doi: 10.3934/dcdsb.2021308
Online First

Online First articles are published articles within a journal that have not yet been assigned to a formal issue. This means they do not yet have a volume number, issue number, or page numbers assigned to them, however, they can still be found and cited using their DOI (Digital Object Identifier). Online First publication benefits the research community by making new scientific discoveries known as quickly as possible.

Readers can access Online First articles via the “Online First” tab for the selected journal.

Emergent behaviors of discrete Lohe aggregation flows

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

* Corresponding author: Hyungjun Choi

Received  July 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 Lohe sphere model and the Lohe matrix model are prototype continuous aggregation models on the unit sphere and the unitary group, respectively. These models have been extensively investigated in recent literature. In this paper, we propose several discrete counterparts for the continuous Lohe type aggregation models and study their emergent behaviors using the Lyapunov function method. For suitable discretization of the Lohe sphere model, we employ a scheme consisting of two steps. In the first step, we solve the first-order forward Euler scheme, and in the second step, we project the intermediate state onto the unit sphere. For this discrete model, we present a sufficient framework leading to the complete state aggregation in terms of system parameters and initial data. For the discretization of the Lohe matrix model, we use the Lie group integrator method, Lie-Trotter splitting method and Strang splitting method to propose three discrete models. For these models, we also provide several analytical frameworks leading to complete state aggregation and asymptotic state-locking.

Citation: Hyungjun Choi, Seung-Yeal Ha, Hansol Park. Emergent behaviors of discrete Lohe aggregation flows. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021308
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]

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

[4]

A. J. Bernoff and C. M. Topaz, Nonlocal aggregation models: A primer of swarm equilibria, SIAM Rev., 55 (2013), 709-747.  doi: 10.1137/130925669.  Google Scholar

[5]

A. Bielecki, Estimation of the Euler method error on a Riemannian manifold, Comm. Numer. Methods Engrg., 18 (2002), 757-763.  doi: 10.1002/cnm.516.  Google Scholar

[6]

J. C. Bronski, T. E. Carty and S. E. Simpson, A matrix valued Kuramoto model, J. Stat. Phys., 178 (2020), 595–624. Archived as arXiv: 1903.09223. 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.   Google Scholar

[8]

E. CelledoniH. Marthinsen and B. Owren, An introduction to lie group integrators-basics, new developments and applications, J. Comput. Phys., 257 (2014), 1040-1061.  doi: 10.1016/j.jcp.2012.12.031.  Google Scholar

[9]

D. Chi, S.-H. Choi and S.-Y. Ha, Emergent behaviors of a holonomic particle system on a sphere, J. Math. Phys., 55 (2014), 052703, 18 pp. doi: 10.1063/1.4878117.  Google Scholar

[10]

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

[11]

Y.-P. Choi and S.-Y. Ha, A simple proof of the complete consensus of discrete-time dynamical networks with time-varying couplings, Int. J. Numer. Anal. Model. Ser. B, 1 (2010), 58-69.   Google Scholar

[12]

Y.-P. ChoiS.-Y. HaS. 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.  Google Scholar

[13]

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

[14]

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

[15]

P. DegondA. Frouvelle and S. Merino-Aceituno, A new flocking model through body attitude coordination, Math. Models Methods Appl. Sci., 27 (2017), 1005-1049.  doi: 10.1142/S0218202517400085.  Google Scholar

[16]

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

[17]

M. P. do Carmo, Riemannian Geometry, Mathematics: Theory and Applications, Birkhäuser. Boston, Boston, MA, 1992.  Google Scholar

[18]

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

[19]

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

[20]

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

[21]

S.-Y. Ha, D. Kim, J. Kim and X. Zhang, Uniform-in-time transition from discrete to continuous dynamics in the Kuramoto synchronization, J. of Math. Phys., 60 (2019), 051508, 21 pp. doi: 10.1063/1.5051788.  Google Scholar

[22]

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. D, 416 (2021), Paper No. 132781, 26 pp. doi: 10.1016/j.physd.2020.132781.  Google Scholar

[23]

S.-Y. HaH. 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.  Google Scholar

[24]

S.-Y. HaD. KoJ. Park and X. Zhang, Collective synchronization of classical and quantum oscillators, EMS Surveys in Mathematical Sciences, 3 (2016), 209-267.  doi: 10.4171/EMSS/17.  Google Scholar

[25]

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

[26]

S.-Y. HaD. Ko and S. W. Ryoo, Emergent dynamics of a generalized Lohe model on some class of Lie groups, J. Stat. Phys., 168 (2017), 171-207.  doi: 10.1007/s10955-017-1797-8.  Google Scholar

[27]

S.-Y. HaZ. Li and X. Xue, Formation of phase-locked states in a population of locally interacting Kuramoto oscillators, J. Differential Equations, 255 (2013), 3053-3070.  doi: 10.1016/j.jde.2013.07.013.  Google Scholar

[28]

S.-Y. Ha and H. Park, From the Lohe tensor model to the complex Lohe sphere model and emergent dynamics, SIAM J. Appl. Dyn. Syst., 19 (2020), 1312-1342.  doi: 10.1137/19M1288553.  Google Scholar

[29]

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

[30]

S.-Y. Ha and S. W. Ryoo, On the emergence and orbital Stability of phase-locked states for the Lohe model, J. Stat. Phys., 163 (2016), 411-439.  doi: 10.1007/s10955-016-1481-4.  Google Scholar

[31]

N. H. Ibragimov, Elementary Lie group analysis and ordinary differential equations, Wiley New York, 197, 1999.  Google Scholar

[32]

A. IserlesH. Z. Munthe-KaasS. P. Nørsett and A. Zanna, Lie-group methods, Acta Numerica, 9 (2000), 215-365.  doi: 10.1017/S0962492900002154.  Google Scholar

[33]

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.  Google Scholar

[34]

T. Jahnke and C. Lubich, Error bounds for exponential operator splittings, BIT Numerical Mathematics, 40 (2000), 735-744.  doi: 10.1023/A:1022396519656.  Google Scholar

[35]

Y. Kuramoto, Chemical Oscillations, Waves and Turbulence, Springer-Verlag, Berlin, 1984. doi: 10.1007/978-3-642-69689-3.  Google Scholar

[36]

Y. Kuramoto, International symposium on mathematical problems in mathematical physics, Lecture Notes Theor. Phys., 30 (1975), 420.   Google Scholar

[37]

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.  Google Scholar

[38]

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.  Google Scholar

[39]

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.  Google Scholar

[40]

J. MarkdahlJ. 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.  Google Scholar

[41]

H. Munthe-Kaas, Runge-kutta methods on lie groups, BIT Numerical Mathematics, 38 (1998), 92-111.  doi: 10.1007/BF02510919.  Google Scholar

[42]

R. Olfati-Saber, Swarms on sphere: A programmable swarm with synchronous behaviors like oscillator networks, IEEE 45th Conference on Decision and Control (CDC), (2006), 5060–5066. Google Scholar

[43]

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

[44] A. PikovskyM. Rosenblum and J. Kurths, Synchronization: A universal Concept in Nonlinear Sciences, Cambridge University Press, Cambridge, 2001.  doi: 10.1017/CBO9780511755743.  Google Scholar
[45]

W. Shim, On the generic complete synchronization of the discrete Kuramoto model, Kinetic and Related Models, 13 (2020), 979-1005.  doi: 10.3934/krm.2020034.  Google Scholar

[46]

G. Strang, On the construction and comparison of difference schemes, SIAM J. Numer. Anal., 5 (1968), 506-517.  doi: 10.1137/0705041.  Google Scholar

[47]

S. H. Strogatz, From Kuramoto to Crawford: Exploring the onset of synchronization in populations of coupled oscillators, Physica D, 143 (2000), 1-20.  doi: 10.1016/S0167-2789(00)00094-4.  Google Scholar

[48]

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

[49]

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

[50]

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

[51]

H. F. Trotter, On the product of semi-groups of operators, Proc. Amer. Math. Soc., 10 (1959), 545-551.  doi: 10.1090/S0002-9939-1959-0108732-6.  Google Scholar

[52]

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

[53]

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

[54]

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

[55]

X. Zhang and T. Zhu, Emergent behaviors of the discrete-time Kuramoto model for generic initial configuration, Commun. Math. Sci., 18 (2020), 535-570.  doi: 10.4310/CMS.2020.v18.n2.a11.  Google Scholar

[56]

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.  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]

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

[4]

A. J. Bernoff and C. M. Topaz, Nonlocal aggregation models: A primer of swarm equilibria, SIAM Rev., 55 (2013), 709-747.  doi: 10.1137/130925669.  Google Scholar

[5]

A. Bielecki, Estimation of the Euler method error on a Riemannian manifold, Comm. Numer. Methods Engrg., 18 (2002), 757-763.  doi: 10.1002/cnm.516.  Google Scholar

[6]

J. C. Bronski, T. E. Carty and S. E. Simpson, A matrix valued Kuramoto model, J. Stat. Phys., 178 (2020), 595–624. Archived as arXiv: 1903.09223. 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.   Google Scholar

[8]

E. CelledoniH. Marthinsen and B. Owren, An introduction to lie group integrators-basics, new developments and applications, J. Comput. Phys., 257 (2014), 1040-1061.  doi: 10.1016/j.jcp.2012.12.031.  Google Scholar

[9]

D. Chi, S.-H. Choi and S.-Y. Ha, Emergent behaviors of a holonomic particle system on a sphere, J. Math. Phys., 55 (2014), 052703, 18 pp. doi: 10.1063/1.4878117.  Google Scholar

[10]

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

[11]

Y.-P. Choi and S.-Y. Ha, A simple proof of the complete consensus of discrete-time dynamical networks with time-varying couplings, Int. J. Numer. Anal. Model. Ser. B, 1 (2010), 58-69.   Google Scholar

[12]

Y.-P. ChoiS.-Y. HaS. 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.  Google Scholar

[13]

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

[14]

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

[15]

P. DegondA. Frouvelle and S. Merino-Aceituno, A new flocking model through body attitude coordination, Math. Models Methods Appl. Sci., 27 (2017), 1005-1049.  doi: 10.1142/S0218202517400085.  Google Scholar

[16]

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

[17]

M. P. do Carmo, Riemannian Geometry, Mathematics: Theory and Applications, Birkhäuser. Boston, Boston, MA, 1992.  Google Scholar

[18]

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

[19]

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

[20]

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

[21]

S.-Y. Ha, D. Kim, J. Kim and X. Zhang, Uniform-in-time transition from discrete to continuous dynamics in the Kuramoto synchronization, J. of Math. Phys., 60 (2019), 051508, 21 pp. doi: 10.1063/1.5051788.  Google Scholar

[22]

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. D, 416 (2021), Paper No. 132781, 26 pp. doi: 10.1016/j.physd.2020.132781.  Google Scholar

[23]

S.-Y. HaH. 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.  Google Scholar

[24]

S.-Y. HaD. KoJ. Park and X. Zhang, Collective synchronization of classical and quantum oscillators, EMS Surveys in Mathematical Sciences, 3 (2016), 209-267.  doi: 10.4171/EMSS/17.  Google Scholar

[25]

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

[26]

S.-Y. HaD. Ko and S. W. Ryoo, Emergent dynamics of a generalized Lohe model on some class of Lie groups, J. Stat. Phys., 168 (2017), 171-207.  doi: 10.1007/s10955-017-1797-8.  Google Scholar

[27]

S.-Y. HaZ. Li and X. Xue, Formation of phase-locked states in a population of locally interacting Kuramoto oscillators, J. Differential Equations, 255 (2013), 3053-3070.  doi: 10.1016/j.jde.2013.07.013.  Google Scholar

[28]

S.-Y. Ha and H. Park, From the Lohe tensor model to the complex Lohe sphere model and emergent dynamics, SIAM J. Appl. Dyn. Syst., 19 (2020), 1312-1342.  doi: 10.1137/19M1288553.  Google Scholar

[29]

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

[30]

S.-Y. Ha and S. W. Ryoo, On the emergence and orbital Stability of phase-locked states for the Lohe model, J. Stat. Phys., 163 (2016), 411-439.  doi: 10.1007/s10955-016-1481-4.  Google Scholar

[31]

N. H. Ibragimov, Elementary Lie group analysis and ordinary differential equations, Wiley New York, 197, 1999.  Google Scholar

[32]

A. IserlesH. Z. Munthe-KaasS. P. Nørsett and A. Zanna, Lie-group methods, Acta Numerica, 9 (2000), 215-365.  doi: 10.1017/S0962492900002154.  Google Scholar

[33]

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.  Google Scholar

[34]

T. Jahnke and C. Lubich, Error bounds for exponential operator splittings, BIT Numerical Mathematics, 40 (2000), 735-744.  doi: 10.1023/A:1022396519656.  Google Scholar

[35]

Y. Kuramoto, Chemical Oscillations, Waves and Turbulence, Springer-Verlag, Berlin, 1984. doi: 10.1007/978-3-642-69689-3.  Google Scholar

[36]

Y. Kuramoto, International symposium on mathematical problems in mathematical physics, Lecture Notes Theor. Phys., 30 (1975), 420.   Google Scholar

[37]

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.  Google Scholar

[38]

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.  Google Scholar

[39]

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.  Google Scholar

[40]

J. MarkdahlJ. 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.  Google Scholar

[41]

H. Munthe-Kaas, Runge-kutta methods on lie groups, BIT Numerical Mathematics, 38 (1998), 92-111.  doi: 10.1007/BF02510919.  Google Scholar

[42]

R. Olfati-Saber, Swarms on sphere: A programmable swarm with synchronous behaviors like oscillator networks, IEEE 45th Conference on Decision and Control (CDC), (2006), 5060–5066. Google Scholar

[43]

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

[44] A. PikovskyM. Rosenblum and J. Kurths, Synchronization: A universal Concept in Nonlinear Sciences, Cambridge University Press, Cambridge, 2001.  doi: 10.1017/CBO9780511755743.  Google Scholar
[45]

W. Shim, On the generic complete synchronization of the discrete Kuramoto model, Kinetic and Related Models, 13 (2020), 979-1005.  doi: 10.3934/krm.2020034.  Google Scholar

[46]

G. Strang, On the construction and comparison of difference schemes, SIAM J. Numer. Anal., 5 (1968), 506-517.  doi: 10.1137/0705041.  Google Scholar

[47]

S. H. Strogatz, From Kuramoto to Crawford: Exploring the onset of synchronization in populations of coupled oscillators, Physica D, 143 (2000), 1-20.  doi: 10.1016/S0167-2789(00)00094-4.  Google Scholar

[48]

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

[49]

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

[50]

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

[51]

H. F. Trotter, On the product of semi-groups of operators, Proc. Amer. Math. Soc., 10 (1959), 545-551.  doi: 10.1090/S0002-9939-1959-0108732-6.  Google Scholar

[52]

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

[53]

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

[54]

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

[55]

X. Zhang and T. Zhu, Emergent behaviors of the discrete-time Kuramoto model for generic initial configuration, Commun. Math. Sci., 18 (2020), 535-570.  doi: 10.4310/CMS.2020.v18.n2.a11.  Google Scholar

[56]

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.  Google Scholar

[1]

Seung-Yeal Ha, Hansol Park. Emergent behaviors of the generalized Lohe matrix model. Discrete & Continuous Dynamical Systems - B, 2021, 26 (8) : 4227-4261. doi: 10.3934/dcdsb.2020286

[2]

Paolo Antonelli, Seung-Yeal Ha, Dohyun Kim, Pierangelo Marcati. The Wigner-Lohe model for quantum synchronization and its emergent dynamics. Networks & Heterogeneous Media, 2017, 12 (3) : 403-416. doi: 10.3934/nhm.2017018

[3]

Seung-Yeal Ha, Dohyun Kim, Jaeseung Lee, Se Eun Noh. Emergence of aggregation in the swarm sphere model with adaptive coupling laws. Kinetic & Related Models, 2019, 12 (2) : 411-444. doi: 10.3934/krm.2019018

[4]

Seung-Yeal Ha, Jaeseung Lee, Zhuchun Li. Emergence of local synchronization in an ensemble of heterogeneous Kuramoto oscillators. Networks & Heterogeneous Media, 2017, 12 (1) : 1-24. doi: 10.3934/nhm.2017001

[5]

Hyungjin Huh. Remarks on the Schrödinger-Lohe model. Networks & Heterogeneous Media, 2019, 14 (4) : 759-769. doi: 10.3934/nhm.2019030

[6]

Seung-Yeal Ha, Myeongju Kang, Hansol Park. Collective behaviors of the Lohe Hermitian sphere model with inertia. Communications on Pure & Applied Analysis, 2021, 20 (7&8) : 2613-2641. doi: 10.3934/cpaa.2021046

[7]

Pedro Aceves-Sanchez, Benjamin Aymard, Diane Peurichard, Pol Kennel, Anne Lorsignol, Franck Plouraboué, Louis Casteilla, Pierre Degond. A new model for the emergence of blood capillary networks. Networks & Heterogeneous Media, 2021, 16 (1) : 91-138. doi: 10.3934/nhm.2021001

[8]

Ondrej Budáč, Michael Herrmann, Barbara Niethammer, Andrej Spielmann. On a model for mass aggregation with maximal size. Kinetic & Related Models, 2011, 4 (2) : 427-439. doi: 10.3934/krm.2011.4.427

[9]

Dong Li, Xiaoyi Zhang. On a nonlocal aggregation model with nonlinear diffusion. Discrete & Continuous Dynamical Systems, 2010, 27 (1) : 301-323. doi: 10.3934/dcds.2010.27.301

[10]

Seung-Yeal Ha, Jinyeong Park, Sang Woo Ryoo. Emergence of phase-locked states for the Winfree model in a large coupling regime. Discrete & Continuous Dynamical Systems, 2015, 35 (8) : 3417-3436. doi: 10.3934/dcds.2015.35.3417

[11]

Floriane Lignet, Vincent Calvez, Emmanuel Grenier, Benjamin Ribba. A structural model of the VEGF signalling pathway: Emergence of robustness and redundancy properties. Mathematical Biosciences & Engineering, 2013, 10 (1) : 167-184. doi: 10.3934/mbe.2013.10.167

[12]

Seung-Yeal Ha, Jinwook Jung, Peter Kuchling. Emergence of anomalous flocking in the fractional Cucker-Smale model. Discrete & Continuous Dynamical Systems, 2019, 39 (9) : 5465-5489. doi: 10.3934/dcds.2019223

[13]

Woojoo Shim. On the generic complete synchronization of the discrete Kuramoto model. Kinetic & Related Models, 2020, 13 (5) : 979-1005. doi: 10.3934/krm.2020034

[14]

Vincent Calvez, Benoȋt Perthame, Shugo Yasuda. Traveling wave and aggregation in a flux-limited Keller-Segel model. Kinetic & Related Models, 2018, 11 (4) : 891-909. doi: 10.3934/krm.2018035

[15]

Casimir Emako-Kazianou, Jie Liao, Nicolas Vauchelet. Synchronising and non-synchronising dynamics for a two-species aggregation model. Discrete & Continuous Dynamical Systems - B, 2017, 22 (6) : 2121-2146. doi: 10.3934/dcdsb.2017088

[16]

Lambertus A. Peletier, Xi-Ling Jiang, Snehal Samant, Stephan Schmidt. Analysis of a complex physiology-directed model for inhibition of platelet aggregation by clopidogrel. Discrete & Continuous Dynamical Systems, 2017, 37 (2) : 945-961. doi: 10.3934/dcds.2017039

[17]

Xin Xu. Existence of monotone positive solutions of a neighbour based chemotaxis model and aggregation phenomenon. Communications on Pure & Applied Analysis, 2020, 19 (9) : 4327-4348. doi: 10.3934/cpaa.2020195

[18]

Marcello Delitala, Tommaso Lorenzi. Emergence of spatial patterns in a mathematical model for the co-culture dynamics of epithelial-like and mesenchymal-like cells. Mathematical Biosciences & Engineering, 2017, 14 (1) : 79-93. doi: 10.3934/mbe.2017006

[19]

Igor Chueshov, Peter E. Kloeden, Meihua Yang. Synchronization in coupled stochastic sine-Gordon wave model. Discrete & Continuous Dynamical Systems - B, 2016, 21 (9) : 2969-2990. doi: 10.3934/dcdsb.2016082

[20]

Xiaoxue Zhao, Zhuchun Li. Synchronization of a Kuramoto-like model for power grids with frustration. Networks & Heterogeneous Media, 2020, 15 (3) : 543-553. doi: 10.3934/nhm.2020030

2020 Impact Factor: 1.327

Metrics

  • PDF downloads (42)
  • HTML views (29)
  • Cited by (0)

Other articles
by authors

[Back to Top]