September  2021, 16(3): 459-492. doi: 10.3934/nhm.2021013

Emergent behaviors of Lohe Hermitian sphere particles under time-delayed interactions

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. 

Department of Mathematical Sciences and Research Institute of Mathematics, Seoul National University, Seoul 08826, Republic of Korea

* Corresponding author: Gyuyoung Hwang

Received  January 2021 Revised  April 2021 Published  September 2021 Early access  May 2021

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 Basic Science Research Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Education (2019R1I1A1A01059585)

We study emergent behaviors of the Lohe Hermitian sphere(LHS) model with a time-delay for a homogeneous and heterogeneous ensemble. The LHS model is a complex counterpart of the Lohe sphere(LS) aggregation model on the unit sphere in Euclidean space, and it describes the aggregation of particles on the unit Hermitian sphere in $ \mathbb C^d $ with $ d \geq 2 $. Recently it has been introduced by two authors of this work as a special case of the Lohe tensor model. When the coupling gain pair satisfies a specific linear relation, namely the Stuart-Landau(SL) coupling gain pair, it can be embedded into the LS model on $ \mathbb R^{2d} $. In this work, we show that if the coupling gain pair is close to the SL coupling pair case, the dynamics of the LHS model exhibits an emergent aggregate phenomenon via the interplay between time-delayed interactions and nonlinear coupling between states. For this, we present several frameworks for complete aggregation and practical aggregation in terms of initial data and system parameters using the Lyapunov functional approach.

Citation: Seung-Yeal Ha, Gyuyoung Hwang, Hansol Park. Emergent behaviors of Lohe Hermitian sphere particles under time-delayed interactions. Networks and Heterogeneous Media, 2021, 16 (3) : 459-492. doi: 10.3934/nhm.2021013
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. 

[2]

G. AlbiN. BellomoL. FermoS.-Y. HaJ. KimL. PareschiD. 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. Barbalat, Systemes dequations differentielles d oscillations non lineaires, Rev. Math. Pures Appl., 4 (1959), 267-270. 

[4]

N. Bellomo and S.-Y. Ha, A quest toward a mathematical theory of the dynamics of swarms, Math. Models Methods Appl. Sci., 27 (2017), 745-770.  doi: 10.1142/S0218202517500154.

[5]

D. BenedettoE. 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. BronskiT. Carty and S. 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), 562-564. 

[8]

J. Byeon, S. -Y. Ha and H. Park, Asymptotic interplay of states and adapted coupling gains in the Lohe Hermitian sphere model, Submitted.

[9]

S.-H. Choi and S.-Y. Ha, Time-delayed interactions and synchronization of identical Lohe oscillators, Quart. Appl. Math., 74 (2016), 297-319.  doi: 10.1090/qam/1417.

[10]

S. -H. Choi and S. -Y. Ha, Large-time dynamics of the asymptotic Lohe model with a small time-delay, J. Phys. A, 48 (2015), 425101 34 pp. doi: 10.1088/1751-8113/48/42/425101.

[11]

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.

[12]

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.

[13]

P. DegondA. FrouvelleS. Merino-Aceituno and A. Trescases, Quaternions in collective dynamics, Multiscale Model. Simul., 16 (2018), 28-77.  doi: 10.1137/17M1135207.

[14]

L. DeVille, Synchronization and stability for quantum Kuramoto, J. Stat. Phys., 174 (2019), 160-187.  doi: 10.1007/s10955-018-2168-9.

[15]

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.

[16]

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.

[17]

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.

[18]

S. -Y. Ha, D. Kim, D. Kim, H. Park and W. Shim, Emergent dynamics of the Lohe matrix ensemble on a network under time-delayed interactions, J. Math. Phys., 61 (2020), 012702. doi: 10.1063/1.5123257.

[19]

S.-Y. HaD. KoJ. Park and X. Zhang, Collective synchronization of classical and quantum oscillators, EMS Surv. Math. Sci., 3 (2016), 209-267.  doi: 10.4171/EMSS/17.

[20]

S.-Y. HaD. 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.

[21]

S.-Y. HaJ. LeeZ. 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.

[22]

S.-Y. HaS. 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.

[23]

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.

[24]

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.

[25]

J. Hale, Theory of Functional Differential Equations, 2nd ed., Springer-Verlag, New York-Heidelberg, 1977.

[26]

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.

[27]

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.

[28] Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, Academic Press, Inc. Boston, MA, 1993. 
[29]

Y. Kuramoto, Self-Entrainment of a Population of Coupled Non-Linear Oscillators, International Symposium on Mathematical Problems in Mathematical Physics, Lecture Notes in Phys., Vol. 39, Springer, Berlin, 1975, 420-422.

[30]

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.

[31]

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

[32]

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

[33]

J. MarkdahlJ. Thunberg and J. Goncalves, Almost global consensus on the n-sphere, IEEE Trans. Automat. Control, 63 (2018), 1664-1675.  doi: 10.1109/TAC.2017.2752799.

[34]

C. S. Peskin, Mathematical Aspects of Heart Physiology, Courant Institute of Mathematical Sciences, New York University, New York, 1975.

[35] A. PikovskyM. Rosenblum and J. Kurths, Synchronization: A Universal Concept in Nonlinear Sciences, Cambridge University Press, Cambridge, 2001.  doi: 10.1017/CBO9780511755743.
[36]

S. H. Strogatz, From Kuramoto to Crawford: Exploring the Onset of Synchronization in Populations of Coupled Oscillators, Phys. D, 143 (2000), 1-20.  doi: 10.1016/S0167-2789(00)00094-4.

[37]

J. ThunbergJ. MarkdahlF. 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.

[38]

C. M. TopazA. 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.

[39]

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.

[40]

T. Vicsek and A. Zefeiris, Collective motion, Phys. Rep., 517 (2012), 71-140. 

[41]

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

[42]

A. T. Winfree, The Geometry of Biological Time, Springer-Verlag, Berlin-New York, 1980.

[43]

J. Zhu, Synchronization of Kuramoto model in a high-dimensional linear space, Phys. Lett. A, 377 (2013), 2939-2943.  doi: 10.1016/j.physleta.2013.09.010.

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. 

[2]

G. AlbiN. BellomoL. FermoS.-Y. HaJ. KimL. PareschiD. 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. Barbalat, Systemes dequations differentielles d oscillations non lineaires, Rev. Math. Pures Appl., 4 (1959), 267-270. 

[4]

N. Bellomo and S.-Y. Ha, A quest toward a mathematical theory of the dynamics of swarms, Math. Models Methods Appl. Sci., 27 (2017), 745-770.  doi: 10.1142/S0218202517500154.

[5]

D. BenedettoE. 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. BronskiT. Carty and S. 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), 562-564. 

[8]

J. Byeon, S. -Y. Ha and H. Park, Asymptotic interplay of states and adapted coupling gains in the Lohe Hermitian sphere model, Submitted.

[9]

S.-H. Choi and S.-Y. Ha, Time-delayed interactions and synchronization of identical Lohe oscillators, Quart. Appl. Math., 74 (2016), 297-319.  doi: 10.1090/qam/1417.

[10]

S. -H. Choi and S. -Y. Ha, Large-time dynamics of the asymptotic Lohe model with a small time-delay, J. Phys. A, 48 (2015), 425101 34 pp. doi: 10.1088/1751-8113/48/42/425101.

[11]

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.

[12]

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.

[13]

P. DegondA. FrouvelleS. Merino-Aceituno and A. Trescases, Quaternions in collective dynamics, Multiscale Model. Simul., 16 (2018), 28-77.  doi: 10.1137/17M1135207.

[14]

L. DeVille, Synchronization and stability for quantum Kuramoto, J. Stat. Phys., 174 (2019), 160-187.  doi: 10.1007/s10955-018-2168-9.

[15]

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.

[16]

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.

[17]

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.

[18]

S. -Y. Ha, D. Kim, D. Kim, H. Park and W. Shim, Emergent dynamics of the Lohe matrix ensemble on a network under time-delayed interactions, J. Math. Phys., 61 (2020), 012702. doi: 10.1063/1.5123257.

[19]

S.-Y. HaD. KoJ. Park and X. Zhang, Collective synchronization of classical and quantum oscillators, EMS Surv. Math. Sci., 3 (2016), 209-267.  doi: 10.4171/EMSS/17.

[20]

S.-Y. HaD. 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.

[21]

S.-Y. HaJ. LeeZ. 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.

[22]

S.-Y. HaS. 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.

[23]

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.

[24]

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.

[25]

J. Hale, Theory of Functional Differential Equations, 2nd ed., Springer-Verlag, New York-Heidelberg, 1977.

[26]

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.

[27]

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.

[28] Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, Academic Press, Inc. Boston, MA, 1993. 
[29]

Y. Kuramoto, Self-Entrainment of a Population of Coupled Non-Linear Oscillators, International Symposium on Mathematical Problems in Mathematical Physics, Lecture Notes in Phys., Vol. 39, Springer, Berlin, 1975, 420-422.

[30]

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.

[31]

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

[32]

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

[33]

J. MarkdahlJ. Thunberg and J. Goncalves, Almost global consensus on the n-sphere, IEEE Trans. Automat. Control, 63 (2018), 1664-1675.  doi: 10.1109/TAC.2017.2752799.

[34]

C. S. Peskin, Mathematical Aspects of Heart Physiology, Courant Institute of Mathematical Sciences, New York University, New York, 1975.

[35] A. PikovskyM. Rosenblum and J. Kurths, Synchronization: A Universal Concept in Nonlinear Sciences, Cambridge University Press, Cambridge, 2001.  doi: 10.1017/CBO9780511755743.
[36]

S. H. Strogatz, From Kuramoto to Crawford: Exploring the Onset of Synchronization in Populations of Coupled Oscillators, Phys. D, 143 (2000), 1-20.  doi: 10.1016/S0167-2789(00)00094-4.

[37]

J. ThunbergJ. MarkdahlF. 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.

[38]

C. M. TopazA. 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.

[39]

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.

[40]

T. Vicsek and A. Zefeiris, Collective motion, Phys. Rep., 517 (2012), 71-140. 

[41]

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

[42]

A. T. Winfree, The Geometry of Biological Time, Springer-Verlag, Berlin-New York, 1980.

[43]

J. Zhu, Synchronization of Kuramoto model in a high-dimensional linear space, Phys. Lett. A, 377 (2013), 2939-2943.  doi: 10.1016/j.physleta.2013.09.010.

Figure 1.  Exponential aggregation for $ \tau>0 $, $ N = 4 $ and $ d = 2 $
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