doi: 10.3934/krm.2021007

Emergent dynamics of a thermodynamic Cucker-Smale ensemble on complete Riemannian manifolds

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. 

School of Mathematics, Korea Institute for Advanced Study, Seoul 02455, Republic of Korea

4. 

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

* Corresponding author: Woojoo Shim

Received  September 2020 Revised  December 2020 Published  January 2021

Fund Project: The work of S.-Y. Ha was supported by National Research Foundation of Korea(NRF-2020R1A2C3A01003881). The author would like to thank Prof. Marshall Slemrod for the suggestion of LaSalle type arguments in Corollary 1, Corollary 2 and Theorem 5.2

We study emergent collective behaviors of a thermodynamic Cucker-Smale (TCS) ensemble on complete smooth Riemannian manifolds. For this, we extend the TCS model on the Euclidean space to a complete smooth Riemannian manifold by adopting the work [30] for a CS ensemble, and provide a sufficient framework to achieve velocity alignment and thermal equilibrium. Compared to the model proposed in [30], our model has an extra thermodynamic observable denoted by temperature, which is assumed to be nonidentical for each particle. However, for isothermal case, our model reduces to the previous CS model in [30] on a manifold in a small velocity regime. As a concrete example, we study emergent dynamics of the TCS model on the unit $ d $-sphere $ \mathbb{S}^d $. We show that the asymptotic emergent dynamics of the proposed TCS model on the unit $ d $-sphere exhibits a dichotomy, either convergence to zero velocity or asymptotic approach toward a common great circle. We also provide several numerical examples illustrating the aforementioned dichotomy on the asymptotic dynamics of the TCS particles on $ \mathbb{S}^2 $.

Citation: Hyunjin Ahn, Seung-Yeal Ha, Woojoo Shim. Emergent dynamics of a thermodynamic Cucker-Smale ensemble on complete Riemannian manifolds. Kinetic & Related Models, doi: 10.3934/krm.2021007
References:
[1]

J. A. AcebrónL. L. BonillaC. J. 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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A. Aydo$\breve{g}$duS. T. McQuade and N. Pouradier Duteil, Opinion dynamics on a general compact Riemannian manifold, Netw. Heterog. Media, 12 (2017), 489-523.  doi: 10.3934/nhm.2017021.  Google Scholar

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I. Barbalat, Systèmes déquations différentielles doscillations non Linéaires, Rev. Math. Pures Appl., 4 (1959), 267-270.   Google Scholar

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J. Buck and E. Buck, Biology of synchronous flashing of fireflies, Nature, 211 (1966), 562-564.  doi: 10.1038/211562a0.  Google Scholar

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J. C. BronskiT. 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.  Google Scholar

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H. ChatéF. GinelliG. GrégoireF. Peruani and F. Raynaud, Modeling collective motion: variations on the Vicsek model, The European Physical Journal B, 64 (2008), 451-456.   Google Scholar

[8]

Y.-P. ChoiS.-Y. HaJ. Jung and J. Kim, Global dynamics of the thermomechanical Cucker-Smale ensemble immersed in incompressible viscous fluids, Nonlinearity, 32 (2019), 1597-1640.  doi: 10.1088/1361-6544/aafaae.  Google Scholar

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Y.-P. ChoiS.-Y. HaJ. Jung and J. Kim, On the coupling of kinetic thermomechanical Cucker-Smale equation and compressible viscous fluid system, J. Math. Fluid Mech., 22 (2020), 4-38.  doi: 10.1007/s00021-019-0466-x.  Google Scholar

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Y.-P. ChoiS.-Y. Ha and J. Kim, Propagation of regularity and finite-time collisions for the thermomechanical Cucker-Smale model with a singular communication, Netw. Heterog. Media, 13 (2018), 379-407.  doi: 10.3934/nhm.2018017.  Google Scholar

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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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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

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J.-G. Dong, S.-Y. Ha and D. Kim, From discrete Cucker-Smale model to continuous Cucker-Smale model in a temperature field, J. Math. Phys., 60 (2019), 072705, 22 pp. doi: 10.1063/1.5084770.  Google Scholar

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J.-G. DongS.-Y. Ha and D. Kim, On the Cucker-Smale with q-closest neighbors in a self-consistent temperature field, SIAM J. Control and Optimization, 58 (2020), 368-392.  doi: 10.1137/18M1195462.  Google Scholar

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J.-G. DongS.-Y. HaD. Kim and J. Kim, Time-delay effect on the flocking in an ensemble of thermomechanical Cucker-Smale particles, J. Differential Equations, 266 (2019), 2373-2407.  doi: 10.1016/j.jde.2018.08.034.  Google Scholar

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R. Fetecau, H. Park and F. S. Patacchini, Well-posedness and asymptotic behaviour of an aggregation model with intrinsic interactions on sphere and other manifolds, Analysis and Applications. Google Scholar

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A. Frouvelle and J.-G. Liu, Long-time dynamics for a simple aggregation equation on the sphere, in International Workshop on Stochastic Dynamics out of Equilibrium, Springer, Cham, 282 (2017), 457-479. doi: 10.1007/978-3-030-15096-9_16.  Google Scholar

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R. C. Fetecau and B. Zhang, Self-organization on Riemannian manifolds, J. Geom. Mech., 11 (2019), 397-426.  doi: 10.3934/jgm.2019020.  Google Scholar

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S.-Y. Ha, S. Hwang, D. Kim, S.-C. Kim and C. Min, Emergent behaviors of a first-order particle swarm model on the hyperbolic space, J. Math. Phys., 61 (2020), 042701, 23 pp. doi: 10.1063/1.5066255.  Google Scholar

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S.-Y. Ha and D. Kim, A second-order particle swarm model on a sphere and emergent dynamics, SIAM J. Appl. Dyn. Syst., 18 (2019), 80-116.  doi: 10.1137/18M1205996.  Google Scholar

[23]

S.-Y. Ha and D. Kim, Emergent behavior of a second-order Lohe matrix model on the unitary group, J. Stat. Phys., 175 (2019), 904-931.  doi: 10.1007/s10955-019-02270-y.  Google Scholar

[24]

S.-Y. HaJ. KimC. MinT. Ruggeri and X. Zhang, Uniform stability and mean-field limit of thermodynamic Cucker-Smale model, Quart. Appl. Math., 77 (2019), 131-176.  doi: 10.1090/qam/1517.  Google Scholar

[25]

S.-Y. HaJ. KimJ. Park and X. Zhang, Complete cluster predictability of the Cucker-Smale flocking model on the real line, Arch. Ration. Mech. Anal., 231 (2019), 319-365.  doi: 10.1007/s00205-018-1281-x.  Google Scholar

[26]

S.-Y. HaJ. Kim and T. Ruggeri, Emergent behaviors of thermodynamic Cucker-Smale particles, SIAM J. Math. Anal., 50 (2018), 3092-3121.  doi: 10.1137/17M111064X.  Google Scholar

[27]

S.-Y. HaJ. Kim and T. Ruggeri, From the relativistic mixture of gases to the relativistic cucker-smale flocking, Arch. Rational Mech. Anal., 235 (2020), 1661-1706.  doi: 10.1007/s00205-019-01452-y.  Google Scholar

[28]

S.-Y. HaD. Ko and S. 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

[29]

S.-Y. HaD. Ko and S. 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

[30]

S.-Y. Ha, D. Kim and F. W. Schlöder, Emergent behaviors of Cucker-Smale flocks on Riemannian manifolds, IEEE Trans. Automat. Control, (2020). doi: 10.1109/TAC.2020.3014096.  Google Scholar

[31]

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.  Google Scholar

[32]

S.-Y. Ha and T. Ruggeri, Emergent dynamics of a thermodynamically consistent particle model, Arch. Ration. Mech. Anal, 223 (2017), 1397-1425.  doi: 10.1007/s00205-016-1062-3.  Google Scholar

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S.-Y. Ha and E. Tadmor, From particle to kinetic and hydrodynamic description of flocking, Kinet. Relat. Models, 1 (2008), 415-435.  doi: 10.3934/krm.2008.1.415.  Google Scholar

[35]

J. J$\ddot{u}$rgen, Riemannian Geometry and Geometric Analysis, Universitext. Springer 2011. doi: 10.1007/978-3-642-21298-7.  Google Scholar

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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

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J. Markdahl, Synchronization on Riemannian manifolds: Multiply connected implies multistable, IEEE Trans. Automat. Control, (2019). doi: 10.1109/TAC.2020.3030849.  Google Scholar

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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

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S. Motsch and E. Tadmor, Heterophilious dynamics enhances consensus, SIAM Review, 56 (2014), 577-621.  doi: 10.1137/120901866.  Google Scholar

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S. Motsch and E. Tadmor, A new model for self-organized dynamics and its flocking behavior, J. Stat. Phys., 144 (2011), 923-947.  doi: 10.1007/s10955-011-0285-9.  Google Scholar

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R. Olfati-SaberJ. A. Fax and R. M. Murray, Consensus and cooperation in networked multi-agent systems, Proc. IEEE, 95 (2007), 215-233.  doi: 10.1109/JPROC.2006.887293.  Google Scholar

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L. M. Ritchie, M. A. Lohe and A. G. Williams, Synchronization of relativistic particles in the hyperbolic Kuramoto model, Chaos, 28 (2018), 053116, 11 pp. doi: 10.1063/1.5021701.  Google Scholar

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A. SarletteS. Bonnabel and R. Sepulchre, Coordinated motion design on Lie groups, IEEE Trans. Automat. Control, 55 (2010), 1047-1058.  doi: 10.1109/TAC.2010.2042003.  Google Scholar

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show all references

References:
[1]

J. A. AcebrónL. L. BonillaC. J. 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]

A. Aydo$\breve{g}$duS. T. McQuade and N. Pouradier Duteil, Opinion dynamics on a general compact Riemannian manifold, Netw. Heterog. Media, 12 (2017), 489-523.  doi: 10.3934/nhm.2017021.  Google Scholar

[4]

I. Barbalat, Systèmes déquations différentielles doscillations non Linéaires, Rev. Math. Pures Appl., 4 (1959), 267-270.   Google Scholar

[5]

J. Buck and E. Buck, Biology of synchronous flashing of fireflies, Nature, 211 (1966), 562-564.  doi: 10.1038/211562a0.  Google Scholar

[6]

J. C. BronskiT. 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.  Google Scholar

[7]

H. ChatéF. GinelliG. GrégoireF. Peruani and F. Raynaud, Modeling collective motion: variations on the Vicsek model, The European Physical Journal B, 64 (2008), 451-456.   Google Scholar

[8]

Y.-P. ChoiS.-Y. HaJ. Jung and J. Kim, Global dynamics of the thermomechanical Cucker-Smale ensemble immersed in incompressible viscous fluids, Nonlinearity, 32 (2019), 1597-1640.  doi: 10.1088/1361-6544/aafaae.  Google Scholar

[9]

Y.-P. ChoiS.-Y. HaJ. Jung and J. Kim, On the coupling of kinetic thermomechanical Cucker-Smale equation and compressible viscous fluid system, J. Math. Fluid Mech., 22 (2020), 4-38.  doi: 10.1007/s00021-019-0466-x.  Google Scholar

[10]

Y.-P. ChoiS.-Y. Ha and J. Kim, Propagation of regularity and finite-time collisions for the thermomechanical Cucker-Smale model with a singular communication, Netw. Heterog. Media, 13 (2018), 379-407.  doi: 10.3934/nhm.2018017.  Google Scholar

[11]

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

[12]

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

[13]

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

[14]

J.-G. Dong, S.-Y. Ha and D. Kim, From discrete Cucker-Smale model to continuous Cucker-Smale model in a temperature field, J. Math. Phys., 60 (2019), 072705, 22 pp. doi: 10.1063/1.5084770.  Google Scholar

[15]

J.-G. DongS.-Y. Ha and D. Kim, On the Cucker-Smale with q-closest neighbors in a self-consistent temperature field, SIAM J. Control and Optimization, 58 (2020), 368-392.  doi: 10.1137/18M1195462.  Google Scholar

[16]

J.-G. Dong, S.-Y. Ha and D. Kim, Emergence of mono-cluster flocking in the thermomechanical Cucker-Smale model under switching topologies, Analysis and Applications, (2020), 1-38. doi: 10.1142/S0219530520500025.  Google Scholar

[17]

J.-G. DongS.-Y. HaD. Kim and J. Kim, Time-delay effect on the flocking in an ensemble of thermomechanical Cucker-Smale particles, J. Differential Equations, 266 (2019), 2373-2407.  doi: 10.1016/j.jde.2018.08.034.  Google Scholar

[18]

R. Fetecau, H. Park and F. S. Patacchini, Well-posedness and asymptotic behaviour of an aggregation model with intrinsic interactions on sphere and other manifolds, Analysis and Applications. Google Scholar

[19]

A. Frouvelle and J.-G. Liu, Long-time dynamics for a simple aggregation equation on the sphere, in International Workshop on Stochastic Dynamics out of Equilibrium, Springer, Cham, 282 (2017), 457-479. doi: 10.1007/978-3-030-15096-9_16.  Google Scholar

[20]

R. C. Fetecau and B. Zhang, Self-organization on Riemannian manifolds, J. Geom. Mech., 11 (2019), 397-426.  doi: 10.3934/jgm.2019020.  Google Scholar

[21]

S.-Y. Ha, S. Hwang, D. Kim, S.-C. Kim and C. Min, Emergent behaviors of a first-order particle swarm model on the hyperbolic space, J. Math. Phys., 61 (2020), 042701, 23 pp. doi: 10.1063/1.5066255.  Google Scholar

[22]

S.-Y. Ha and D. Kim, A second-order particle swarm model on a sphere and emergent dynamics, SIAM J. Appl. Dyn. Syst., 18 (2019), 80-116.  doi: 10.1137/18M1205996.  Google Scholar

[23]

S.-Y. Ha and D. Kim, Emergent behavior of a second-order Lohe matrix model on the unitary group, J. Stat. Phys., 175 (2019), 904-931.  doi: 10.1007/s10955-019-02270-y.  Google Scholar

[24]

S.-Y. HaJ. KimC. MinT. Ruggeri and X. Zhang, Uniform stability and mean-field limit of thermodynamic Cucker-Smale model, Quart. Appl. Math., 77 (2019), 131-176.  doi: 10.1090/qam/1517.  Google Scholar

[25]

S.-Y. HaJ. KimJ. Park and X. Zhang, Complete cluster predictability of the Cucker-Smale flocking model on the real line, Arch. Ration. Mech. Anal., 231 (2019), 319-365.  doi: 10.1007/s00205-018-1281-x.  Google Scholar

[26]

S.-Y. HaJ. Kim and T. Ruggeri, Emergent behaviors of thermodynamic Cucker-Smale particles, SIAM J. Math. Anal., 50 (2018), 3092-3121.  doi: 10.1137/17M111064X.  Google Scholar

[27]

S.-Y. HaJ. Kim and T. Ruggeri, From the relativistic mixture of gases to the relativistic cucker-smale flocking, Arch. Rational Mech. Anal., 235 (2020), 1661-1706.  doi: 10.1007/s00205-019-01452-y.  Google Scholar

[28]

S.-Y. HaD. Ko and S. 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

[29]

S.-Y. HaD. Ko and S. 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

[30]

S.-Y. Ha, D. Kim and F. W. Schlöder, Emergent behaviors of Cucker-Smale flocks on Riemannian manifolds, IEEE Trans. Automat. Control, (2020). doi: 10.1109/TAC.2020.3014096.  Google Scholar

[31]

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.  Google Scholar

[32]

S.-Y. Ha and T. Ruggeri, Emergent dynamics of a thermodynamically consistent particle model, Arch. Ration. Mech. Anal, 223 (2017), 1397-1425.  doi: 10.1007/s00205-016-1062-3.  Google Scholar

[33] M. W. HirschS. Smale and R. L. Devaney, Differential Equations, Dynamical systems, and an Introduction to Chaos, Third edition, Elsevier/Academic Press, Amsterdam, 2013.  doi: 10.1016/B978-0-12-382010-5.00001-4.  Google Scholar
[34]

S.-Y. Ha and E. Tadmor, From particle to kinetic and hydrodynamic description of flocking, Kinet. Relat. Models, 1 (2008), 415-435.  doi: 10.3934/krm.2008.1.415.  Google Scholar

[35]

J. J$\ddot{u}$rgen, Riemannian Geometry and Geometric Analysis, Universitext. Springer 2011. doi: 10.1007/978-3-642-21298-7.  Google Scholar

[36]

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

[37]

J. Markdahl, Synchronization on Riemannian manifolds: Multiply connected implies multistable, IEEE Trans. Automat. Control, (2019). doi: 10.1109/TAC.2020.3030849.  Google Scholar

[38]

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

[39]

S. Motsch and E. Tadmor, Heterophilious dynamics enhances consensus, SIAM Review, 56 (2014), 577-621.  doi: 10.1137/120901866.  Google Scholar

[40]

S. Motsch and E. Tadmor, A new model for self-organized dynamics and its flocking behavior, J. Stat. Phys., 144 (2011), 923-947.  doi: 10.1007/s10955-011-0285-9.  Google Scholar

[41]

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.  Google Scholar

[42]

R. Olfati-SaberJ. A. Fax and R. M. Murray, Consensus and cooperation in networked multi-agent systems, Proc. IEEE, 95 (2007), 215-233.  doi: 10.1109/JPROC.2006.887293.  Google Scholar

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

C. W. Reynolds, Flocks, herds, and schools: A distributed behavioral model, Comput. Graph, 21 (1987), 25-34.  doi: 10.1145/280811.281008.  Google Scholar

[45]

L. M. Ritchie, M. A. Lohe and A. G. Williams, Synchronization of relativistic particles in the hyperbolic Kuramoto model, Chaos, 28 (2018), 053116, 11 pp. doi: 10.1063/1.5021701.  Google Scholar

[46]

A. SarletteS. Bonnabel and R. Sepulchre, Coordinated motion design on Lie groups, IEEE Trans. Automat. Control, 55 (2010), 1047-1058.  doi: 10.1109/TAC.2010.2042003.  Google Scholar

[47]

A. Sarlette and R. Sepulchre, Consensus optimization on manifolds, SIAM J. Control Optim., 48 (2009), 56-76.  doi: 10.1137/060673400.  Google Scholar

[48]

R. TronB. Afsari and R. Vidal, Riemannian consensus for manifolds with bounded curvature, IEEE Trans. Automat. Contr., 58 (2013), 921-934.  doi: 10.1109/TAC.2012.2225533.  Google Scholar

[49]

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Figure 1.  Configurations of TCS particles at time $ t = 0 $ and $ t = 200 $
Figure 2.  Temporal evolutions of $ T_i $'s and $ \ln \mathcal{E} $
Figure 3.  TCS particles at $ t = 0 $ and $ t\geq 50 $
Figure 4.  Exponential convergence of $ \mathcal{E} $ to zero
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