doi: 10.3934/cpaa.2021140
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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 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 & Applied Analysis, 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.   Google Scholar

[2]

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[4]

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

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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.  Google Scholar

[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.  Google Scholar

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[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

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

[20]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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). Google Scholar

[25]

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

[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.  Google Scholar

[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.  Google Scholar

[28]

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

[29]

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. 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.   Google Scholar

[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.  Google Scholar

[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]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

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

[20]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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). Google Scholar

[25]

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

[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.  Google Scholar

[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.  Google Scholar

[28]

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

[29]

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

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