November  2021, 20(11): 3975-4006. doi: 10.3934/cpaa.2021140

Interplay of random inputs and adaptive couplings in the Winfree model

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, Republic of Korea

2. 

School of Mathematics, Korea Institute for Advanced Study, Hoegiro 85, Seoul 02455, Republic of Korea

3. 

Department of Mathematics, Research Institute for Natural Sciences, Hanyang University, Seoul 04763, Republic of Korea

* Corresponding author

Received  April 2021 Revised  July 2021 Published  November 2021 Early access  August 2021

Fund Project: The work of S. Y. Ha was supported by the NRF grant (2020R1A2C3A01003881), the work of D. Kim was supported by a KIAS Individual Grant (MG073901) at Korea Institute for Advanced Study, 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) and the Ministry of Science and ICT (NRF-2020R1A4A3079066)

We study a structural robustness of the complete oscillator death state in the Winfree model with random inputs and adaptive couplings. For this, we present a sufficient framework formulated in terms of initial data, natural frequencies and adaptive coupling strengths. In our proposed framework, we derive propagation of infinitesimal variations in random space and asymptotic disappearance of random effects which exhibits the robustness of the complete oscillator death state for the random Winfree model.

Citation: Seung-Yeal Ha, Doheon Kim, Bora Moon. Interplay of random inputs and adaptive couplings in the Winfree model. Communications on Pure and Applied Analysis, 2021, 20 (11) : 3975-4006. doi: 10.3934/cpaa.2021140
References:
[1]

J. A. AcebronL. L. BonillaC. J. P. Pérez VicenteF. Ritort and R. Spigler, The Kuramoto model: A simple paradigm for synchronization phenomena, Rev. Mod. Phys., 77 (2005), 137-185. 

[2]

G. Albi, L. Pareschi and M. Zanella, Uncertainty quantification in control problems for flocking models, Math. Probl. Eng., (2015) Art. 850124, 14 pp. doi: 10.1155/2015/850124.

[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]

J. Buck and E. Buck, Biology of synchronous flashing of fireflies, Nature, 211 (1966), 562-564. 

[5]

Y. P. ChoiS. Y. HaS. 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.

[6]

J. A. CarrilloL. Pareschi and M. Zanella, Particle based gPC methods for mean-field models of swarming with uncertainty, Commun. Comput. Phys., 25 (2019), 508-531.  doi: 10.4208/cicp.oa-2017-0244.

[7]

G. Q. Chen and B. Perthame, Well-posedness for non-isotropic degenerate parabolic-hyperbolic equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 20 (2003), 645–668. doi: 10.1016/S0294-1449(02)00014-8.

[8]

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.

[9]

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.

[10]

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.

[11]

S. Y. Ha and S. Jin, Local sensitivity analysis for the Cucker-Smale model with random inputs, Kinet. Relat. Models, 11 (2018), 859-889.  doi: 10.3934/krm.2018034.

[12]

S. Y. Ha, S. Jin and J. Jung, A local sensitivity analysis for the kinetic Cucker-Smale equation with random inputs, J. Differ. Equ., 265 (2018), 3618–3649. doi: 10.1016/j.jde.2018.05.013.

[13]

S. Y. Ha, S. Jin, J. Jung and W. Shim, A local sensitivity analysis for the hydrodynamic Cucker-Smale model with random inputs, J. Differ. Equ., 268 (2020), 636–679. doi: 10.1016/j.jde.2019.08.031.

[14]

S. Y. HaD. KoJ. 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.

[15]

S. Y. HaD. KoJ. Park and S. W. Ryoo, Emergent dynamics of Winfree oscillators on locally coupled networks, J. Differ. Equ., 260 (2016), 4203-4236.  doi: 10.1016/j.jde.2015.11.008.

[16]

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. doi: 10.1063/1.5017063.

[17]

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.

[18]

S. Jin and L. Pareschi, Uncertainty Quantification for Hyperbolic and Kinetic Equations, SEMA SIMAI Springer Series Book 14, Springer, 2018. doi: 10.1007/978-3-319-67110-9_6.

[19]

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

[20]

Y. Kuramoto, International symposium on mathematical problems in mathematical physics, Lecture notes in theoretical physics, 30 (1975), 420.

[21]

S. Louca and F. M. Atay, Spatially structured networks of pulse-coupled phase oscillators on metric spaces,, Discrete Contin. Dyn. Syst., 34 (2014), 3703–3745. doi: 10.3934/dcds.2014.34.3703.

[22]

W. OukilA. Kessi and Ph. Thieullen, Synchronization hypothesis in the Winfree model, Dyn. Syst., 32 (2017), 326-339.  doi: 10.1080/14689367.2016.1227303.

[23]

D. D. Quinn, R. H. Rand and S. Strogatz, Singular unlocking transition in the Winfree model of coupled oscillators, Physical Rev. E, 75 (2007), 036218. doi: 10.1103/PhysRevE.75.036218.

[24]

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).

[25]

Q. Ren and J. Zhao, Adaptive coupling and enhanced synchronization in coupled phase oscillators, Phys. Rev. E, 76 (2007), 016207.

[26]

A. Saltelli, M. Ratto, T. Andres, F. Campolongo, J. Cariboni, D. Gatelli, M. Saisana and S. Tarantola, Introduction to sensitivity analysis, Global sensitivity analysis. The Primer, (2008), 1–51.

[27]

P. Seliger, S. C. Young and L. S. Tsimring, Plasticity and learning in a network of coupled phase oscillators, Phys. Rev. E, 65 (2002), 041906. doi: 10.1103/PhysRevE.65.041906.

[28]

R. C. Smith, Uncertainty quantification: Theory, Implementation, and Applications, 2013.

[29]

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. AcebronL. L. BonillaC. J. P. Pérez VicenteF. Ritort and R. Spigler, The Kuramoto model: A simple paradigm for synchronization phenomena, Rev. Mod. Phys., 77 (2005), 137-185. 

[2]

G. Albi, L. Pareschi and M. Zanella, Uncertainty quantification in control problems for flocking models, Math. Probl. Eng., (2015) Art. 850124, 14 pp. doi: 10.1155/2015/850124.

[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]

J. Buck and E. Buck, Biology of synchronous flashing of fireflies, Nature, 211 (1966), 562-564. 

[5]

Y. P. ChoiS. Y. HaS. 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.

[6]

J. A. CarrilloL. Pareschi and M. Zanella, Particle based gPC methods for mean-field models of swarming with uncertainty, Commun. Comput. Phys., 25 (2019), 508-531.  doi: 10.4208/cicp.oa-2017-0244.

[7]

G. Q. Chen and B. Perthame, Well-posedness for non-isotropic degenerate parabolic-hyperbolic equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 20 (2003), 645–668. doi: 10.1016/S0294-1449(02)00014-8.

[8]

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.

[9]

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.

[10]

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.

[11]

S. Y. Ha and S. Jin, Local sensitivity analysis for the Cucker-Smale model with random inputs, Kinet. Relat. Models, 11 (2018), 859-889.  doi: 10.3934/krm.2018034.

[12]

S. Y. Ha, S. Jin and J. Jung, A local sensitivity analysis for the kinetic Cucker-Smale equation with random inputs, J. Differ. Equ., 265 (2018), 3618–3649. doi: 10.1016/j.jde.2018.05.013.

[13]

S. Y. Ha, S. Jin, J. Jung and W. Shim, A local sensitivity analysis for the hydrodynamic Cucker-Smale model with random inputs, J. Differ. Equ., 268 (2020), 636–679. doi: 10.1016/j.jde.2019.08.031.

[14]

S. Y. HaD. KoJ. 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.

[15]

S. Y. HaD. KoJ. Park and S. W. Ryoo, Emergent dynamics of Winfree oscillators on locally coupled networks, J. Differ. Equ., 260 (2016), 4203-4236.  doi: 10.1016/j.jde.2015.11.008.

[16]

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. doi: 10.1063/1.5017063.

[17]

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.

[18]

S. Jin and L. Pareschi, Uncertainty Quantification for Hyperbolic and Kinetic Equations, SEMA SIMAI Springer Series Book 14, Springer, 2018. doi: 10.1007/978-3-319-67110-9_6.

[19]

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

[20]

Y. Kuramoto, International symposium on mathematical problems in mathematical physics, Lecture notes in theoretical physics, 30 (1975), 420.

[21]

S. Louca and F. M. Atay, Spatially structured networks of pulse-coupled phase oscillators on metric spaces,, Discrete Contin. Dyn. Syst., 34 (2014), 3703–3745. doi: 10.3934/dcds.2014.34.3703.

[22]

W. OukilA. Kessi and Ph. Thieullen, Synchronization hypothesis in the Winfree model, Dyn. Syst., 32 (2017), 326-339.  doi: 10.1080/14689367.2016.1227303.

[23]

D. D. Quinn, R. H. Rand and S. Strogatz, Singular unlocking transition in the Winfree model of coupled oscillators, Physical Rev. E, 75 (2007), 036218. doi: 10.1103/PhysRevE.75.036218.

[24]

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).

[25]

Q. Ren and J. Zhao, Adaptive coupling and enhanced synchronization in coupled phase oscillators, Phys. Rev. E, 76 (2007), 016207.

[26]

A. Saltelli, M. Ratto, T. Andres, F. Campolongo, J. Cariboni, D. Gatelli, M. Saisana and S. Tarantola, Introduction to sensitivity analysis, Global sensitivity analysis. The Primer, (2008), 1–51.

[27]

P. Seliger, S. C. Young and L. S. Tsimring, Plasticity and learning in a network of coupled phase oscillators, Phys. Rev. E, 65 (2002), 041906. doi: 10.1103/PhysRevE.65.041906.

[28]

R. C. Smith, Uncertainty quantification: Theory, Implementation, and Applications, 2013.

[29]

A. Winfree, Biological rhythms and the behavior of populations of coupled oscillators, J. Theoret. Biol., 16 (1967), 15-42. 

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