The Winfree model is the first phase model for synchronization and it exhibits diverse asymptotic patterns that cannot be observed in the Kuramoto model. In this paper, we propose a Winfree type model describing the aggregation of particles on the surface of an infinite cylinder. For a special case, our proposed model is in fact equivalent to the complex Winfree model. For the proposed model, we present a sufficient framework leading to the complete oscillator death and uniform $ \ell_p $-stability in a large coupling regime. We also derive the corresponding kinetic model via uniform-in-time mean-field limit. In addition, we also provide several numerical simulations for the particle and compare them with analytical results.
Citation: |
[1] |
J. A. Acebron, L. L. Bonilla, C. J. P. Pérez Vicente, F. Ritort and R. Spigler, The Kuramoto model: A simple paradigm for synchronization phenomena, Rev. Mod. Phys., 77 (2005), 137-185.
![]() |
[2] |
G. Albi, N. Bellomo, L. Fermo, S.-Y. Ha, J. Kim, L. Pareschi, D. Poyato and J. Soler, Vehicular traffic, crowds, and swarms: From kinetic theory and multiscale methods to applications and research perspectives, Math. Models Methods Appl. Sci., 29 (2019), 1901-2005.
doi: 10.1142/S0218202519500374.![]() ![]() ![]() |
[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.
![]() |
[4] |
M. Ballerini, N. Cabibbo, R. Candelier, A. Cavagna, E. Cisbani, I. Giardina, V. Lecomte, A. Orlandi, G. Parisi, A. Procaccini, M. Viale and V. Zdravkovic, Interaction ruling animal collective behavior depends on topological rather than metric distance: Evidence from a field study, Proc. Natl. Acad. Sci. USA, 105 (2008), 1232-1237.
![]() |
[5] |
J. Buck and E. Buck, Biology of synchronous flashing of fireflies, Nature, 211 (1966), 562.
![]() |
[6] |
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.![]() ![]() ![]() |
[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] |
F. Cucker and S. Smale, Emergent behavior in flocks, IEEE Trans. Automat. Control, 52 (2007), 852-862.
doi: 10.1109/TAC.2007.895842.![]() ![]() ![]() |
[9] |
P. Degond and S. Motsch, Large scale dynamics of the persistent turning walker model of fish behavior, J. Stat. Phys., 131 (2008), 989-1021.
doi: 10.1007/s10955-008-9529-8.![]() ![]() ![]() |
[10] |
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.![]() ![]() ![]() |
[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. Giannuzzi, D. Marinazzo, G. Nardulli, M. Pellicoro and S. Stramaglia, Phase diagram of a generalized Winfree model, Physical Review E, 75 (2007), 051104.
![]() |
[13] |
S.-Y. Ha, M.-J. Kang, C. Lattanzio and B. Rubino, A class of interacting particle systems on the infinite cylinder with flocking phenomena, Math. Models Methods Appl. Sci., 22 (2012), 1250008, 25 pp.
doi: 10.1142/S021820251250008X.![]() ![]() ![]() |
[14] |
S.-Y. Ha, M. Kang and B. Moon, On the emerging asymptotic patterns for the Winfree model with frustrations, submitted.
![]() |
[15] |
S.-Y. Ha and D. Kim, Robustness and asymptotic stability for the Winfree model on a general network under the effect of time-delay, J. Math. Phys., 59 (2018), 112702, 20 pp.
doi: 10.1063/1.5017063.![]() ![]() ![]() |
[16] |
S.-Y. Ha, D. Kim and B. Moon, Interplay of random inputs and adaptive couplings in the Winfree model, submitted.
![]() |
[17] |
S.-Y. Ha, J. 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.![]() ![]() ![]() |
[18] |
S.-Y. Ha, D. Ko, J. Park and S. W. Ryoo, Emergence of partial locking states from the ensemble of Winfree oscillators, Quart. Appl. Math., 75 (2017), 39-68.
doi: 10.1090/qam/1448.![]() ![]() ![]() |
[19] |
S.-Y. Ha, D. Ko, J. Park and S. W. Ryoo, Emergent dynamics of Winfree oscillators on locally coupled networks, J. Differential Equations, 260 (2016), 4203-4236.
doi: 10.1016/j.jde.2015.11.008.![]() ![]() ![]() |
[20] |
S.-Y. Ha, D. Ko, J. 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.![]() ![]() ![]() |
[21] |
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.![]() ![]() ![]() |
[22] |
S.-Y. Ha, J. Park and S. W. Ryoo, Emergence of phase-locked states for the Winfree model in a large coupling regime, Discrete Contin. Dyn. Syst., 35 (2015), 3417-3436.
doi: 10.3934/dcds.2015.35.3417.![]() ![]() ![]() |
[23] |
S.-Y. Ha, J. Park and X. T. Zhang, A global well-posedness and asymptotic dynamics of the kinetic Winfree equation, Discrete Contin. Dyn. Syst. Ser. B, 25 (2020), 1317-1344.
![]() |
[24] |
Y. Kuramoto, International symposium on mathematical problems in mathematical physics, Lecture Notes in Theoretical Physics, 30 (1975), 420.
![]() |
[25] |
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.![]() ![]() ![]() |
[26] |
P. V. Mieghem, A complex variant of the Kuramoto model, preprint, (2009), available at: https://www.nas.ewi.tudelft.nl/people/Piet/papers.
![]() |
[27] |
H. Neunzert, An introduction to the nonlinear Boltzmann-Vlasov equation, Kinetic Theories and the Boltzmann Equation, Lecture Notes in Math., Springer, Berlin, 1048 (1984), 60-110.
doi: 10.1007/BFb0071878.![]() ![]() ![]() |
[28] |
W. Oukil, A. Kessi and Ph. Thieullen, Synchronization hypothesis in the Winfree model, Dyn. Syst., 32 (2017), 326-339.
doi: 10.1080/14689367.2016.1227303.![]() ![]() ![]() |
[29] |
W. Oukil, Ph. Thieullen and A. Kessi, Invariant cone and synchronization state stability of the mean field models, Dyn. Syst., 34 (2019), 422-433.
doi: 10.1080/14689367.2018.1547683.![]() ![]() ![]() |
[30] |
D. A. Paley, N. E. Leonard, R. Sepulchre, D. Grunbaum and J. K. Parrish, Oscillator models and collective motion, IEEE Control Systems, 27 (2007), 89-105.
![]() |
[31] |
D. D. Quinn, R. H. Rand and S. 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 (CD-ROM), (2005).
![]() |
[33] |
J. Toner and Y. Tu, Flocks, herds, and schools: A quantitative theory of flocking, Phys. Rev. E, 58 (1998), 4828-4858.
doi: 10.1103/PhysRevE.58.4828.![]() ![]() ![]() |
[34] |
C. Villani, Optimal Transport: Old and New, Grundlehren der Mathematischen Wissenschaften, 338. Springer-Verlag, Berlin, 2009.
doi: 10.1007/978-3-540-71050-9.![]() ![]() ![]() |
[35] |
A. Winfree, Biological rhythms and the behavior of populations of coupled oscillators, J. Theoret. Biol., 16 (1967), 15-42.
![]() |