American Institute of Mathematical Sciences

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May  2021, 26(5): 2749-2779. doi: 10.3934/dcdsb.2020204

Collective behaviors of a Winfree ensemble on an infinite cylinder

Seung-Yeal Ha 1, , Myeongju Kang 2, and  Bora Moon 3,,

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)

* Corresponding author: Bora Moon

Received  February 2020 Revised  April 2020 Published  May 2021 Early access  June 2020

Fund Project: The work of S.-Y. Ha is supported by the NRF grant (2017R1A2B2001864) and the work of B. Moon was supported by Basic Science Research Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Education (2019R1I1A1A01059585)

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.

Keywords: Complete oscillator death, synchronization, mean-field limit, Winfree model.
Mathematics Subject Classification: Primary:70F99, 92B25;Secondary:35Q70.
Citation: Seung-Yeal Ha, Myeongju Kang, Bora Moon. Collective behaviors of a Winfree ensemble on an infinite cylinder. Discrete and Continuous Dynamical Systems - B, 2021, 26 (5) : 2749-2779. doi: 10.3934/dcdsb.2020204
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. 

[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. 

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. 

[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. 

Figure 1.  Time evolution of $ \frac{X(t)}{t} $ for two different initial data
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Figure 2.  Initial and terminal configurations of $ (X_1, Y_1) $ and $ (X_2, Y_2) $
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Figure 3.  Uniform $ \ell_1 $-stability
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