doi: 10.3934/dcdsb.2021131

Aggregation and disaggregation of active particles on the unit sphere with time-dependent frequencies

1. 

School of Mathematics, Statistics and Data Science, Sungshin Women's University, Seoul 02844, Republic of Korea

2. 

Department of Mathematics, Research Institute for Natural Sciences, Hanyang University, Seoul 04763, Republic of Korea

* Corresponding author: Jeongho Kim

Received  September 2020 Revised  March 2021 Published  April 2021

Fund Project: The work of J. Kim was supported by the Basic Science Research Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Science and ICT (NRF-2020R1A4A3079066)

We introduce an active swarming model on the sphere which contains additional temporal dynamics for the natural frequency, inspired from the recently introduced modified Kuramoto model, where the natural frequency has its own dynamics. For the attractive interacting particle system, we provide a sufficient framework that leads to the asymptotic aggregation, i.e., all the particles are aggregated to the single point and the natural frequencies also tend to a common value. On the other hand, for the repulsive interacting particle system, we present a sufficient condition for the disaggregation, i.e., the order parameter of the system decays to 0, which implies that the particles are uniformly distributed over the sphere asymptotically. Finally, we also provide several numerical simulation results that support the theoretical results of the paper.

Citation: Dohyun Kim, Jeongho Kim. Aggregation and disaggregation of active particles on the unit sphere with time-dependent frequencies. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021131
References:
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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.  Google Scholar

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[32]

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J. Markdahl and J. Gonçalves, Global convergence properties of a consensus protocol on the $n$-sphere, 2016 55th IEEE Conference on Decision and Control (CDC), (2016), pp. 2487–2492. doi: 10.1109/CDC.2016.7798792.  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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J. Markdahl, D. Proverbio and J. Gonçalves, Robust synchronization of heterogeneous robot swarms on the sphere, 2020 59th IEEE Conference on Decision and Control (CDC), (2020), pp. 5798–5803. doi: 10.1109/CDC42340.2020.9304268.  Google Scholar

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

[39]

R. Olfati-Saber, Flocking for multi-agent dynamic systems: Algorithms and theory, IEEE Trans. Automat. Control, 51 (2006), 401-420.  doi: 10.1109/TAC.2005.864190.  Google Scholar

[40]

L. PereaG. Gomez and P. Elosegui, Extension of the Cucker-Smale control law to space flight formations, J. Guid. Control, 32 (2009), 527-537.  doi: 10.2514/1.36269.  Google Scholar

[41]

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

[42]

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

[43]

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.  Google Scholar

[44]

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.  Google Scholar

[45]

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

[46]

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

show all references

References:
[1]

A. AydoǧS. 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.  Google Scholar

[2]

I. Barbǎlat, Syst$\grave{e}$mes d'$\acute{e}$quations diff$\acute{e}$rentielles d'oscillations non lin$\acute{e}$aires, Rev. Math. Pures Appl., 4 (1959), 267–270.  Google Scholar

[3]

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

[4]

Z. Cai and R. Li, Numerical regularized moment method of arbitrary order for Boltzmann-BGK equation, SIAM J. Sci. Comput., 32 (2010), 2875-2907.  doi: 10.1137/100785466.  Google Scholar

[5]

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.  Google Scholar

[6]

J. A. CarrilloY.-P. ChoiC. Totzeck and O. Tse, An analytical framework for consensus-based global optimization method, Math. Models Methods Appl. Sci., 28 (2018), 1037-1066.  doi: 10.1142/S0218202518500276.  Google Scholar

[7]

J. A. Carrillo, S. Jin, L. Li and Y. Zhu, A consensus-based global optimization method for high dimensional machine learning problems, ESAIM Control Optim. Calc. Var., 27 (2021), Paper No. S5, 22 pp. doi: 10.1051/cocv/2020046.  Google Scholar

[8]

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. doi: 10.1103/PhysRevX.9.011002.  Google Scholar

[9]

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. doi: 10.1063/1.5084965.  Google Scholar

[10]

C. ChenS. LiuX.-q. ShiH. Chaté and Y. Wu, Weak synchronization and large-scale collective oscillation in dense bacterial suspensions, Nature, 542 (2017), 210-214.  doi: 10.1038/nature20817.  Google Scholar

[11]

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. doi: 10.1063/1.4878117.  Google Scholar

[12]

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.  Google Scholar

[13]

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

[14]

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

[15]

D. Cumin and C. P. Unsworth, Generalising the Kuramoto model for the study of neuronal synchronisation in the brain, Phys. D, 226 (2007), 181-196.  doi: 10.1016/j.physd.2006.12.004.  Google Scholar

[16]

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

[17]

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

[18]

P. DegondJ.-G. LiuS. Motsch and V. Panferov, Hydrodynamic models of self-organized dynamics: Derivation and existence theory, Methods Appl. Anal., 20 (2013), 89-114.  doi: 10.4310/MAA.2013.v20.n2.a1.  Google Scholar

[19]

P. Degond and S. Motsch, Continuum limit of self-driven particles with orientation interaction, Math. Models Methods Appl. Sci., 18 (2008), 1193-1215.  doi: 10.1142/S0218202508003005.  Google Scholar

[20]

J. DuanY. Kuang and H. Tang, Model reduction of a two-dimensional kinetic swarming model by operator projections, East Asian J. Appl. Math., 8 (2018), 151-180.  doi: 10.4208/eajam.170617.141117a.  Google Scholar

[21]

Y. FanJ. KoellermeinerJ. LiR. Li and M. Torrilhon, Model reduction of kinetic equations by operator projection, J. Stat. Phys., 162 (2016), 457-486.  doi: 10.1007/s10955-015-1384-9.  Google Scholar

[22]

A. Frouvelle and J.-G. Liu, Long-time dynamics for a simple aggregation equation on the sphere, Springer Proc. Math. Stat., 282 Springer, Cham, 2019,457–479. doi: 10.1007/978-3-030-15096-9_16.  Google Scholar

[23]

I. M. Gamba and M.-J. Kang, Global weak solutions for Kolmogorov-Vicsek type equations with orientational interactions, Arch. Rational Mech. Anal., 222, (2016), 317-–342. doi: 10.1007/s00205-016-1002-2.  Google Scholar

[24]

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.  Google Scholar

[25]

S.-Y. HaS. Jin and D. Kim, Convergence of a first-order consensus-based global optimization algorithm, Math. Models Methods Appl. Sci., 30 (2020), 2417-2444.  doi: 10.1142/S0218202520500463.  Google Scholar

[26]

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.  Google Scholar

[27]

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

[28]

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.  Google Scholar

[29]

D. Kim and J. Kim, Stochastic Lohe matrix model on the Lie group and mean-field limit, J. Stat. Phys., 178 (2020), 1467-1514.  doi: 10.1007/s10955-020-02516-0.  Google Scholar

[30]

J. Koellermeier and M. Torrilhon, Numerical study of partially conservative moment equations in kinetic theory, Commun. Comput. Phys., 21 (2017), 981-1011.  doi: 10.4208/cicp.OA-2016-0053.  Google Scholar

[31]

Y. Kuramoto, Self-entrainment of a population of coupled non-linear oscillators, International Symposium on Mathematical Problems in Mathematical Physics., Lecture Notes in Theoretical Physics 39 1975,420–422. doi: 10.1007/BFb0013365.  Google Scholar

[32]

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

[33]

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

[34]

M. A. Lohe, On the double sphere model of synchronization, Phys. D, 412 (2020), 132642, 13 pp. doi: 10.1016/j.physd.2020.132642.  Google Scholar

[35]

J. Markdahl and J. Gonçalves, Global convergence properties of a consensus protocol on the $n$-sphere, 2016 55th IEEE Conference on Decision and Control (CDC), (2016), pp. 2487–2492. doi: 10.1109/CDC.2016.7798792.  Google Scholar

[36]

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

[37]

J. Markdahl, D. Proverbio and J. Gonçalves, Robust synchronization of heterogeneous robot swarms on the sphere, 2020 59th IEEE Conference on Decision and Control (CDC), (2020), pp. 5798–5803. doi: 10.1109/CDC42340.2020.9304268.  Google Scholar

[38]

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

[39]

R. Olfati-Saber, Flocking for multi-agent dynamic systems: Algorithms and theory, IEEE Trans. Automat. Control, 51 (2006), 401-420.  doi: 10.1109/TAC.2005.864190.  Google Scholar

[40]

L. PereaG. Gomez and P. Elosegui, Extension of the Cucker-Smale control law to space flight formations, J. Guid. Control, 32 (2009), 527-537.  doi: 10.2514/1.36269.  Google Scholar

[41]

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

[42]

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

[43]

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.  Google Scholar

[44]

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.  Google Scholar

[45]

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

[46]

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

Figure 1.  Set of initial data confined in a quadrant of the unit sphere
Figure 2.  Particle trajectories for the attractive case ($ \kappa = 1 $): Simulation for Case 1 (Left), Case 2 (Middle) and Case 3 (Right). The blue marks on the sphere denote the initial positions of particles, while the red marks illustrate the positions of particles at the terminal time $ t = 50 $. The trajectory for Case 3 with $ \gamma = 1 $ blows up in finite time
Figure 3.  The dynamics of the diameter of $ \Omega $ and the order parameters for the attractive case ($ \kappa = 1 $) with Case 1 (Left), Case 2 (Middle) and Case 3 (Right). The graphs for Case 3 with $ \gamma = 1 $ again blows up
Figure 4.  Trajectory of the $ x_1(t) $ for the attractive case ($ \kappa = 1 $). Trajectory of the Case 1 (Left), trajectory for Case 2 (Middle) and trajectory for Case 3 (Right)
Figure 5.  Numerical simulations for the repulsive case ($ \kappa = -1 $): Simulation for Case 1 (Left), Case 2 (Middle) and Case 3 (Right). The blue marks on sphere denote the initial positions of particles, while the red marks illustrate the positions of particles at the terminal time $ t = 50 $
Figure 6.  The dynamics of the diameter of $ \Omega $ and the order parameters for the repulsive case ($ \kappa = -1 $) with Case 1 (Left), Case 2 (Middle) and Case 3 (Right). The graphs for Case 3 with $ \gamma = 1 $ again blows up
Figure 7.  Trajectory of the $ x_1(t) $ for the repulsive case ($ \kappa = -1 $). Trajectory of the Case 1 (Left), trajectory for Case 2 (Middle) and trajectory for case 3 (Right)
Table 1.  Choices for $ (C,\gamma_0) $ for each case
$ C $ $ \gamma_0 $
$ (\mathcal{C}_1) $ $ D({\bf\Omega}^0) $ $ \mu\Gamma_ {\rm{Lip}} $
$ (\mathcal{C}_2) $ $ D({\bf\Omega}^0)\exp\left( \frac{2\mu \|\Omega_c^0\|_ { {\rm{F}}}}{\gamma} \right) $ 0
$ (\mathcal{C}_3) $ $ D({\bf\Omega}^0) $ $ \max\left\{\frac{\mu \|\Psi\|_ { {\rm{F}}} (f(O,O)+\omega_2( D( {\bf\Omega}^0 )) )}{\min_{1\le i\le N}\;\;\|\Omega_i^0\|_ { {\rm{F}}} },2\mu\omega_1(R)\|\Psi\|_ { {\rm{F}}} \right\} $
$ C $ $ \gamma_0 $
$ (\mathcal{C}_1) $ $ D({\bf\Omega}^0) $ $ \mu\Gamma_ {\rm{Lip}} $
$ (\mathcal{C}_2) $ $ D({\bf\Omega}^0)\exp\left( \frac{2\mu \|\Omega_c^0\|_ { {\rm{F}}}}{\gamma} \right) $ 0
$ (\mathcal{C}_3) $ $ D({\bf\Omega}^0) $ $ \max\left\{\frac{\mu \|\Psi\|_ { {\rm{F}}} (f(O,O)+\omega_2( D( {\bf\Omega}^0 )) )}{\min_{1\le i\le N}\;\;\|\Omega_i^0\|_ { {\rm{F}}} },2\mu\omega_1(R)\|\Psi\|_ { {\rm{F}}} \right\} $
[1]

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

[2]

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

[3]

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