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 & 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.   Google Scholar

[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.  Google Scholar

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I. Barbalat, Systemes dequations differentielles d oscillations non lineaires, Rev. Math. Pures Appl., 4 (1959), 267-270.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

J. Buck and E. Buck, Biology of synchronous flashing of fireflies, Nature, 211 (1966), 562-564.   Google Scholar

[8]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

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

[14]

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

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[25]

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

[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.  Google Scholar

[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.  Google Scholar

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

[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.  Google Scholar

[31]

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

[32]

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

[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.  Google Scholar

[34]

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

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[40]

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

[41]

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

[42]

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

[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.  Google Scholar

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.   Google Scholar

[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.  Google Scholar

[3]

I. Barbalat, Systemes dequations differentielles d oscillations non lineaires, Rev. Math. Pures Appl., 4 (1959), 267-270.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

J. Buck and E. Buck, Biology of synchronous flashing of fireflies, Nature, 211 (1966), 562-564.   Google Scholar

[8]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

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

[14]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[25]

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

[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.  Google Scholar

[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.  Google Scholar

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

[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.  Google Scholar

[31]

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

[32]

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

[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.  Google Scholar

[34]

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

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[40]

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

[41]

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

[42]

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

[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.  Google Scholar

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