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Asymptotic interplay of states and adaptive coupling gains in the Lohe Hermitian sphere model
1. | Department of Mathematical Sciences, Seoul National University, Seoul 08826, Republic of Korea |
2. | Department of Mathematical Sciences and Research Institute of Mathematics, Seoul National University, Seoul 08826, Republic of Korea |
3. | Department of Mathematics, Simon Fraser University, 8888 University Dr, Burnaby, BC V5A 1S6, Canada |
We study emergent dynamics of the Lohe Hermitian sphere (LHS) model with the same free flows under the dynamic interplay between state evolution and adaptive couplings. The LHS model is a complex counterpart of the Lohe sphere (LS) model on the unit sphere in Euclidean space, and when particles lie in the Euclidean unit sphere embedded in $ \mathbb C^{d+1} $, it reduces to the Lohe sphere model. In the absence of interactions between states and coupling gains, emergent dynamics have been addressed in [
References:
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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] |
G. Albi, N. Bellomo, L. Fermo, S.-Y. Ha, J. Kim, 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. |
[3] |
I. Aoki,
A simulation study on the schooling mechanism in fish, Bulletin of the Japan Society of Scientific Fisheries, 48 (1982), 1081-1088.
|
[4] |
I. Barbalat,
Systèmes d'équations différentielles d'oscillations non Linéaires, Rev. Math. Pures Appl., 4 (1959), 267-270.
|
[5] |
D. Benedetto, E. Caglioti and U. Montemagno,
On the complete phase synchronization for the Kuramoto model in the mean-field limit, Commun. Math. Sci., 13 (2015), 1775-1786.
doi: 10.4310/CMS.2015.v13.n7.a6. |
[6] |
J. C. Bronski, T. E. Carty and S. E. Simpson,
A matrix valued Kuramoto model, J. Stat. Phys., 178 (2020), 595-624.
doi: 10.1007/s10955-019-02442-w. |
[7] |
J. Buck and E. Buck,
Biology of synchronous flashing of fireflies, Nature, 211 (1966), 562.
|
[8] |
S.-H. Choi and S.-Y. Ha,
Complete entrainment of Lohe oscillators under attractive and repulsive couplings, SIAM J. Appl. Dyn. Syst., 13 (2014), 1417-1441.
doi: 10.1137/140961699. |
[9] |
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), 735-754.
doi: 10.1016/j.physd.2011.11.011. |
[10] |
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. |
[11] |
F. Cucker and S. Smale,
Emergent behavior in flocks, IEEE Trans. Automat. Control, 52 (2007), 852-862.
doi: 10.1109/TAC.2007.895842. |
[12] |
P. Degond, A. Frouvelle, S. Merino-Aceituno and A. Trescases,
Quaternions in collective dynamics, Multiscale Model. Simul., 16 (2018), 28-77.
doi: 10.1137/17M1135207. |
[13] |
L. DeVille, Aggregation and stability for quantum Kuramoto, J. Stat. Phys., 174 (2019), 160–187.
doi: 10.1007/s10955-018-2168-9. |
[14] |
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. |
[15] |
F. Dörfler and F. Bullo,
Synchronization in complex networks of phase oscillators: A survey, Automatica J. IFAC, 50 (2014), 1539-1564.
doi: 10.1016/j.automatica.2014.04.012. |
[16] |
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. |
[17] |
S.-Y. Ha, D. Kim, J. Lee and S. E. Noh,
Emergence of aggregation in the swarm sphere model with adaptive coupling laws, Kinet. Relat. Models, 12 (2019), 411-444.
doi: 10.3934/krm.2019018. |
[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, Commun. Math. Sci., 14 (2016), 1073-1091.
doi: 10.4310/CMS.2016.v14.n4.a10. |
[19] |
S.-Y. Ha, D. Ko and S. W. Ryoo,
On the relaxation dynamics of Lohe oscillators on some Riemannian manifolds, J. Stat. Phys., 172 (2018), 1427-1478.
doi: 10.1007/s10955-018-2091-0. |
[20] |
S.-Y. Ha, J. Lee, Z. Li and J. Park,
Emergent dynamics of Kuramoto oscillators with adaptive couplings: Conservation law and fast learning, SIAM J. Appl. Dyn. Syst., 17 (2018), 1560-1588.
doi: 10.1137/17M1124048. |
[21] |
S.-Y. Ha, S. E. Noh and J. Park,
Synchronization of Kuramoto oscillators with adaptive couplings, SIAM J. Appl. Dyn. Syst., 15 (2016), 162-194.
doi: 10.1137/15M101484X. |
[22] |
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. |
[23] |
S.-Y. Ha and H. Park,
From the Lohe tensor model to the Lohe Hermitian sphere model and emergent dynamics, SIAM J. Appl. Dyn. Syst., 19 (2020), 1312-1342.
doi: 10.1137/19M1288553. |
[24] |
V. Jaćimović and A. Crnkić, Low-dimensional dynamics in non-Abelian Kuramoto model on the 3-sphere, Chaos, 28 (2018), 083105, 8 pp.
doi: 10.1063/1.5029485. |
[25] |
D. Kim,
State-dependent dynamics of the Lohe matrix ensemble on the unitary group under the gradient flow, SIAM J. Appl. Dyn. Syst., 19 (2020), 1080-1123.
doi: 10.1137/19M1294605. |
[26] |
Y. Kuramoto,
International symposium on mathematical problems in mathematical physics, Lecture Notes Theor. Phys., 30 (1975), 420.
|
[27] |
M. A. Lohe, Systems of matrix Riccati equations, linear fractional transformations, partial integrability and synchronization., J. Math. Phys., 60 (2019), 072701, 25 pp.
doi: 10.1063/1.5085248. |
[28] |
M. A. Lohe, Quantum synchronization over quantum networks, J. Phys. A: Math. Theor., 43 (2010), 465301, 20 pp.
doi: 10.1088/1751-8113/43/46/465301. |
[29] |
M. A. Lohe, Non-abelian Kuramoto model and synchronization, J. Phys. A: Math. Theor., 42 (2009), 395101, 25 pp.
doi: 10.1088/1751-8113/42/39/395101. |
[30] |
J. Markdahl, J. Thunberg and J. Gonçalves,
Almost global consensus on the $n$-sphere, IEEE Trans. Automat. Control, 63 (2018), 1664-1675.
doi: 10.1109/TAC.2017.2752799. |
[31] |
C. S. Peskin, Mathematical Aspects of Heart Physiology, Courant Institute of Mathematical Sciences, New York, 1975. |
[32] |
A. Pikovsky, M. Rosenblum and J. Kurths, Synchronization: A universal Concept in Nonlinear Sciences, Cambridge University Press, Cambridge, 2001.
doi: 10.1017/CBO9780511755743.![]() ![]() ![]() |
[33] |
J. Thunberg, J. Markdahl, F. Bernard and J. Goncalves,
A lifting method for analyzing distributed synchronization on the unit sphere, Automatica J. IFAC, 96 (2018), 253-258.
doi: 10.1016/j.automatica.2018.07.007. |
[34] |
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. |
[35] |
C. M. Topaz, A. L. Bertozzi and M. A. Lewis,
A nonlocal continuum model for biological aggregation, Bull. Math. Biol., 68 (2006), 1601-1623.
doi: 10.1007/s11538-006-9088-6. |
[36] |
T. Vicsek and A. Zefeiris,
Collective motion, Phys. Rep., 517 (2012), 71-140.
|
[37] |
A. T. Winfree,
Biological rhythms and the behavior of populations of coupled oscillators, J. Theor. Biol., 16 (1967), 15-42.
|
[38] |
A. T. Winfree, The Geometry of Biological Time, Springer, New York, 1980. |
[39] |
J. Zhu,
Synchronization of Kuramoto model in a high-dimensional linear space, Physics Letters A, 377 (2013), 2939-2943.
doi: 10.1016/j.physleta.2013.09.010. |
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] |
G. Albi, N. Bellomo, L. Fermo, S.-Y. Ha, J. Kim, 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. |
[3] |
I. Aoki,
A simulation study on the schooling mechanism in fish, Bulletin of the Japan Society of Scientific Fisheries, 48 (1982), 1081-1088.
|
[4] |
I. Barbalat,
Systèmes d'équations différentielles d'oscillations non Linéaires, Rev. Math. Pures Appl., 4 (1959), 267-270.
|
[5] |
D. Benedetto, E. Caglioti and U. Montemagno,
On the complete phase synchronization for the Kuramoto model in the mean-field limit, Commun. Math. Sci., 13 (2015), 1775-1786.
doi: 10.4310/CMS.2015.v13.n7.a6. |
[6] |
J. C. Bronski, T. E. Carty and S. E. Simpson,
A matrix valued Kuramoto model, J. Stat. Phys., 178 (2020), 595-624.
doi: 10.1007/s10955-019-02442-w. |
[7] |
J. Buck and E. Buck,
Biology of synchronous flashing of fireflies, Nature, 211 (1966), 562.
|
[8] |
S.-H. Choi and S.-Y. Ha,
Complete entrainment of Lohe oscillators under attractive and repulsive couplings, SIAM J. Appl. Dyn. Syst., 13 (2014), 1417-1441.
doi: 10.1137/140961699. |
[9] |
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), 735-754.
doi: 10.1016/j.physd.2011.11.011. |
[10] |
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. |
[11] |
F. Cucker and S. Smale,
Emergent behavior in flocks, IEEE Trans. Automat. Control, 52 (2007), 852-862.
doi: 10.1109/TAC.2007.895842. |
[12] |
P. Degond, A. Frouvelle, S. Merino-Aceituno and A. Trescases,
Quaternions in collective dynamics, Multiscale Model. Simul., 16 (2018), 28-77.
doi: 10.1137/17M1135207. |
[13] |
L. DeVille, Aggregation and stability for quantum Kuramoto, J. Stat. Phys., 174 (2019), 160–187.
doi: 10.1007/s10955-018-2168-9. |
[14] |
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. |
[15] |
F. Dörfler and F. Bullo,
Synchronization in complex networks of phase oscillators: A survey, Automatica J. IFAC, 50 (2014), 1539-1564.
doi: 10.1016/j.automatica.2014.04.012. |
[16] |
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. |
[17] |
S.-Y. Ha, D. Kim, J. Lee and S. E. Noh,
Emergence of aggregation in the swarm sphere model with adaptive coupling laws, Kinet. Relat. Models, 12 (2019), 411-444.
doi: 10.3934/krm.2019018. |
[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, Commun. Math. Sci., 14 (2016), 1073-1091.
doi: 10.4310/CMS.2016.v14.n4.a10. |
[19] |
S.-Y. Ha, D. Ko and S. W. Ryoo,
On the relaxation dynamics of Lohe oscillators on some Riemannian manifolds, J. Stat. Phys., 172 (2018), 1427-1478.
doi: 10.1007/s10955-018-2091-0. |
[20] |
S.-Y. Ha, J. Lee, Z. Li and J. Park,
Emergent dynamics of Kuramoto oscillators with adaptive couplings: Conservation law and fast learning, SIAM J. Appl. Dyn. Syst., 17 (2018), 1560-1588.
doi: 10.1137/17M1124048. |
[21] |
S.-Y. Ha, S. E. Noh and J. Park,
Synchronization of Kuramoto oscillators with adaptive couplings, SIAM J. Appl. Dyn. Syst., 15 (2016), 162-194.
doi: 10.1137/15M101484X. |
[22] |
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. |
[23] |
S.-Y. Ha and H. Park,
From the Lohe tensor model to the Lohe Hermitian sphere model and emergent dynamics, SIAM J. Appl. Dyn. Syst., 19 (2020), 1312-1342.
doi: 10.1137/19M1288553. |
[24] |
V. Jaćimović and A. Crnkić, Low-dimensional dynamics in non-Abelian Kuramoto model on the 3-sphere, Chaos, 28 (2018), 083105, 8 pp.
doi: 10.1063/1.5029485. |
[25] |
D. Kim,
State-dependent dynamics of the Lohe matrix ensemble on the unitary group under the gradient flow, SIAM J. Appl. Dyn. Syst., 19 (2020), 1080-1123.
doi: 10.1137/19M1294605. |
[26] |
Y. Kuramoto,
International symposium on mathematical problems in mathematical physics, Lecture Notes Theor. Phys., 30 (1975), 420.
|
[27] |
M. A. Lohe, Systems of matrix Riccati equations, linear fractional transformations, partial integrability and synchronization., J. Math. Phys., 60 (2019), 072701, 25 pp.
doi: 10.1063/1.5085248. |
[28] |
M. A. Lohe, Quantum synchronization over quantum networks, J. Phys. A: Math. Theor., 43 (2010), 465301, 20 pp.
doi: 10.1088/1751-8113/43/46/465301. |
[29] |
M. A. Lohe, Non-abelian Kuramoto model and synchronization, J. Phys. A: Math. Theor., 42 (2009), 395101, 25 pp.
doi: 10.1088/1751-8113/42/39/395101. |
[30] |
J. Markdahl, J. Thunberg and J. Gonçalves,
Almost global consensus on the $n$-sphere, IEEE Trans. Automat. Control, 63 (2018), 1664-1675.
doi: 10.1109/TAC.2017.2752799. |
[31] |
C. S. Peskin, Mathematical Aspects of Heart Physiology, Courant Institute of Mathematical Sciences, New York, 1975. |
[32] |
A. Pikovsky, M. Rosenblum and J. Kurths, Synchronization: A universal Concept in Nonlinear Sciences, Cambridge University Press, Cambridge, 2001.
doi: 10.1017/CBO9780511755743.![]() ![]() ![]() |
[33] |
J. Thunberg, J. Markdahl, F. Bernard and J. Goncalves,
A lifting method for analyzing distributed synchronization on the unit sphere, Automatica J. IFAC, 96 (2018), 253-258.
doi: 10.1016/j.automatica.2018.07.007. |
[34] |
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. |
[35] |
C. M. Topaz, A. L. Bertozzi and M. A. Lewis,
A nonlocal continuum model for biological aggregation, Bull. Math. Biol., 68 (2006), 1601-1623.
doi: 10.1007/s11538-006-9088-6. |
[36] |
T. Vicsek and A. Zefeiris,
Collective motion, Phys. Rep., 517 (2012), 71-140.
|
[37] |
A. T. Winfree,
Biological rhythms and the behavior of populations of coupled oscillators, J. Theor. Biol., 16 (1967), 15-42.
|
[38] |
A. T. Winfree, The Geometry of Biological Time, Springer, New York, 1980. |
[39] |
J. Zhu,
Synchronization of Kuramoto model in a high-dimensional linear space, Physics Letters A, 377 (2013), 2939-2943.
doi: 10.1016/j.physleta.2013.09.010. |
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