (1048). doi: 10.1007/BFb0071878. Google Scholar

[26]

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

[27]

P.-A. Raviart, An analysis of particle methods,, in Numerical Methods in Fluid Dynamics (Como, 1127 (1983), 243. doi: 10.1007/BFb0074532. Google Scholar

[28]

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

[29]

H. Sphohn, Large Scale Dynamics of Interacting Particles,, Springer-Verlag, (1991). doi: 10.1007/978-3-642-84371-6. Google Scholar

[30]

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

[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. doi: 10.1103/PhysRevLett.78.2104. Google Scholar

[32]

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

[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. doi: 10.1007/s003329900038. Google Scholar

[34]

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

[35]

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

[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. doi: 10.1103/PhysRevE.57.1563. Google Scholar

[37]

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

[38]

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

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