August  2015, 35(8): 3417-3436. doi: 10.3934/dcds.2015.35.3417

Emergence of phase-locked states for the Winfree model in a large coupling regime

1. 

Department of Mathematical Sciences and Research Institute of Mathematics, Seoul National University, Seoul 151-747

2. 

Department of Mathematical Sciences, Seoul National University, Seoul, 151-747, South Korea, South Korea

Received  November 2014 Revised  January 2015 Published  February 2015

We study the large-time behavior of the globally coupled Winfree model in a large coupling regime. The Winfree model is the first mathematical model for the synchronization phenomenon in an ensemble of weakly coupled limit-cycle oscillators. For the dynamic formation of phase-locked states, we provide a sufficient framework in terms of geometric conditions on the coupling functions and coupling strength. We show that in the proposed framework, the emergent phase-locked state is the unique equilibrium state and it is asymptotically stable in an $l^1$-norm; further, we investigate its configurational structure. We also provide several numerical simulations, and compare them with our analytical results.
Citation: Seung-Yeal Ha, Jinyeong Park, Sang Woo Ryoo. Emergence of phase-locked states for the Winfree model in a large coupling regime. Discrete & Continuous Dynamical Systems, 2015, 35 (8) : 3417-3436. doi: 10.3934/dcds.2015.35.3417
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. doi: 10.1103/RevModPhys.77.137.  Google Scholar

[2]

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

[3]

J. Buck and E. Buck, Biology of synchronous flashing of fireflies, Nature, 211 (1966), 562-564. doi: 10.1038/211562a0.  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. Nat. Acad. Sci., 105 (2008), 1232-1237. doi: 10.1073/pnas.0711437105.  Google Scholar

[5]

N. Bellomo, B. Piccoli and A. Tosin, Modeling crowd dynamics from a complex system viewpoint, Math. Models Methods Appl. Sci., 22 (2012), 1230004, 29pp. doi: 10.1142/S0218202512300049.  Google Scholar

[6]

N. Bellomo and J. Soler, On the mathematical theory of the dynamics of swarms viewed as complex systems, Math. Models Methods Appl. Sci., 22 (2012), 1140006-1140035. doi: 10.1142/S0218202511400069.  Google Scholar

[7]

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.  Google Scholar

[8]

J. A. Carrillo, A. Klar, S. Martin and S. Tiwari, Self-propelled interacting particle systems with roosting force, Math. Mod. Meth. Appl. Sci., 20 (2010), 1533-1552. doi: 10.1142/S0218202510004684.  Google Scholar

[9]

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

[10]

F. Cucker and S. Smale, Emergent behavior in flocks, IEEE Trans. Automat. Control, 52 (2007), 852-862. doi: 10.1109/TAC.2007.895842.  Google Scholar

[11]

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

[12]

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

[13]

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

[14]

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

[15]

M. Fornasier, J. Haskovec and G. Toscani, Fluid dynamic description of flocking via Povzner-Boltzmann equation, Physica D, 240 (2011), 21-31. doi: 10.1016/j.physd.2010.08.003.  Google Scholar

[16]

J. A. Fax and R. M. Murray, Information flow and cooperative control of vehicle formations, IEEE Trans. Automatic Control, 49 (2004), 1465-1476. doi: 10.1109/TAC.2004.834433.  Google Scholar

[17]

S.-Y. Ha, H. K. Kim and J. Park, Remarks on the complete frequency synchronization of Kuramoto oscillators,, Submitted., ().   Google Scholar

[18]

S.-Y. Ha, H. K. Kim and S. W. Ryoo, Emergence of phase-locked states for the Kuramoto model in a large coupling regime,, Submitted., ().   Google Scholar

[19]

S.-Y. Ha, Z. Li and X. Xue, Formation of phase-locked states in a population of locally interacting Kuramoro oscillators, J. Differential Equations, 255 (2013), 3053-3070. doi: 10.1016/j.jde.2013.07.013.  Google Scholar

[20]

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

[21]

S.-Y. Ha and E. Tadmor, From particle to kinetic and hydrodynamic description of flocking, Kinetic and Related Models, 1 (2008), 415-435. doi: 10.3934/krm.2008.1.415.  Google Scholar

[22]

A. Jadbabaie, J. Lin and A. Morse, Coordination of groups of mobile autonomous agents using nearest neighbor rules, IEEE Trans. Autom. Control, 48 (2003), 988-1001. doi: 10.1109/TAC.2003.812781.  Google Scholar

[23]

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

[24]

Y. Kuramoto, Self-entrainment of a population of coupled non-linear oscillators, International symposium on mathematical problems in mathematical physics, Lecture notes in theoretical physics, 39 (1975), 420-422.  Google Scholar

[25]

N. E. Leonard, D. A. Paley, F. Lekien, R. Sepulchre, D. 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

[26]

S. Motsch and E. Tadmor, A new model for self-organized dynamics and its flocking behavior, J. Stat. Phys., 144 (2011), 923-947. doi: 10.1007/s10955-011-0285-9.  Google Scholar

[27]

G. Nardulli, D. Mrinazzo, M. Pellicoro and S. Stramaglia, Phase shifts between synchronized oscillators in the Winfree and Kuramoto models,, Available at, ().   Google Scholar

[28]

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. doi: 10.1109/MCS.2007.384123.  Google Scholar

[29]

J. Park, H. Kim and S.-Y. Ha, Cucker-Smale flocking with inter-particle bonding forces, IEEE Tran. Automatic Control, 55 (2010), 2617-2623. doi: 10.1109/TAC.2010.2061070.  Google Scholar

[30]

L. Perea, G. Gómez and P. Elosegui, Extension of the Cucker-Smale control law to space flight formation, J. Guidance, Control and Dynamics, 32 (2009), 527-537. doi: 10.2514/1.36269.  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, 10pp. 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, August 7-12, 2005 (CD-ROM). Google Scholar

[33]

W. Ren and R. W. Beard, Consensus seeking in multi-agent systems under dynamically changing interaction topologies, IEEE Trans. Automatic Control, 50 (2005), 655-661. doi: 10.1109/TAC.2005.846556.  Google Scholar

[34]

R. O. Saber, J. A. Fax and R. M. Murray, Consensus and cooperation in networked multi- agent systems, Proc. of the IEEE, 95 (2007), 215-233. Google Scholar

[35]

R. O. Saber and R. M. Murray, Consensus problems in networks of agents with switching topology and time-delays, IEEE Trans. Automatic Control, 49 (2004), 1520-1533. doi: 10.1109/TAC.2004.834113.  Google Scholar

[36]

C. M. Topaz and A. L. Bertozzi, Swarming patterns in a two-dimensional kinematic model for biological groups, SIAM J. Appl. Math., 65 (2004), 152-174. doi: 10.1137/S0036139903437424.  Google Scholar

[37]

H. G. Tanner, A. Jadbabaie and G. J. Pappas, Flocking in fixed and switching networks, IEEE Trans. Automatic Control, 52 (2007), 863-868. doi: 10.1109/TAC.2007.895948.  Google Scholar

[38]

J. Toner and Y. Tu, Flocks, herds, and Schools: A quantitative theory of flocking, Physical Review E., 58 (1998), 4828-4858. doi: 10.1103/PhysRevE.58.4828.  Google Scholar

[39]

T. Vicsek, A. Czirók, E. Ben-Jacob, I. Cohen and O. Schochet, Novel type of phase transition in a system of self-driven particles, Phys. Rev. Lett., 75 (1995), 1226-1229. doi: 10.1103/PhysRevLett.75.1226.  Google Scholar

[40]

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., 17, Amer. Math. Soc., Providence, R.I., (1979), 93-126.  Google Scholar

[41]

A. 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. 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. doi: 10.1103/RevModPhys.77.137.  Google Scholar

[2]

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

[3]

J. Buck and E. Buck, Biology of synchronous flashing of fireflies, Nature, 211 (1966), 562-564. doi: 10.1038/211562a0.  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. Nat. Acad. Sci., 105 (2008), 1232-1237. doi: 10.1073/pnas.0711437105.  Google Scholar

[5]

N. Bellomo, B. Piccoli and A. Tosin, Modeling crowd dynamics from a complex system viewpoint, Math. Models Methods Appl. Sci., 22 (2012), 1230004, 29pp. doi: 10.1142/S0218202512300049.  Google Scholar

[6]

N. Bellomo and J. Soler, On the mathematical theory of the dynamics of swarms viewed as complex systems, Math. Models Methods Appl. Sci., 22 (2012), 1140006-1140035. doi: 10.1142/S0218202511400069.  Google Scholar

[7]

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.  Google Scholar

[8]

J. A. Carrillo, A. Klar, S. Martin and S. Tiwari, Self-propelled interacting particle systems with roosting force, Math. Mod. Meth. Appl. Sci., 20 (2010), 1533-1552. doi: 10.1142/S0218202510004684.  Google Scholar

[9]

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

[10]

F. Cucker and S. Smale, Emergent behavior in flocks, IEEE Trans. Automat. Control, 52 (2007), 852-862. doi: 10.1109/TAC.2007.895842.  Google Scholar

[11]

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

[12]

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

[13]

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

[14]

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

[15]

M. Fornasier, J. Haskovec and G. Toscani, Fluid dynamic description of flocking via Povzner-Boltzmann equation, Physica D, 240 (2011), 21-31. doi: 10.1016/j.physd.2010.08.003.  Google Scholar

[16]

J. A. Fax and R. M. Murray, Information flow and cooperative control of vehicle formations, IEEE Trans. Automatic Control, 49 (2004), 1465-1476. doi: 10.1109/TAC.2004.834433.  Google Scholar

[17]

S.-Y. Ha, H. K. Kim and J. Park, Remarks on the complete frequency synchronization of Kuramoto oscillators,, Submitted., ().   Google Scholar

[18]

S.-Y. Ha, H. K. Kim and S. W. Ryoo, Emergence of phase-locked states for the Kuramoto model in a large coupling regime,, Submitted., ().   Google Scholar

[19]

S.-Y. Ha, Z. Li and X. Xue, Formation of phase-locked states in a population of locally interacting Kuramoro oscillators, J. Differential Equations, 255 (2013), 3053-3070. doi: 10.1016/j.jde.2013.07.013.  Google Scholar

[20]

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

[21]

S.-Y. Ha and E. Tadmor, From particle to kinetic and hydrodynamic description of flocking, Kinetic and Related Models, 1 (2008), 415-435. doi: 10.3934/krm.2008.1.415.  Google Scholar

[22]

A. Jadbabaie, J. Lin and A. Morse, Coordination of groups of mobile autonomous agents using nearest neighbor rules, IEEE Trans. Autom. Control, 48 (2003), 988-1001. doi: 10.1109/TAC.2003.812781.  Google Scholar

[23]

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

[24]

Y. Kuramoto, Self-entrainment of a population of coupled non-linear oscillators, International symposium on mathematical problems in mathematical physics, Lecture notes in theoretical physics, 39 (1975), 420-422.  Google Scholar

[25]

N. E. Leonard, D. A. Paley, F. Lekien, R. Sepulchre, D. 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

[26]

S. Motsch and E. Tadmor, A new model for self-organized dynamics and its flocking behavior, J. Stat. Phys., 144 (2011), 923-947. doi: 10.1007/s10955-011-0285-9.  Google Scholar

[27]

G. Nardulli, D. Mrinazzo, M. Pellicoro and S. Stramaglia, Phase shifts between synchronized oscillators in the Winfree and Kuramoto models,, Available at, ().   Google Scholar

[28]

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. doi: 10.1109/MCS.2007.384123.  Google Scholar

[29]

J. Park, H. Kim and S.-Y. Ha, Cucker-Smale flocking with inter-particle bonding forces, IEEE Tran. Automatic Control, 55 (2010), 2617-2623. doi: 10.1109/TAC.2010.2061070.  Google Scholar

[30]

L. Perea, G. Gómez and P. Elosegui, Extension of the Cucker-Smale control law to space flight formation, J. Guidance, Control and Dynamics, 32 (2009), 527-537. doi: 10.2514/1.36269.  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, 10pp. 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, August 7-12, 2005 (CD-ROM). Google Scholar

[33]

W. Ren and R. W. Beard, Consensus seeking in multi-agent systems under dynamically changing interaction topologies, IEEE Trans. Automatic Control, 50 (2005), 655-661. doi: 10.1109/TAC.2005.846556.  Google Scholar

[34]

R. O. Saber, J. A. Fax and R. M. Murray, Consensus and cooperation in networked multi- agent systems, Proc. of the IEEE, 95 (2007), 215-233. Google Scholar

[35]

R. O. Saber and R. M. Murray, Consensus problems in networks of agents with switching topology and time-delays, IEEE Trans. Automatic Control, 49 (2004), 1520-1533. doi: 10.1109/TAC.2004.834113.  Google Scholar

[36]

C. M. Topaz and A. L. Bertozzi, Swarming patterns in a two-dimensional kinematic model for biological groups, SIAM J. Appl. Math., 65 (2004), 152-174. doi: 10.1137/S0036139903437424.  Google Scholar

[37]

H. G. Tanner, A. Jadbabaie and G. J. Pappas, Flocking in fixed and switching networks, IEEE Trans. Automatic Control, 52 (2007), 863-868. doi: 10.1109/TAC.2007.895948.  Google Scholar

[38]

J. Toner and Y. Tu, Flocks, herds, and Schools: A quantitative theory of flocking, Physical Review E., 58 (1998), 4828-4858. doi: 10.1103/PhysRevE.58.4828.  Google Scholar

[39]

T. Vicsek, A. Czirók, E. Ben-Jacob, I. Cohen and O. Schochet, Novel type of phase transition in a system of self-driven particles, Phys. Rev. Lett., 75 (1995), 1226-1229. doi: 10.1103/PhysRevLett.75.1226.  Google Scholar

[40]

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., 17, Amer. Math. Soc., Providence, R.I., (1979), 93-126.  Google Scholar

[41]

A. 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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