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On the fractional susceptibility function of piecewise expanding maps
Generalization of the Winfree model to the high-dimensional sphere and its emergent dynamics
Department of Mathematical Sciences, Seoul National University, Seoul 08826, Republic of Korea |
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 $-stability and the existence of the equilibrium solution.
References:
[1] |
J. A. Acebron, L. L. Bonilla, C. J. P. Pérez Vicente, F. Ritort and R. Spigler,
The kuramoto model: A simple paradigm for synchronization phenomena, Rev. Mod. Phys., 77 (2005), 137-185.
|
[2] |
D. Aeyels and J. Rogge,
Stability of phase locking and existence of frequency in networks of globally coupled oscillators, Prog. Theor. Phys., 112 (2004), 921-941.
|
[3] |
J. T. Ariaratnam and S. H. Strogatz,
Phase diagram for the winfree model of coupled nonlinearoscillators, Phys. Rev. Lett., 86 (2001), 4278-4281.
|
[4] |
G. Albi, N. Bellomo, L. Fermo, S.-Y. Ha, L. Pareschi, D. Poyato and J. Soler,
Vehicular traffic, crowds, and swarms: From kinetic theory and multiscale methods to applications and research perspectives, Math. Models Methods Appl. Sci., 29 (2019), 1901-2005.
doi: 10.1142/S0218202519500374. |
[5] |
I. Aoki,
A simulation study on the schooling mechanism in fish, Bulletin of the Japan Society of Scientific Fisheries, 48 (1982), 1081-1088.
|
[6] |
I. Barb$\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.
|
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