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April  2020, 40(4): 2037-2060. doi: 10.3934/dcds.2020105

Emergent dynamics of an orientation flocking model for multi-agent system

Seung-Yeal Ha 1,2, , Dohyun Kim 3, , Jaeseung Lee 4, and  Se Eun Noh 5,,

1. 

Department of Mathematical Sciences and Research Institute of Mathematics, Seoul National University, Seoul 08826, Republic of Korea
2. 

Korea Institute for Advanced Study, Hoegiro 85, Seoul, 02455, Republic of Korea
3. 

National Institute for Mathematical Sciences, 70, Yuseong-daero 1689 beon-gil, Yuseong-gu, Daejeon 34047, Republic of Korea
4. 

The Research Institute of Basic Sciences, Seoul National University, Seoul 08826, Republic of Korea
5. 

Department of Mathematics, Myongji University, 116 Myongjiro, Yong-In 17058, Republic of Korea

* Corresponding author: Se Eun Noh

Received  October 2018 Revised  September 2019 Published  January 2020

Fund Project: The work of S.-Y. Ha was supported by the National Research Foundation of Korea (NRF-2017R1A5A1015626), the work of D. Kim was supported from the National Institute for Mathematical Sciences (NIMS) grant funded by the Korea government (MIST) (No.B19610000), and the work of S. E. Noh was supported by the National Research Foundation of Korea (NRF-2017R1C1B5018312).

We study the orientation flocking for the deterministic counterpart of a stochastic agent-based model introduced by Degond, Frouvelle and Merino-Aceituno in 2017, where the orientation is defined as a $ {\rm SO}(3) $ matrix. Their proposed model can be reduced to the other collective dynamics models such as the Lohe matrix model and the Viscek-type model as special cases. In this work, we study the emergent dynamics of the orientation flocking model in two frameworks. First, we present sufficient conditions leading to the orientation flocking when the natural frequency matrices are identical. To be precise, we prove that all orientation matrices asymptotically converge to the common one, and the spatial position diameter remains uniformly bounded. Second, we show the emergence of orientation-locked states for non-identical natural frequency matrices, that is, the difference of any two orientation matrices tends to the definite constant matrix. On the other hand, we establish the finite-in-time stability with respect to initial data of the proposed orientation flocking model. We also present the numerical results consistent with our rigorous analysis. Our work remains valid even for dimensions greater than three.

Keywords: Body attitude coordination, emergence, Lohe model, orientation flocking, stability.
Mathematics Subject Classification: Primary: 82C10, 82C22, 35B37, 34C15.
Citation: Seung-Yeal Ha, Dohyun Kim, Jaeseung Lee, Se Eun Noh. Emergent dynamics of an orientation flocking model for multi-agent system. Discrete and Continuous Dynamical Systems, 2020, 40 (4) : 2037-2060. doi: 10.3934/dcds.2020105
References:
[1]

J. A. Acebrón, L. L. Bonilla, C. J. Pérez Vicente, F. Ritort and R. Spigler, The Kuramoto model: A simple paradigm for synchronization phenomena, Rev. Mod. Phys., 77 (2005), 137-185.  doi: 10.1103/RevModPhys.77.137.

[2]

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.

[3]

D. Benedetto, E. Caglioti and U. Montemagno, Exponential dephasing of oscillators in the kinetic Kuramoto model, J. Stat. Phys., 162 (2016), 813-823.  doi: 10.1007/s10955-015-1426-3.

[4]

J. C. Bronski, L. DeVille and M. J. Park, Fully synchronous solutions and the synchronization phase transition for the finite-$N$ Kuramoto model, Chaos, 22 (2012), 033133, 17 pp. doi: 10.1063/1.4745197.

[5]

S. Chandra, M. Girvan and E. Ott, Complexity reduction ansatz for systems of interacting orientable agents: Beyond the Kuramoto model, Chaos, 29 (2019), 053107, 8 pp. doi: 10.1063/1.5093038.

[6]

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.

[7]

S.-H. Choi and S.-Y. Ha, Emergence of flocking for a multi-agent system moving with constant speed, Commun. Math. Sci., 14 (2016), 953-972.  doi: 10.4310/CMS.2016.v14.n4.a4.

[8]

Y.-P. Choi, S.-Y. Ha, S. Jung and Y. Kim, Asymptotic formation and orbital stability of phase-locked states for the Kuramoto model, Phys. D, 241 (2012), 735-754.  doi: 10.1016/j.physd.2011.11.011.

[9]

N. Chopra and M. W. Spong, On exponential synchronization of Kuramoto oscillators, IEEE Trans. Automat. Control, 54 (2009), 353-357.  doi: 10.1109/TAC.2008.2007884.

[10]

F. Cucker and S. Smale, Emergent behavior in flocks, IEEE Trans. Automat. Control, 52 (2007), 852-862.  doi: 10.1109/TAC.2007.895842.

[11]

P. Degond, A. 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.

[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, Synchronization 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. P. 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 and J.-G. Liu, A simple proof of Cucker-Smale flocking dynamics and mean field limit, Commun. Math. Sci., 7 (2009), 297-325.  doi: 10.4310/CMS.2009.v7.n2.a2.

[18]

S.-Y. Ha, T. Ha and J.-H. Kim, On the complete synchronization for the globally coupled Kuramoto model, Phys. D, 239 (2010), 1692-1700.  doi: 10.1016/j.physd.2010.05.003.

[19]

S.-Y. Ha, M.-J. Kang and E. Jeong, Emergent behaviour of a generalized Viscek-type flocking model, Nonlinearity, 23 (2010), 3139-3156.  doi: 10.1088/0951-7715/23/12/008.

[20]

S.-Y. Ha, D. Kim, J. Lee and S. E. Noh, Particle and kinetic models for swarming particles on a sphere and stability properties, J. Stat. Phys., 174 (2019), 622-655.  doi: 10.1007/s10955-018-2169-8.

[21]

S.-Y. Ha, H. K. Kim and J. Park, Remarks on the complete synchronization of Kuramoto oscillators, Nonlinearity, 28 (2015), 1441-1462.  doi: 10.1088/0951-7715/28/5/1441.

[22]

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

[23]

S.-Y. Ha, H. K. Kim and S. W. Ryoo, Emergence of phase-locked states for the Kuramoto model in a large coupling strength, Discrete Contin. Dyn. Syst., 35 (2015), 3417-3436. 

[24]

S.-Y. Ha, D. 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.

[25]

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.

[26]

S.-Y. Ha, D. Ko and Y. L. Zhang, Remarks on the critical coupling strength for the Cucker-Smale model with unit speed, Discrete Contin. Dyn. Syst., 38 (2018), 2763-2793.  doi: 10.3934/dcds.2018116.

[27]

S.-Y. Ha, J. Kim and X. T. Zhang, Uniform stability of the Cucker-Smale model and its application to the mean-field limit, Kinet. Relat. Mod., 11 (2018), 1157-1181.  doi: 10.3934/krm.2018045.

[28]

S.-Y. Ha, Z. C. Li and X. P. 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.

[29]

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.

[30]

Y. Kuramoto, Chemical Oscillations, Waves and Turbulence, Springer Series in Synergetics, 19. Springer-Verlag, Berlin, 1984. doi: 10.1007/978-3-642-69689-3.

[31]

Y. Kuramoto, Self-entrainment of population of coupled non-linear oscillators, International Symposium on Mathematical Problems in Theoretical Physics, Lecture Notes in Phys., Springer, Berlin, 39 (1975), 420-422. 

[32]

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.

[33]

R. Olfati-Saber, Swarms on sphere: A programmable swarm with synchronous behaviors like oscillator networks, Proc. of the 45th IEEE conference on Decision and Control, (2006), 5060–5066. doi: 10.1109/CDC.2006.376811.

[34] A. Pikovsky, M. Rosenblum and J. Kurths, Synchronization: A Universal Concept in Nonlinear Sciences, Cambridge Nonlinear Science Series, 12. Cambridge University Press, Cambridge, 2001.  doi: 10.1017/CBO9780511755743.
[35]

A. Sarlette, R. Sepulchre and N. E. Leonard, Autonomous rigid body attitude synchronization, Automatica, 45 (2009), 572-577. 

[36]

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.

[37]

R. Tron, B. Afsari and R. Vidal, Intrinsic consensus on SO(3) with almost-global convergence, 2012 IEEE Conference on Decision and Control, (2012), 2052–2058. doi: 10.1109/CDC.2012.6426677.

[38]

T. Vicsek, A. Czirók, E. Ben-Jacob, I. CohenI and O. Shochet, Novel type of phase transition in a system of self-drivenparticles, Phys. Rev. Lett., 75 (1995), 1226-1229.  doi: 10.1103/PhysRevLett.75.1226.

show all references

References:
[1]

J. A. Acebrón, L. L. Bonilla, C. J. Pérez Vicente, F. Ritort and R. Spigler, The Kuramoto model: A simple paradigm for synchronization phenomena, Rev. Mod. Phys., 77 (2005), 137-185.  doi: 10.1103/RevModPhys.77.137.

[2]

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.

[3]

D. Benedetto, E. Caglioti and U. Montemagno, Exponential dephasing of oscillators in the kinetic Kuramoto model, J. Stat. Phys., 162 (2016), 813-823.  doi: 10.1007/s10955-015-1426-3.

[4]

J. C. Bronski, L. DeVille and M. J. Park, Fully synchronous solutions and the synchronization phase transition for the finite-$N$ Kuramoto model, Chaos, 22 (2012), 033133, 17 pp. doi: 10.1063/1.4745197.

[5]

S. Chandra, M. Girvan and E. Ott, Complexity reduction ansatz for systems of interacting orientable agents: Beyond the Kuramoto model, Chaos, 29 (2019), 053107, 8 pp. doi: 10.1063/1.5093038.

[6]

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.

[7]

S.-H. Choi and S.-Y. Ha, Emergence of flocking for a multi-agent system moving with constant speed, Commun. Math. Sci., 14 (2016), 953-972.  doi: 10.4310/CMS.2016.v14.n4.a4.

[8]

Y.-P. Choi, S.-Y. Ha, S. Jung and Y. Kim, Asymptotic formation and orbital stability of phase-locked states for the Kuramoto model, Phys. D, 241 (2012), 735-754.  doi: 10.1016/j.physd.2011.11.011.

[9]

N. Chopra and M. W. Spong, On exponential synchronization of Kuramoto oscillators, IEEE Trans. Automat. Control, 54 (2009), 353-357.  doi: 10.1109/TAC.2008.2007884.

[10]

F. Cucker and S. Smale, Emergent behavior in flocks, IEEE Trans. Automat. Control, 52 (2007), 852-862.  doi: 10.1109/TAC.2007.895842.

[11]

P. Degond, A. 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.

[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, Synchronization 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. P. 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 and J.-G. Liu, A simple proof of Cucker-Smale flocking dynamics and mean field limit, Commun. Math. Sci., 7 (2009), 297-325.  doi: 10.4310/CMS.2009.v7.n2.a2.

[18]

S.-Y. Ha, T. Ha and J.-H. Kim, On the complete synchronization for the globally coupled Kuramoto model, Phys. D, 239 (2010), 1692-1700.  doi: 10.1016/j.physd.2010.05.003.

[19]

S.-Y. Ha, M.-J. Kang and E. Jeong, Emergent behaviour of a generalized Viscek-type flocking model, Nonlinearity, 23 (2010), 3139-3156.  doi: 10.1088/0951-7715/23/12/008.

[20]

S.-Y. Ha, D. Kim, J. Lee and S. E. Noh, Particle and kinetic models for swarming particles on a sphere and stability properties, J. Stat. Phys., 174 (2019), 622-655.  doi: 10.1007/s10955-018-2169-8.

[21]

S.-Y. Ha, H. K. Kim and J. Park, Remarks on the complete synchronization of Kuramoto oscillators, Nonlinearity, 28 (2015), 1441-1462.  doi: 10.1088/0951-7715/28/5/1441.

[22]

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

[23]

S.-Y. Ha, H. K. Kim and S. W. Ryoo, Emergence of phase-locked states for the Kuramoto model in a large coupling strength, Discrete Contin. Dyn. Syst., 35 (2015), 3417-3436. 

[24]

S.-Y. Ha, D. 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.

[25]

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.

[26]

S.-Y. Ha, D. Ko and Y. L. Zhang, Remarks on the critical coupling strength for the Cucker-Smale model with unit speed, Discrete Contin. Dyn. Syst., 38 (2018), 2763-2793.  doi: 10.3934/dcds.2018116.

[27]

S.-Y. Ha, J. Kim and X. T. Zhang, Uniform stability of the Cucker-Smale model and its application to the mean-field limit, Kinet. Relat. Mod., 11 (2018), 1157-1181.  doi: 10.3934/krm.2018045.

[28]

S.-Y. Ha, Z. C. Li and X. P. 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.

[29]

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.

[30]

Y. Kuramoto, Chemical Oscillations, Waves and Turbulence, Springer Series in Synergetics, 19. Springer-Verlag, Berlin, 1984. doi: 10.1007/978-3-642-69689-3.

[31]

Y. Kuramoto, Self-entrainment of population of coupled non-linear oscillators, International Symposium on Mathematical Problems in Theoretical Physics, Lecture Notes in Phys., Springer, Berlin, 39 (1975), 420-422. 

[32]

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.

[33]

R. Olfati-Saber, Swarms on sphere: A programmable swarm with synchronous behaviors like oscillator networks, Proc. of the 45th IEEE conference on Decision and Control, (2006), 5060–5066. doi: 10.1109/CDC.2006.376811.

[34] A. Pikovsky, M. Rosenblum and J. Kurths, Synchronization: A Universal Concept in Nonlinear Sciences, Cambridge Nonlinear Science Series, 12. Cambridge University Press, Cambridge, 2001.  doi: 10.1017/CBO9780511755743.
[35]

A. Sarlette, R. Sepulchre and N. E. Leonard, Autonomous rigid body attitude synchronization, Automatica, 45 (2009), 572-577. 

[36]

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.

[37]

R. Tron, B. Afsari and R. Vidal, Intrinsic consensus on SO(3) with almost-global convergence, 2012 IEEE Conference on Decision and Control, (2012), 2052–2058. doi: 10.1109/CDC.2012.6426677.

[38]

T. Vicsek, A. Czirók, E. Ben-Jacob, I. CohenI and O. Shochet, Novel type of phase transition in a system of self-drivenparticles, Phys. Rev. Lett., 75 (1995), 1226-1229.  doi: 10.1103/PhysRevLett.75.1226.

Figure 1.  Identical case: $ H_i \equiv H $
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Figure 2.  Non-identical case
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