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doi: 10.3934/dcdsb.2022007
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Asymptotic interplay of states and adaptive coupling gains in the Lohe Hermitian sphere model

Junhyeok Byeon 1,, , Seung-Yeal Ha 2, and  Hansol Park 3,

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

* Corresponding author: Junhyeok Byeon

Received  September 2021 Revised  November 2021 Early access January 2022

Fund Project: The work of S.-Y. Ha was supported by National Research Foundation of Korea(NRF-2020R1A2C3A01003881).
The work of H. Park was supported by Pacific Institute for the Mathematical Science(PIMS), Canada postdoctoral fellowship

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 [23]. In this paper, we further extend earlier results in the aforementioned work to the setting in which the state and coupling gains are dynamically interrelated via two types of coupling laws, namely anti-Hebbian and Hebbian coupling laws. In each case, we present two sufficient frameworks leading to complete aggregation depending on the coupling laws, when the corresponding free flow is the same for all particles.

Keywords: Emergence, Lohe Hermitian sphere model, synchronization, complex vector, tensor.
Mathematics Subject Classification: Primary: 70G60; Secondary: 34D06.
Citation: Junhyeok Byeon, Seung-Yeal Ha, Hansol Park. Asymptotic interplay of states and adaptive coupling gains in the Lohe Hermitian sphere model. Discrete and Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2022007
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.

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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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