• Previous Article
    Global weak solution and smooth solution of the periodic initial value problem for the generalized Landau-Lifshitz-Bloch equation in high dimensions
  • DCDS-B Home
  • This Issue
  • Next Article
    Existence of pullback attractors for the non-autonomous suspension bridge equation with time delay
April  2020, 25(4): 1317-1344. 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  April 2020 Early access  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 and Continuous Dynamical Systems - B, 2020, 25 (4) : 1317-1344. 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.

[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.

[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.

[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.

[5]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[21]

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

[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. 

[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.

[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.

[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.

[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.

[27]

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

[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.

[29]

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

[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.

[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.

[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.

[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.

[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. 

[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.

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.

[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.

[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.

[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.

[5]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[21]

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

[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. 

[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.

[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.

[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.

[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.

[27]

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

[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.

[29]

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

[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.

[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.

[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.

[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.

[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. 

[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.

[1]

Seung-Yeal Ha, Jinwook Jung, Jeongho Kim, Jinyeong Park, Xiongtao Zhang. A mean-field limit of the particle swarmalator model. Kinetic and Related Models, 2021, 14 (3) : 429-468. doi: 10.3934/krm.2021011

[2]

Seung-Yeal Ha, Jeongho Kim, Jinyeong Park, Xiongtao Zhang. Uniform stability and mean-field limit for the augmented Kuramoto model. Networks and Heterogeneous Media, 2018, 13 (2) : 297-322. doi: 10.3934/nhm.2018013

[3]

Matthew Rosenzweig. The mean-field limit of the Lieb-Liniger model. Discrete and Continuous Dynamical Systems, 2022, 42 (6) : 3005-3037. doi: 10.3934/dcds.2022006

[4]

Rong Yang, Li Chen. Mean-field limit for a collision-avoiding flocking system and the time-asymptotic flocking dynamics for the kinetic equation. Kinetic and Related Models, 2014, 7 (2) : 381-400. doi: 10.3934/krm.2014.7.381

[5]

Thierry Paul, Mario Pulvirenti. Asymptotic expansion of the mean-field approximation. Discrete and Continuous Dynamical Systems, 2019, 39 (4) : 1891-1921. doi: 10.3934/dcds.2019080

[6]

Seung-Yeal Ha, Jeongho Kim, Peter Pickl, Xiongtao Zhang. A probabilistic approach for the mean-field limit to the Cucker-Smale model with a singular communication. Kinetic and Related Models, 2019, 12 (5) : 1045-1067. doi: 10.3934/krm.2019039

[7]

Seung-Yeal Ha, Jeongho Kim, Xiongtao Zhang. Uniform stability of the Cucker-Smale model and its application to the Mean-Field limit. Kinetic and Related Models, 2018, 11 (5) : 1157-1181. doi: 10.3934/krm.2018045

[8]

Hyunjin Ahn, Seung-Yeal Ha, Jeongho Kim. Uniform stability of the relativistic Cucker-Smale model and its application to a mean-field limit. Communications on Pure and Applied Analysis, 2021, 20 (12) : 4209-4237. doi: 10.3934/cpaa.2021156

[9]

Patrick Gerard, Christophe Pallard. A mean-field toy model for resonant transport. Kinetic and Related Models, 2010, 3 (2) : 299-309. doi: 10.3934/krm.2010.3.299

[10]

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

[11]

Azmy S. Ackleh, Vinodh K. Chellamuthu, Kazufumi Ito. Finite difference approximations for measure-valued solutions of a hierarchically size-structured population model. Mathematical Biosciences & Engineering, 2015, 12 (2) : 233-258. doi: 10.3934/mbe.2015.12.233

[12]

Gerasimenko Viktor. Heisenberg picture of quantum kinetic evolution in mean-field limit. Kinetic and Related Models, 2011, 4 (1) : 385-399. doi: 10.3934/krm.2011.4.385

[13]

Michael Herty, Mattia Zanella. Performance bounds for the mean-field limit of constrained dynamics. Discrete and Continuous Dynamical Systems, 2017, 37 (4) : 2023-2043. doi: 10.3934/dcds.2017086

[14]

Nastassia Pouradier Duteil. Mean-field limit of collective dynamics with time-varying weights. Networks and Heterogeneous Media, 2022, 17 (2) : 129-161. doi: 10.3934/nhm.2022001

[15]

Ammari Zied, Liard Quentin. On uniqueness of measure-valued solutions to Liouville's equation of Hamiltonian PDEs. Discrete and Continuous Dynamical Systems, 2018, 38 (2) : 723-748. doi: 10.3934/dcds.2018032

[16]

Cleopatra Christoforou, Myrto Galanopoulou, Athanasios E. Tzavaras. Measure-valued solutions for the equations of polyconvex adiabatic thermoelasticity. Discrete and Continuous Dynamical Systems, 2019, 39 (11) : 6175-6206. doi: 10.3934/dcds.2019269

[17]

Maria Michaela Porzio, Flavia Smarrazzo, Alberto Tesei. Radon measure-valued solutions of unsteady filtration equations. Discrete and Continuous Dynamical Systems, 2022  doi: 10.3934/dcds.2022040

[18]

Franco Flandoli, Enrico Priola, Giovanni Zanco. A mean-field model with discontinuous coefficients for neurons with spatial interaction. Discrete and Continuous Dynamical Systems, 2019, 39 (6) : 3037-3067. doi: 10.3934/dcds.2019126

[19]

Young-Pil Choi, Samir Salem. Cucker-Smale flocking particles with multiplicative noises: Stochastic mean-field limit and phase transition. Kinetic and Related Models, 2019, 12 (3) : 573-592. doi: 10.3934/krm.2019023

[20]

Joachim Crevat. Mean-field limit of a spatially-extended FitzHugh-Nagumo neural network. Kinetic and Related Models, 2019, 12 (6) : 1329-1358. doi: 10.3934/krm.2019052

2020 Impact Factor: 1.327

Metrics

  • PDF downloads (263)
  • HTML views (158)
  • Cited by (1)

Other articles
by authors

[Back to Top]