April  2020, 13(2): 401-434. doi: 10.3934/krm.2020014

Asymptotic behavior of a second-order swarm sphere model and its kinetic limit

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

Received  May 2019 Revised  September 2019 Published  January 2020

Fund Project: The work of D. Kim was supported by National Institute for Mathematical Sciences (NIMS) grant funded by the Korea government (MSIT) (No.B19610000).

We study the asymptotic behavior of a second-order swarm model on the unit sphere in both particle and kinetic regimes for the identical cases. For the emergent behaviors of the particle model, we show that a solution to the particle system with identical oscillators always converge to the equilibrium by employing the gradient-like flow approach. Moreover, we establish the uniform-in-time $ \ell_2 $-stability using the complete aggregation estimate. By applying such uniform stability result, we can perform a rigorous mean-field limit, which is valid for all time, to derive the Vlasov-type kinetic equation on the phase space. For the proposed kinetic equation, we present the global existence of measure-valued solutions and emergent behaviors.

Citation: Dohyun Kim. Asymptotic behavior of a second-order swarm sphere model and its kinetic limit. Kinetic and Related Models, 2020, 13 (2) : 401-434. doi: 10.3934/krm.2020014
References:
[1]

A. Aydoǧ, S. T. McQuade and N. P. Duteil, Opinion dynamics on a general compact Riemannian manifold, Netw. Heterog. Media, 12 (2017), 489–523. doi: 10.3934/nhm.2017021.

[2]

I. Barbǎlat, Systèmes d'équations différentielles d'oscillations non linéaires, Rev. Math. Pures Appl, 4 (1959), 267-270. 

[3]

A. BricardJ.-B. CaussinN. DesreumauxO. Dauchot and D. Bartolo, Emergence of macroscopic directed motion in populations of motile colloids, Nature, 503 (2013), 95-98.  doi: 10.1038/nature12673.

[4]

M. CaponigroA. C. Lai and B. Piccoli, A nonlinear model of opinion formation on the sphere, Discrete Contin. Dyn. Syst, 35 (2015), 4241-4268.  doi: 10.3934/dcds.2015.35.4241.

[5]

S. Chandra, M. Girvan and E. Ott, Continuous versus discontinuous transitions in the D-dimensional generalized Kuramoto model: Odd D is different, Phys. Rev. X, 9 (2019), 011002.

[6]

S. Chandra and E. Ott, Observing microscopic transitions from macroscopic bursts: Instability-mediated resetting in the incoherent regime of the D-dimensional generalized Kuramoto model, Chaos, 29 (2019), 033124, 13pp. doi: 10.1063/1.5084965.

[7]

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, 18pp. doi: 10.1063/1.4878117.

[8]

J. ChoS.-Y. HaF. HuangC. Jin and D. Ko, Emergence of bi-cluster flocking for the Cucker-Smale model, Math. Models Methods Appl. Sci, 26 (2016), 1191-1218.  doi: 10.1142/S0218202516500287.

[9]

Y.-P. ChoiS.-Y. Ha and J. Morales, Emergent dynamics of the Kuramoto model under the effect of inertia, Discrete Contin. Dyn. Syst, 38 (2018), 4875-4913.  doi: 10.3934/dcds.2018213.

[10]

Y.-P. ChoiS.-Y. Ha and S.-B. Yun, Complete synchronization of Kuramoto oscillators with finite inertia, Phys. D, 240 (2011), 32-44.  doi: 10.1016/j.physd.2010.08.004.

[11]

Y.-P. Choi and Z. Li, Synchronization of nonuniform Kuramoto oscillators for power grids with general connectivity and dampings, Nonlinearity, 32 (2019), 559-583.  doi: 10.1088/1361-6544/aaec94.

[12]

Y.-P. ChoiZ. LiS.-Y. HaX. Xue and S.-B. Yun, Complete entrainment of Kuramoto oscillators with inertia on networks via gradient-like flow, J. Differential Equations, 257 (2014), 2591-2621.  doi: 10.1016/j.jde.2014.05.054.

[13]

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

[14]

T. DaninoO. Mondragon-PalominoL. Tsimring and J. Hasty, A synchronized quorum of genetic clocks, Nature, 463 (2010), 326-330.  doi: 10.1038/nature08753.

[15]

G. B. Ermentrout, An adaptive model for synchrony in the firefly Pteroptyx malaccae, J. Math. Biol, 29 (1991), 571-585.  doi: 10.1007/BF00164052.

[16]

A. Frouvelle and J.-G. Liu, Long-time dynamics for a simple aggregation equation on the sphere, in Stochastic Dynamics Out of Equilibrium. IHPStochDyn 2017, (eds. G. Giacomin, S. Olla, E. Saada, H. Spohn, G. Stoltz), Springer Proceedings in Mathematica & Statistic, Springer, Cham, 282 (2019), 457–479.

[17]

T. GregorK. FujimotoN. Masaki and S. Sawai, The onset of collective behavior in social amoebae, Science, 328 (2010), 1021-1025.  doi: 10.1126/science.1183415.

[18]

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.

[19]

S.-Y. HaD. KimJ. Lee and S. E. No, 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.

[20]

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

[21]

S.-Y. HaD. Ko and S. Ryoo, On the relaxation dynamics of Lohe oscillators on the Riemannian manifold, J. Stat. Phys, 172 (2018), 1427-1478.  doi: 10.1007/s10955-018-2091-0.

[22]

S.-Y. Ha and J.-G. Liu, A simple proof of the Cucker-Smale flocking dynamics and mean-field limit, Commu. Math. Sci, 7 (2007), 297-325.  doi: 10.4310/CMS.2009.v7.n2.a2.

[23]

A. Haraux and M. A. Jendoubi, Convergence of solutions of second-order gradient-like systems with analytic nonlinearities, J. Differential Equations, 144 (1998), 313-320.  doi: 10.1006/jdeq.1997.3393.

[24]

S. M. Hung and S. N. Givigi, A q-learning approach to flocking with UAVs in a stochastic environment, IEEE Trans. Cybern, 47 (2017), 186-197.  doi: 10.1109/TCYB.2015.2509646.

[25]

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

[26]

M. A. Lohe, Non-Abelian Kuramoto model and synchronization, J. Phys. A, 42 (2009), 395101, 25pp. doi: 10.1088/1751-8113/42/39/395101.

[27]

M. A. Lohe, High-dimensional generalizations of the Watanabe-Strogatz transform for vector models of synchronization, J. Phys. A, 51 (2018), 225101, 24pp. doi: 10.1088/1751-8121/aac030.

[28]

M. C. MarchettiJ. F. JoannyS. RamaswamyT. B. LiverpoolJ. ProstM. Rao and R. Aditi Simha, Hydrodynamics of soft active matter, Rev. Mod. Phys, 85 (2013), 1143-1189. 

[29]

J. Markdahl and J. Gonçalves, Global convergence properties of a consensus protocol on the n-sphere, Proc. of the 55th IEEE conference on Decision and Control, (2016), 2487–2492.

[30]

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

[31]

F. J. NédélecT. SurryA. C. Maggas and S. Leibler, Self-organization of microtubules and motors, Nature, 389 (1997), 305-308. 

[32]

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.

[33]

L. PereaP. Elosegui and G. Gomez, Extension of the Cucker-Smale control law to space flight formations, J. Guid. Control, 32 (2009), 526-537.  doi: 10.2514/1.36269.

[34]

M. RubensteinA. Cornejo and R. Nagapal, Programmable self-assembly in a thousand-robot swarm, Science, 345 (2014), 795-799.  doi: 10.1126/science.1254295.

[35]

R. Sknepnek and S. Henkes, Active swarms on a sphere, Phys. Rev. E, 91 (2015), 022306. doi: 10.1103/PhysRevE.91.022306.

[36]

Y. Sun, W. Li and D. Zhao, Realization of consensus of multi-agent systems with stochastically mixed interactions, Chaos, 26 (2016), 073112, 8pp. doi: 10.1063/1.4958927.

[37]

T. VicsekA. CzirókE. Ben-JacobI. Cohen and O. Shochet, Novel type of phase transition in a system of self-driven particles, Phys. Rev. Lett, 75 (1995), 1226-1229.  doi: 10.1103/PhysRevLett.75.1226.

[38]

A. T. Winfree, Biological rhythms and the behavior of populations of coupled oscillators, J. Theor. Biol, 16 (1967), 15-42.  doi: 10.1016/0022-5193(67)90051-3.

[39]

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.

[40]

J. ZhuJ. Zhu and C. Qian, On equilibria and consensus of the Lohe model with identical oscillators, SIAM J. Appl. Dyns. Syst, 17 (2018), 1716-1741.  doi: 10.1137/17M112765X.

show all references

References:
[1]

A. Aydoǧ, S. T. McQuade and N. P. Duteil, Opinion dynamics on a general compact Riemannian manifold, Netw. Heterog. Media, 12 (2017), 489–523. doi: 10.3934/nhm.2017021.

[2]

I. Barbǎlat, Systèmes d'équations différentielles d'oscillations non linéaires, Rev. Math. Pures Appl, 4 (1959), 267-270. 

[3]

A. BricardJ.-B. CaussinN. DesreumauxO. Dauchot and D. Bartolo, Emergence of macroscopic directed motion in populations of motile colloids, Nature, 503 (2013), 95-98.  doi: 10.1038/nature12673.

[4]

M. CaponigroA. C. Lai and B. Piccoli, A nonlinear model of opinion formation on the sphere, Discrete Contin. Dyn. Syst, 35 (2015), 4241-4268.  doi: 10.3934/dcds.2015.35.4241.

[5]

S. Chandra, M. Girvan and E. Ott, Continuous versus discontinuous transitions in the D-dimensional generalized Kuramoto model: Odd D is different, Phys. Rev. X, 9 (2019), 011002.

[6]

S. Chandra and E. Ott, Observing microscopic transitions from macroscopic bursts: Instability-mediated resetting in the incoherent regime of the D-dimensional generalized Kuramoto model, Chaos, 29 (2019), 033124, 13pp. doi: 10.1063/1.5084965.

[7]

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, 18pp. doi: 10.1063/1.4878117.

[8]

J. ChoS.-Y. HaF. HuangC. Jin and D. Ko, Emergence of bi-cluster flocking for the Cucker-Smale model, Math. Models Methods Appl. Sci, 26 (2016), 1191-1218.  doi: 10.1142/S0218202516500287.

[9]

Y.-P. ChoiS.-Y. Ha and J. Morales, Emergent dynamics of the Kuramoto model under the effect of inertia, Discrete Contin. Dyn. Syst, 38 (2018), 4875-4913.  doi: 10.3934/dcds.2018213.

[10]

Y.-P. ChoiS.-Y. Ha and S.-B. Yun, Complete synchronization of Kuramoto oscillators with finite inertia, Phys. D, 240 (2011), 32-44.  doi: 10.1016/j.physd.2010.08.004.

[11]

Y.-P. Choi and Z. Li, Synchronization of nonuniform Kuramoto oscillators for power grids with general connectivity and dampings, Nonlinearity, 32 (2019), 559-583.  doi: 10.1088/1361-6544/aaec94.

[12]

Y.-P. ChoiZ. LiS.-Y. HaX. Xue and S.-B. Yun, Complete entrainment of Kuramoto oscillators with inertia on networks via gradient-like flow, J. Differential Equations, 257 (2014), 2591-2621.  doi: 10.1016/j.jde.2014.05.054.

[13]

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

[14]

T. DaninoO. Mondragon-PalominoL. Tsimring and J. Hasty, A synchronized quorum of genetic clocks, Nature, 463 (2010), 326-330.  doi: 10.1038/nature08753.

[15]

G. B. Ermentrout, An adaptive model for synchrony in the firefly Pteroptyx malaccae, J. Math. Biol, 29 (1991), 571-585.  doi: 10.1007/BF00164052.

[16]

A. Frouvelle and J.-G. Liu, Long-time dynamics for a simple aggregation equation on the sphere, in Stochastic Dynamics Out of Equilibrium. IHPStochDyn 2017, (eds. G. Giacomin, S. Olla, E. Saada, H. Spohn, G. Stoltz), Springer Proceedings in Mathematica & Statistic, Springer, Cham, 282 (2019), 457–479.

[17]

T. GregorK. FujimotoN. Masaki and S. Sawai, The onset of collective behavior in social amoebae, Science, 328 (2010), 1021-1025.  doi: 10.1126/science.1183415.

[18]

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.

[19]

S.-Y. HaD. KimJ. Lee and S. E. No, 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.

[20]

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

[21]

S.-Y. HaD. Ko and S. Ryoo, On the relaxation dynamics of Lohe oscillators on the Riemannian manifold, J. Stat. Phys, 172 (2018), 1427-1478.  doi: 10.1007/s10955-018-2091-0.

[22]

S.-Y. Ha and J.-G. Liu, A simple proof of the Cucker-Smale flocking dynamics and mean-field limit, Commu. Math. Sci, 7 (2007), 297-325.  doi: 10.4310/CMS.2009.v7.n2.a2.

[23]

A. Haraux and M. A. Jendoubi, Convergence of solutions of second-order gradient-like systems with analytic nonlinearities, J. Differential Equations, 144 (1998), 313-320.  doi: 10.1006/jdeq.1997.3393.

[24]

S. M. Hung and S. N. Givigi, A q-learning approach to flocking with UAVs in a stochastic environment, IEEE Trans. Cybern, 47 (2017), 186-197.  doi: 10.1109/TCYB.2015.2509646.

[25]

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

[26]

M. A. Lohe, Non-Abelian Kuramoto model and synchronization, J. Phys. A, 42 (2009), 395101, 25pp. doi: 10.1088/1751-8113/42/39/395101.

[27]

M. A. Lohe, High-dimensional generalizations of the Watanabe-Strogatz transform for vector models of synchronization, J. Phys. A, 51 (2018), 225101, 24pp. doi: 10.1088/1751-8121/aac030.

[28]

M. C. MarchettiJ. F. JoannyS. RamaswamyT. B. LiverpoolJ. ProstM. Rao and R. Aditi Simha, Hydrodynamics of soft active matter, Rev. Mod. Phys, 85 (2013), 1143-1189. 

[29]

J. Markdahl and J. Gonçalves, Global convergence properties of a consensus protocol on the n-sphere, Proc. of the 55th IEEE conference on Decision and Control, (2016), 2487–2492.

[30]

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

[31]

F. J. NédélecT. SurryA. C. Maggas and S. Leibler, Self-organization of microtubules and motors, Nature, 389 (1997), 305-308. 

[32]

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.

[33]

L. PereaP. Elosegui and G. Gomez, Extension of the Cucker-Smale control law to space flight formations, J. Guid. Control, 32 (2009), 526-537.  doi: 10.2514/1.36269.

[34]

M. RubensteinA. Cornejo and R. Nagapal, Programmable self-assembly in a thousand-robot swarm, Science, 345 (2014), 795-799.  doi: 10.1126/science.1254295.

[35]

R. Sknepnek and S. Henkes, Active swarms on a sphere, Phys. Rev. E, 91 (2015), 022306. doi: 10.1103/PhysRevE.91.022306.

[36]

Y. Sun, W. Li and D. Zhao, Realization of consensus of multi-agent systems with stochastically mixed interactions, Chaos, 26 (2016), 073112, 8pp. doi: 10.1063/1.4958927.

[37]

T. VicsekA. CzirókE. Ben-JacobI. Cohen and O. Shochet, Novel type of phase transition in a system of self-driven particles, Phys. Rev. Lett, 75 (1995), 1226-1229.  doi: 10.1103/PhysRevLett.75.1226.

[38]

A. T. Winfree, Biological rhythms and the behavior of populations of coupled oscillators, J. Theor. Biol, 16 (1967), 15-42.  doi: 10.1016/0022-5193(67)90051-3.

[39]

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.

[40]

J. ZhuJ. Zhu and C. Qian, On equilibria and consensus of the Lohe model with identical oscillators, SIAM J. Appl. Dyns. Syst, 17 (2018), 1716-1741.  doi: 10.1137/17M112765X.

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