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 - A, 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.  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.  doi: 10.1103/PhysRevLett.86.4278.  Google Scholar

[3]

J. Buck and E. Buck, Biology of synchronous flashing of fireflies,, Nature, 211 (1966), 562.  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.  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).  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.  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.  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.  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.  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.  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.  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.  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.  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.  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.  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.  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.  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.  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.  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.  doi: 10.1109/TAC.2003.812781.  Google Scholar

[23]

Y. Kuramoto, Chemical Oscillations, Waves and Turbulence,, Springer-Verlag, (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.   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.  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.  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.  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.  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, 32 (2009), 527.  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).  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, (2005), 7.   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.  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.   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.  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.  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.  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.  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.  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., (1979), 93.   Google Scholar

[41]

A. Winfree, Biological rhythms and the behavior of populations of coupled oscillators,, J. Theoret. Biol., 16 (1967), 15.  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.  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.  doi: 10.1103/PhysRevLett.86.4278.  Google Scholar

[3]

J. Buck and E. Buck, Biology of synchronous flashing of fireflies,, Nature, 211 (1966), 562.  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.  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).  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.  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.  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.  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.  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.  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.  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.  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.  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.  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.  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.  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.  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.  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.  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.  doi: 10.1109/TAC.2003.812781.  Google Scholar

[23]

Y. Kuramoto, Chemical Oscillations, Waves and Turbulence,, Springer-Verlag, (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.   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.  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.  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.  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.  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, 32 (2009), 527.  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).  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, (2005), 7.   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.  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.   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.  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.  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.  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.  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.  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., (1979), 93.   Google Scholar

[41]

A. Winfree, Biological rhythms and the behavior of populations of coupled oscillators,, J. Theoret. Biol., 16 (1967), 15.  doi: 10.1016/0022-5193(67)90051-3.  Google Scholar

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