
-
Previous Article
Random attractors for stochastic Navier-Stokes equation on a 2D rotating sphere with stable Lévy noise
- DCDS-B Home
- This Issue
-
Next Article
On predation effort allocation strategy over two patches
Collective behaviors of a Winfree ensemble on an infinite cylinder
1. | Department of Mathematical Sciences and Research Institute of Mathematics, Seoul National University, Seoul 08826 and, Korea Institute for Advanced Study, Hoegiro 85, Seoul 02455, Korea (Republic of) |
2. | Department of Mathematical Sciences, Seoul National University, Seoul 08826, Korea (Republic of) |
3. | Research Institute of Mathematics, Seoul National University, Seoul 08826, Korea (Republic of) |
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.
References:
[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. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[5] |
J. Buck and E. Buck, Biology of synchronous flashing of fireflies, Nature, 211 (1966), 562. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[24] |
Y. Kuramoto, International symposium on mathematical problems in mathematical physics, Lecture Notes in Theoretical Physics, 30 (1975), 420. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[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). Google Scholar |
[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. Google Scholar |
show all references
References:
[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. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[5] |
J. Buck and E. Buck, Biology of synchronous flashing of fireflies, Nature, 211 (1966), 562. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[24] |
Y. Kuramoto, International symposium on mathematical problems in mathematical physics, Lecture Notes in Theoretical Physics, 30 (1975), 420. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[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). Google Scholar |
[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. Google Scholar |


[1] |
Illés Horváth, Kristóf Attila Horváth, Péter Kovács, Miklós Telek. Mean-field analysis of a scaling MAC radio protocol. Journal of Industrial & Management Optimization, 2021, 17 (1) : 279-297. doi: 10.3934/jimo.2019111 |
[2] |
Jingrui Sun, Hanxiao Wang. Mean-field stochastic linear-quadratic optimal control problems: Weak closed-loop solvability. Mathematical Control & Related Fields, 2021, 11 (1) : 47-71. doi: 10.3934/mcrf.2020026 |
[3] |
Laura Aquilanti, Simone Cacace, Fabio Camilli, Raul De Maio. A Mean Field Games model for finite mixtures of Bernoulli and categorical distributions. Journal of Dynamics & Games, 2020 doi: 10.3934/jdg.2020033 |
[4] |
Shuang Chen, Jinqiao Duan, Ji Li. Effective reduction of a three-dimensional circadian oscillator model. Discrete & Continuous Dynamical Systems - B, 2020 doi: 10.3934/dcdsb.2020349 |
[5] |
Daniele Bartolucci, Changfeng Gui, Yeyao Hu, Aleks Jevnikar, Wen Yang. Mean field equations on tori: Existence and uniqueness of evenly symmetric blow-up solutions. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3093-3116. doi: 10.3934/dcds.2020039 |
[6] |
Alain Bensoussan, Xinwei Feng, Jianhui Huang. Linear-quadratic-Gaussian mean-field-game with partial observation and common noise. Mathematical Control & Related Fields, 2021, 11 (1) : 23-46. doi: 10.3934/mcrf.2020025 |
[7] |
Yangjian Sun, Changjian Liu. The Poincaré bifurcation of a SD oscillator. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1565-1577. doi: 10.3934/dcdsb.2020173 |
[8] |
Yi-Long Luo, Yangjun Ma. Low Mach number limit for the compressible inertial Qian-Sheng model of liquid crystals: Convergence for classical solutions. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 921-966. doi: 10.3934/dcds.2020304 |
[9] |
Björn Augner, Dieter Bothe. The fast-sorption and fast-surface-reaction limit of a heterogeneous catalysis model. Discrete & Continuous Dynamical Systems - S, 2021, 14 (2) : 533-574. doi: 10.3934/dcdss.2020406 |
[10] |
Jiannan Zhang, Ping Chen, Zhuo Jin, Shuanming Li. Open-loop equilibrium strategy for mean-variance portfolio selection: A log-return model. Journal of Industrial & Management Optimization, 2021, 17 (2) : 765-777. doi: 10.3934/jimo.2019133 |
[11] |
Leslaw Skrzypek, Yuncheng You. Feedback synchronization of FHN cellular neural networks. Discrete & Continuous Dynamical Systems - B, 2020 doi: 10.3934/dcdsb.2021001 |
[12] |
Liupeng Wang, Yunqing Huang. Error estimates for second-order SAV finite element method to phase field crystal model. Electronic Research Archive, 2021, 29 (1) : 1735-1752. doi: 10.3934/era.2020089 |
[13] |
Qianqian Han, Xiao-Song Yang. Qualitative analysis of a generalized Nosé-Hoover oscillator. Discrete & Continuous Dynamical Systems - B, 2020 doi: 10.3934/dcdsb.2020346 |
[14] |
Xiangrui Meng, Jian Gao. Complete weight enumerator of torsion codes. Advances in Mathematics of Communications, 2020 doi: 10.3934/amc.2020124 |
[15] |
Jie Li, Xiangdong Ye, Tao Yu. Mean equicontinuity, complexity and applications. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 359-393. doi: 10.3934/dcds.2020167 |
[16] |
Nguyen Thi Kim Son, Nguyen Phuong Dong, Le Hoang Son, Alireza Khastan, Hoang Viet Long. Complete controllability for a class of fractional evolution equations with uncertainty. Evolution Equations & Control Theory, 2020 doi: 10.3934/eect.2020104 |
[17] |
Dandan Wang, Xiwang Cao, Gaojun Luo. A class of linear codes and their complete weight enumerators. Advances in Mathematics of Communications, 2021, 15 (1) : 73-97. doi: 10.3934/amc.2020044 |
[18] |
Xinfu Chen, Huiqiang Jiang, Guoqing Liu. Boundary spike of the singular limit of an energy minimizing problem. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3253-3290. doi: 10.3934/dcds.2020124 |
[19] |
Hai-Liang Li, Tong Yang, Mingying Zhong. Diffusion limit of the Vlasov-Poisson-Boltzmann system. Kinetic & Related Models, , () : -. doi: 10.3934/krm.2021003 |
[20] |
Hideki Murakawa. Fast reaction limit of reaction-diffusion systems. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 1047-1062. doi: 10.3934/dcdss.2020405 |
2019 Impact Factor: 1.27
Tools
Metrics
Other articles
by authors
[Back to Top]