doi: 10.3934/dcdsb.2019229

A global well-posedness and asymptotic dynamics of the kinetic Winfree equation

1. 

Department of Mathematical Sciences and Research Institute of Mathematics, Seoul National University, Seoul, 08826, Korea

2. 

Korea Institute for Advanced Study, Hoegiro 85, Seoul, 02455, Korea

3. 

Department of Mathematics and Research Institute of Natural Sciences, Hanyang University, 222 Wangsimni-ro, Seongdong-gu, Seoul 04763, Korea

4. 

Center for Mathematical sciences, Huazhong University of Science and Technology, Luoyu road 1037, Wuhan 430074, China

* Corresponding author: Xiongtao Zhang

Received  April 2018 Published  November 2019

Fund Project: The work of S.-Y. Ha is supported by National Research Foundation of Korea(NRF-2017R1A2B2001864), and the work of J. Park has been supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (NRF-2018R1C1B5043861). The work of X. Zhang is supported by the National Natural Science Foundation of China (Grant No. 11801194)

We study a global well-posedness and asymptotic dynamics of measure-valued solutions to the kinetic Winfree equation. For this, we introduce a second-order extension of the first-order Winfree model on an extended phase-frequency space. We present the uniform(-in-time) $ \ell_p $-stability estimate with respect to initial data and the equivalence relation between the original Winfree model and its second-order extension. For this extended model, we present uniform-in-time mean-field limit and large-time behavior of measure-valued solution for the second-order Winfree model. Using stability and asymptotic estimates for the extended model and the equivalence relation, we recover the uniform mean-field limit and large-time asymptotics for the Winfree model. 200 words.

Citation: Seung-Yeal Ha, Jinyeong Park, Xiongtao Zhang. A global well-posedness and asymptotic dynamics of the kinetic Winfree equation. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2019229
References:
[1]

J. A. AcebrónL. L. BonillaC. J. P. VicenteF. 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.  Google Scholar

[2]

L. AngeliniG. LattanziR. MaestriD. MarinazzoG. NardulliL. NittiM. PellicoroG. D. Pinna and S. Stramaglia, Phase shifts of synchronized oscillators and the systolic-diastolic blood pressure relation, Physical Review E, 69 (2004), 061923.  doi: 10.1103/PhysRevE.69.061923.  Google Scholar

[3]

J. T. Ariaratnam and S. H. Strogatz, Phase diagram for the Winfree model of coupled nonlinear oscillators, Phys. Rev. Lett., 86 (2001), 4278-4281.  doi: 10.1103/PhysRevLett.86.4278.  Google Scholar

[4]

D. BenedettoE. Caglioti and U. Montemagno, On the complete phase synchronization for the Kuramoto model in the mean-field limit, Commun. Math. Sci., 13 (2015), 1775-1786.  doi: 10.4310/CMS.2015.v13.n7.a6.  Google Scholar

[5]

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

[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.  Google Scholar

[7]

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

[8]

P. Degond and S. Motsch, Continuum limit of self-driven particles with orientation interaction, Math. Mod. Meth. Appl. Sci., 18 (2008), 1193-1215.  doi: 10.1142/S0218202508003005.  Google Scholar

[9]

P. Degond and S. Motsch, Large scale dynamics of the persistent turing walker model of fish behavior, J. Stat. Phys., 131 (2008), 989-1021.  doi: 10.1007/s10955-008-9529-8.  Google Scholar

[10]

J.-G. Dong and X. P. Xue, Synchronization analysis of Kuramoto oscillators, Commun. Math. Sci., 11 (2013), 465-480.  doi: 10.4310/CMS.2013.v11.n2.a7.  Google Scholar

[11]

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

[12]

F. Dörfler and F. Bullo, Synchronization in complex network of phase oscillators: A survey, Automatica J. IFAC, 50 (2014), 1539-1564.  doi: 10.1016/j.automatica.2014.04.012.  Google Scholar

[13]

F. GiannuzziD. MarinazzoG. NardulliM. Pellicoro and S. Stramaglia, Phase diagram of a generalized Winfree model, Physical Review E, 75 (2007), 051104.  doi: 10.1103/PhysRevE.75.051104.  Google Scholar

[14]

S.-Y. HaH. K. Kim and S. W. Ryoo, Emergence of phase-locked states for the Kuramoto model in a large coupling regime, Commun. Math. Sci., 14 (2016), 1073-1091.  doi: 10.4310/CMS.2016.v14.n4.a10.  Google Scholar

[15]

S.-Y. HaJ. Kim and X. T. Zhang, Uniform stability of the Cucker-Smale model and its application to the mean-field limit, Kinet. Relat. Models, 11 (2018), 1157-1181.  doi: 10.3934/krm.2018045.  Google Scholar

[16]

S.-Y. HaD. KoJ. Park and S. W. Ryoo, Emergent dynamics of Winfree oscillators on locally coupled networks, J. Diff. Equations, 260 (2016), 4203-4236.  doi: 10.1016/j.jde.2015.11.008.  Google Scholar

[17]

S.-Y. HaD. KoJ. Park and X. T. Zhang, Collective synchronization of classical and quantum oscillators, EMS Surv. Math. Sci., 3 (2016), 209-267.  doi: 10.4171/EMSS/17.  Google Scholar

[18]

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

[19]

S.-Y. HaJ. Park and S. W. Ryoo, Emergence of phase-locked states for the Winfree model in a large coupling regime, Discrete and Continuous Dynamical Systems-A, 35 (2015), 3417-3436.  doi: 10.3934/dcds.2015.35.3417.  Google Scholar

[20]

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

[21]

Y. Kuramoto, International symposium on mathematical problems in mathematical physics, Lecture Notes in Theoretical Physics, 30 (1975), 420.   Google Scholar

[22]

Y. Kuramoto, Self-entrainment of a population of coupled non-linear oscillators, International Symposium on Mathematical Problems in Theoretical Physics, Lecture Notes in Phys., Springer, Berlin, 39 (1975), 420-422.   Google Scholar

[23]

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

[24]

N. E. LeonardD. A. PaleyF. LekienR. SepulchreD. M. Fratantoni and R. E. Davis, Collective motion, sensor networks and ocean sampling, Proc. IEEE, 95 (2007), 48-74.  doi: 10.1109/JPROC.2006.887295.  Google Scholar

[25]

S. Louca and F. M. Atay, Spatially structured networks of pulse-coupled phase oscillators on metric spaces, Discrete and Continuous Dynamical Systems-B, 34 (2014), 3703-3745.  doi: 10.3934/dcds.2014.34.3703.  Google Scholar

[26]

H. Neunzert, An introduction to the nonlinear Boltzmann-Vlasov equation, Kinetic Theories and the Boltzmann equation, Lecture Notes in Mathematics, Lecture Notes in Math., Springer, Berlin, 1048 (1984), 60–110. doi: 10.1007/BFb0071878.  Google Scholar

[27]

W. Oukil, Synchronization in abstract mean field models, preprint, arXiv: 1703.07692. Google Scholar

[28]

W. OukilA. Kessi and P. Thieullen, Synchronization hypothesis in the Winfree model, Dyn. Syst., 32 (2017), 326-339.  doi: 10.1080/14689367.2016.1227303.  Google Scholar

[29]

D. A. PaleyN. E. LeonardR. SepulchreD. Grunbaum and J. K. Parrish, Oscillator models and collective motion, IEEE Control Systems., 27 (2007), 89-105.   Google Scholar

[30]

L. PereaG. Gómez and P. Elosegui, Extension of the Cucker-Smale control law to space flight formation, J. Guidance, Control and Dynamics, 32 (2009), 526-536.  doi: 10.2514/1.36269.  Google Scholar

[31]

D. D. Quinn, R. H. Rand and S. H. Strogatz, Singular unlocking transition in the Winfree model of coupled oscillators, Physical Review E, 75 (2007), 036218, 10 pp. doi: 10.1103/PhysRevE.75.036218.  Google Scholar

[32]

D. D. Quinn, R. H. Rand and S. Strogatz, Synchronization in the Winfree model of coupled nonlinear interactions, A. ENOC 2005 Conference, Eindhoven, Netherlands, 2005. Google Scholar

[33]

C. Villani, Optimal Transport, Old and New, Grundlehren der Mathematischen Wissenschaften, 338, Springer-Verlag, Berlin, 2009. doi: 10.1007/978-3-540-71050-9.  Google Scholar

[34]

A. Winfree, 24 hard problems about the mathematics of 24 hour rhythms, Nonlinear Oscillations in Biology (Proc. Tenth Summer Sem. Appl. Math., Univ. Utah, Salt Lake City, Utah, 1978), Lectures in Appl. Math., Amer. Math. Soc., Providence, R.I., 17 (1979), 93-126.   Google Scholar

[35]

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

show all references

References:
[1]

J. A. AcebrónL. L. BonillaC. J. P. VicenteF. 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.  Google Scholar

[2]

L. AngeliniG. LattanziR. MaestriD. MarinazzoG. NardulliL. NittiM. PellicoroG. D. Pinna and S. Stramaglia, Phase shifts of synchronized oscillators and the systolic-diastolic blood pressure relation, Physical Review E, 69 (2004), 061923.  doi: 10.1103/PhysRevE.69.061923.  Google Scholar

[3]

J. T. Ariaratnam and S. H. Strogatz, Phase diagram for the Winfree model of coupled nonlinear oscillators, Phys. Rev. Lett., 86 (2001), 4278-4281.  doi: 10.1103/PhysRevLett.86.4278.  Google Scholar

[4]

D. BenedettoE. Caglioti and U. Montemagno, On the complete phase synchronization for the Kuramoto model in the mean-field limit, Commun. Math. Sci., 13 (2015), 1775-1786.  doi: 10.4310/CMS.2015.v13.n7.a6.  Google Scholar

[5]

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

[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.  Google Scholar

[7]

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

[8]

P. Degond and S. Motsch, Continuum limit of self-driven particles with orientation interaction, Math. Mod. Meth. Appl. Sci., 18 (2008), 1193-1215.  doi: 10.1142/S0218202508003005.  Google Scholar

[9]

P. Degond and S. Motsch, Large scale dynamics of the persistent turing walker model of fish behavior, J. Stat. Phys., 131 (2008), 989-1021.  doi: 10.1007/s10955-008-9529-8.  Google Scholar

[10]

J.-G. Dong and X. P. Xue, Synchronization analysis of Kuramoto oscillators, Commun. Math. Sci., 11 (2013), 465-480.  doi: 10.4310/CMS.2013.v11.n2.a7.  Google Scholar

[11]

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

[12]

F. Dörfler and F. Bullo, Synchronization in complex network of phase oscillators: A survey, Automatica J. IFAC, 50 (2014), 1539-1564.  doi: 10.1016/j.automatica.2014.04.012.  Google Scholar

[13]

F. GiannuzziD. MarinazzoG. NardulliM. Pellicoro and S. Stramaglia, Phase diagram of a generalized Winfree model, Physical Review E, 75 (2007), 051104.  doi: 10.1103/PhysRevE.75.051104.  Google Scholar

[14]

S.-Y. HaH. K. Kim and S. W. Ryoo, Emergence of phase-locked states for the Kuramoto model in a large coupling regime, Commun. Math. Sci., 14 (2016), 1073-1091.  doi: 10.4310/CMS.2016.v14.n4.a10.  Google Scholar

[15]

S.-Y. HaJ. Kim and X. T. Zhang, Uniform stability of the Cucker-Smale model and its application to the mean-field limit, Kinet. Relat. Models, 11 (2018), 1157-1181.  doi: 10.3934/krm.2018045.  Google Scholar

[16]

S.-Y. HaD. KoJ. Park and S. W. Ryoo, Emergent dynamics of Winfree oscillators on locally coupled networks, J. Diff. Equations, 260 (2016), 4203-4236.  doi: 10.1016/j.jde.2015.11.008.  Google Scholar

[17]

S.-Y. HaD. KoJ. Park and X. T. Zhang, Collective synchronization of classical and quantum oscillators, EMS Surv. Math. Sci., 3 (2016), 209-267.  doi: 10.4171/EMSS/17.  Google Scholar

[18]

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

[19]

S.-Y. HaJ. Park and S. W. Ryoo, Emergence of phase-locked states for the Winfree model in a large coupling regime, Discrete and Continuous Dynamical Systems-A, 35 (2015), 3417-3436.  doi: 10.3934/dcds.2015.35.3417.  Google Scholar

[20]

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

[21]

Y. Kuramoto, International symposium on mathematical problems in mathematical physics, Lecture Notes in Theoretical Physics, 30 (1975), 420.   Google Scholar

[22]

Y. Kuramoto, Self-entrainment of a population of coupled non-linear oscillators, International Symposium on Mathematical Problems in Theoretical Physics, Lecture Notes in Phys., Springer, Berlin, 39 (1975), 420-422.   Google Scholar

[23]

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

[24]

N. E. LeonardD. A. PaleyF. LekienR. SepulchreD. M. Fratantoni and R. E. Davis, Collective motion, sensor networks and ocean sampling, Proc. IEEE, 95 (2007), 48-74.  doi: 10.1109/JPROC.2006.887295.  Google Scholar

[25]

S. Louca and F. M. Atay, Spatially structured networks of pulse-coupled phase oscillators on metric spaces, Discrete and Continuous Dynamical Systems-B, 34 (2014), 3703-3745.  doi: 10.3934/dcds.2014.34.3703.  Google Scholar

[26]

H. Neunzert, An introduction to the nonlinear Boltzmann-Vlasov equation, Kinetic Theories and the Boltzmann equation, Lecture Notes in Mathematics, Lecture Notes in Math., Springer, Berlin, 1048 (1984), 60–110. doi: 10.1007/BFb0071878.  Google Scholar

[27]

W. Oukil, Synchronization in abstract mean field models, preprint, arXiv: 1703.07692. Google Scholar

[28]

W. OukilA. Kessi and P. Thieullen, Synchronization hypothesis in the Winfree model, Dyn. Syst., 32 (2017), 326-339.  doi: 10.1080/14689367.2016.1227303.  Google Scholar

[29]

D. A. PaleyN. E. LeonardR. SepulchreD. Grunbaum and J. K. Parrish, Oscillator models and collective motion, IEEE Control Systems., 27 (2007), 89-105.   Google Scholar

[30]

L. PereaG. Gómez and P. Elosegui, Extension of the Cucker-Smale control law to space flight formation, J. Guidance, Control and Dynamics, 32 (2009), 526-536.  doi: 10.2514/1.36269.  Google Scholar

[31]

D. D. Quinn, R. H. Rand and S. H. Strogatz, Singular unlocking transition in the Winfree model of coupled oscillators, Physical Review E, 75 (2007), 036218, 10 pp. doi: 10.1103/PhysRevE.75.036218.  Google Scholar

[32]

D. D. Quinn, R. H. Rand and S. Strogatz, Synchronization in the Winfree model of coupled nonlinear interactions, A. ENOC 2005 Conference, Eindhoven, Netherlands, 2005. Google Scholar

[33]

C. Villani, Optimal Transport, Old and New, Grundlehren der Mathematischen Wissenschaften, 338, Springer-Verlag, Berlin, 2009. doi: 10.1007/978-3-540-71050-9.  Google Scholar

[34]

A. Winfree, 24 hard problems about the mathematics of 24 hour rhythms, Nonlinear Oscillations in Biology (Proc. Tenth Summer Sem. Appl. Math., Univ. Utah, Salt Lake City, Utah, 1978), Lectures in Appl. Math., Amer. Math. Soc., Providence, R.I., 17 (1979), 93-126.   Google Scholar

[35]

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

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