American Institute of Mathematical Sciences

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February  2022, 42(2): 707-735. doi: 10.3934/dcds.2021134

Generalization of the Winfree model to the high-dimensional sphere and its emergent dynamics

Hansol Park ,

Department of Mathematical Sciences, Seoul National University, Seoul 08826, Republic of Korea

*Corresponding author: Hansol Park

Received  February 2021 Revised  June 2021 Published  February 2022 Early access  September 2021

Fund Project: The work of H. Park is supported by National Research Foundation of Korea (NRF-2020R1A2C3A01003881)

We present a high-dimensional Winfree model in this paper. The Winfree model is a mathematical model for synchronization on the unit circle. We generalize this model compare to the high-dimensional sphere and we call it the Winfree sphere model. We restricted the support of the influence function in the neighborhood of the attraction point to a small diameter to mimic the influence function as the Dirac delta distribution. We can obtain several new conditions of the complete phase-locking states for the identical Winfree sphere model from restricting the support of the influence function. We also prove the complete oscillator death(COD) state from the exponential $ \ell^1 $-stability and the existence of the equilibrium solution.

Keywords: Winfree model, aggregation, complete synchronization, high-dimensional sphere.
Mathematics Subject Classification: Primary: 70G60, 34D06; Secondary: 70F10.
Citation: Hansol Park. Generalization of the Winfree model to the high-dimensional sphere and its emergent dynamics. Discrete and Continuous Dynamical Systems, 2022, 42 (2) : 707-735. doi: 10.3934/dcds.2021134
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]

D. Aeyels and J. Rogge, Stability of phase locking and existence of frequency in networks of globally coupled oscillators, Prog. Theor. Phys., 112 (2004), 921-941. 

[3]

J. T. Ariaratnam and S. H. Strogatz, Phase diagram for the winfree model of coupled nonlinearoscillators, Phys. Rev. Lett., 86 (2001), 4278-4281. 

[4]

G. Albi, N. Bellomo, L. Fermo, S.-Y. Ha, 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.

[5]

I. Aoki, A simulation study on the schooling mechanism in fish, Bulletin of the Japan Society of Scientific Fisheries, 48 (1982), 1081-1088. 

[6]

I. Barb$\check{a}$lat, Syst$\grave{e}$mes d$\acute{e}$quations diff$\acute{e}$rentielles d'oscillations non Lin$\acute{e}$aires, Rev. Math. Pures Appl., 4 (1959), 267-270. 

[7]

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. 

[8]

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

[9]

P. Degond, A. Frouvelle and J.-G. Liu, Phase transitions, hysteresis, and hyperbolicity for self-organized alignment dynamics, Arch. Ration. Mech. Anal., 216 (2015), 63-115.  doi: 10.1007/s00205-014-0800-7.

[10]

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.

[11]

J.-G. Dong and X. Xue, Synchronization analysis of Kuramoto oscillators, Commun. Math. Sci., 11 (2013), 465-480.  doi: 10.4310/CMS.2013.v11.n2.a7.

[12]

I. M. Gamba and M.-J. Kang, Global weak solutions for Kolmogorov-Vicsek type equations with orientational interactions, Arch. Ration. Mech. Anal., 222 (2016), 317-342.  doi: 10.1007/s00205-016-1002-2.

[13]

S-.Y. Ha, J. Y. 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.

[14]

S.-Y. Ha, D. Kim, H. Park and S. W. Ryoo, Constants of motions for the finite-dimensional Lohe type models with frustration and applications to emergent dynamics, Phys., 416 (2021), 132781. doi: 10.1016/j.physd.2020.132781.

[15]

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), 4203-4326.  doi: 10.1016/j.jde.2015.11.008.

[16]

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.

[17]

S.-Y. Ha and H. Park, Emergent behaviors of Lohe tensor flocks, J. Stat. Phys., 178 (2020), 1268-1292.  doi: 10.1007/s10955-020-02505-3.

[18]

S.-Y. Ha and H. Park, From the Lohe tensor model to the Hermitian Lohe sphere model and emergent dynamics, SIAM Journal on Applied Dynamical Systems, 19 (2020), 1312-1342.  doi: 10.1137/19M1288553.

[19]

S.-Y. Ha, J. Park and X. Zhang, A global well-posedness and asymptotic dynamics of the kinetic Winfree equation, Discrete Contin. Dyn. Syst. Ser. B, 25 (2020), 1317-1344.  doi: 10.3934/dcdsb.2019229.

[20]

S.-Y. Ha, M. Kang and B. Moon, Collective behaviors of a Winfree ensemble on an infinite cylinder, Discrete Contin. Dyn. Syst. Ser. B, 26 (2021), 2749-2779.  doi: 10.3934/dcdsb.2020204.

[21]

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.

[22]

S.-Y. Ha, H. W. Kim and S. W. Ryoo, Emergence of phase-locked states for the Kuramoto model in a large coupling regime, Commun. Math. Sci., 14 (2016), 1073-1091.  doi: 10.4310/CMS.2016.v14.n4.a10.

[23]

V. Jaćimović and A. Crnkić, Low-dimensional dynamics in non-Abelian Kuramoto model on the 3-sphere, Chaos, 28 (2018), 083105.  doi: 10.1063/1.5029485.

[24]

Y. Kuramoto, Chemical Oscillations, Waves and Turbulence, Springer-Verlag, Berlin, 1984.

[25]

Y. Kuramoto, Self-entrainment of a population of coupled non-linear oscillators, Lecture Notes in Theoretical Physics, 39 (1975), 420-422. 

[26]

M. A. Lohe, Systems of matrix Riccati equations, linear fractional transformations, partial integrability and synchronization, J. Math. Phys., 60 (2019), 072701.  doi: 10.1063/1.5085248.

[27]

M. A. Lohe, Quantum synchronization over quantum networks, J. Phys. A: Math. Theor., 43 (2010), 465301.  doi: 10.1088/1751-8113/43/46/465301.

[28]

M. A. Lohe, Non-abelian Kuramoto model and synchronization, J. Phys. A: Math. Theor., 42 (2009), 395101.  doi: 10.1088/1751-8113/42/39/395101.

[29]

G. Nardulli, D. Marinazzo, M. Pellicoro and S. Stramaglia, Phase shifts between synchronized oscillators in the Winfree and Kuramoto models, Available at http://www.necsi.edu/events/iccs/openconf/author/papers/708.pdf.

[30]

R. Olfati-Saber, Swarms on sphere: A programmable swarm with synchronous behaviors like oscillator networks, IEEE conference on Decision & Control, 45 (2006), 5060-5066. 

[31]

H. Park, The Watanabe-Strogatz transform and constant of motion functionals for kinetic vector models, preprint.

[32]

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.  doi: 10.1103/PhysRevE.75.036218.

[33]

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, (2005), 7–12.

[34]

R. Sknepnek and S. Henkes, Active swarms on a sphere, Physical Review E, 2 (2015), 022306. 

[35]

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.

[36]

A. Winfree, Biological rhythms and the behavior of populations of coupled oscillators, J. Theoret. Bio., 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]

D. Aeyels and J. Rogge, Stability of phase locking and existence of frequency in networks of globally coupled oscillators, Prog. Theor. Phys., 112 (2004), 921-941. 

[3]

J. T. Ariaratnam and S. H. Strogatz, Phase diagram for the winfree model of coupled nonlinearoscillators, Phys. Rev. Lett., 86 (2001), 4278-4281. 

[4]

G. Albi, N. Bellomo, L. Fermo, S.-Y. Ha, 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.

[5]

I. Aoki, A simulation study on the schooling mechanism in fish, Bulletin of the Japan Society of Scientific Fisheries, 48 (1982), 1081-1088. 

[6]

I. Barb$\check{a}$lat, Syst$\grave{e}$mes d$\acute{e}$quations diff$\acute{e}$rentielles d'oscillations non Lin$\acute{e}$aires, Rev. Math. Pures Appl., 4 (1959), 267-270. 

[7]

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. 

[8]

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

[9]

P. Degond, A. Frouvelle and J.-G. Liu, Phase transitions, hysteresis, and hyperbolicity for self-organized alignment dynamics, Arch. Ration. Mech. Anal., 216 (2015), 63-115.  doi: 10.1007/s00205-014-0800-7.

[10]

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.

[11]

J.-G. Dong and X. Xue, Synchronization analysis of Kuramoto oscillators, Commun. Math. Sci., 11 (2013), 465-480.  doi: 10.4310/CMS.2013.v11.n2.a7.

[12]

I. M. Gamba and M.-J. Kang, Global weak solutions for Kolmogorov-Vicsek type equations with orientational interactions, Arch. Ration. Mech. Anal., 222 (2016), 317-342.  doi: 10.1007/s00205-016-1002-2.

[13]

S-.Y. Ha, J. Y. 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.

[14]

S.-Y. Ha, D. Kim, H. Park and S. W. Ryoo, Constants of motions for the finite-dimensional Lohe type models with frustration and applications to emergent dynamics, Phys., 416 (2021), 132781. doi: 10.1016/j.physd.2020.132781.

[15]

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), 4203-4326.  doi: 10.1016/j.jde.2015.11.008.

[16]

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.

[17]

S.-Y. Ha and H. Park, Emergent behaviors of Lohe tensor flocks, J. Stat. Phys., 178 (2020), 1268-1292.  doi: 10.1007/s10955-020-02505-3.

[18]

S.-Y. Ha and H. Park, From the Lohe tensor model to the Hermitian Lohe sphere model and emergent dynamics, SIAM Journal on Applied Dynamical Systems, 19 (2020), 1312-1342.  doi: 10.1137/19M1288553.

[19]

S.-Y. Ha, J. Park and X. Zhang, A global well-posedness and asymptotic dynamics of the kinetic Winfree equation, Discrete Contin. Dyn. Syst. Ser. B, 25 (2020), 1317-1344.  doi: 10.3934/dcdsb.2019229.

[20]

S.-Y. Ha, M. Kang and B. Moon, Collective behaviors of a Winfree ensemble on an infinite cylinder, Discrete Contin. Dyn. Syst. Ser. B, 26 (2021), 2749-2779.  doi: 10.3934/dcdsb.2020204.

[21]

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.

[22]

S.-Y. Ha, H. W. Kim and S. W. Ryoo, Emergence of phase-locked states for the Kuramoto model in a large coupling regime, Commun. Math. Sci., 14 (2016), 1073-1091.  doi: 10.4310/CMS.2016.v14.n4.a10.

[23]

V. Jaćimović and A. Crnkić, Low-dimensional dynamics in non-Abelian Kuramoto model on the 3-sphere, Chaos, 28 (2018), 083105.  doi: 10.1063/1.5029485.

[24]

Y. Kuramoto, Chemical Oscillations, Waves and Turbulence, Springer-Verlag, Berlin, 1984.

[25]

Y. Kuramoto, Self-entrainment of a population of coupled non-linear oscillators, Lecture Notes in Theoretical Physics, 39 (1975), 420-422. 

[26]

M. A. Lohe, Systems of matrix Riccati equations, linear fractional transformations, partial integrability and synchronization, J. Math. Phys., 60 (2019), 072701.  doi: 10.1063/1.5085248.

[27]

M. A. Lohe, Quantum synchronization over quantum networks, J. Phys. A: Math. Theor., 43 (2010), 465301.  doi: 10.1088/1751-8113/43/46/465301.

[28]

M. A. Lohe, Non-abelian Kuramoto model and synchronization, J. Phys. A: Math. Theor., 42 (2009), 395101.  doi: 10.1088/1751-8113/42/39/395101.

[29]

G. Nardulli, D. Marinazzo, M. Pellicoro and S. Stramaglia, Phase shifts between synchronized oscillators in the Winfree and Kuramoto models, Available at http://www.necsi.edu/events/iccs/openconf/author/papers/708.pdf.

[30]

R. Olfati-Saber, Swarms on sphere: A programmable swarm with synchronous behaviors like oscillator networks, IEEE conference on Decision & Control, 45 (2006), 5060-5066. 

[31]

H. Park, The Watanabe-Strogatz transform and constant of motion functionals for kinetic vector models, preprint.

[32]

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.  doi: 10.1103/PhysRevE.75.036218.

[33]

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, (2005), 7–12.

[34]

R. Sknepnek and S. Henkes, Active swarms on a sphere, Physical Review E, 2 (2015), 022306. 

[35]

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.

[36]

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

Figure 1.  Geometric visualization of $\mathcal{D}_{\frac{\pi}{2}-\alpha} $ when $d = 2 $
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Figure 2.  Examples of $ \tilde{I} $
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Figure 3.  Geometric visualization of $n_e(x_j)$ in Lemma 4.1
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Figure 4.  Geometric visualization of the condition for $ \Omega_i$ when $d = 2 $
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