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

Dohyun Kim; Jeongho Kim

doi:10.3934/dcdsb.2021131

Article Contents

2022, Volume 27, Issue 4: 2247-2273. Doi: 10.3934/dcdsb.2021131

This issue Previous Article On existence and numerical approximation in phase-lag thermoelasticity with two temperatures Next Article Dynamics of a depletion-type Gierer-Meinhardt model with Langmuir-Hinshelwood reaction scheme

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

Dohyun Kim^1, and
Jeongho Kim^2, ,

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
^* Corresponding author: Jeongho Kim

Received: August 31, 2020

Revised: February 28, 2021

Early access: April 2021

Published: April 2022

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)

Abstract Full Text(HTML) Figure(7) / Table(1) Related Papers Cited by

Abstract

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.

Keywords:

Mathematics Subject Classification: Primary: 34C15, 34D06; Secondary: 34C40.

Citation:

Full Text(HTML)

Figure 1. Set of initial data confined in a quadrant of the unit sphere

Download: Full-size image PowerPoint slide

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

Download: Full-size image PowerPoint slide

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

Download: Full-size image PowerPoint slide

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)

Download: Full-size image PowerPoint slide

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 $

Download: Full-size image PowerPoint slide

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

Download: Full-size image PowerPoint slide

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)

Download: Full-size image PowerPoint slide

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\} $

| Show Table

DownLoad: CSV

Related Papers

Cited by

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, 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.
[3]	J. Buck and E. Buck, Biology of synchronous flashing of fireflies, Nature, 211 (1966), 562-564. doi: 10.1038/211562a0.
[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.
[5]	M. Caponigro, A. 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.
[6]	J. A. Carrillo, Y.-P. Choi, C. 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.
[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.
[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.
[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.
[10]	C. Chen, S. Liu, X.-q. Shi, H. Chaté and Y. Wu, Weak synchronization and large-scale collective oscillation in dense bacterial suspensions, Nature, 542 (2017), 210-214. doi: 10.1038/nature20817.
[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.
[12]	J. Cho, S.-Y. Ha, F. Huang, C. 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.
[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.
[14]	F. Cucker and S. Smale, Emergent behavior in flocks, IEEE Trans. Automat. Control, 52 (2007), 852-862. doi: 10.1109/TAC.2007.895842.
[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.
[16]	T. Danino, O. Mondragon-Palomino, L. Tsimring and J. Hasty, A synchronized quorum of genetic clocks, Nature, 463 (2010), 326-330. doi: 10.1038/nature08753.
[17]	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.
[18]	P. Degond, J.-G. Liu, S. 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.
[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.
[20]	J. Duan, Y. 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.
[21]	Y. Fan, J. Koellermeiner, J. Li, R. 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.
[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.
[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.
[24]	T. Gregor, K. Fujimoto, N. Masaki and S. Sawai, The onset of collective behavior in social amoebae, Science, 328 (2010), 1021-1025. doi: 10.1126/science.1183415.
[25]	S.-Y. Ha, S. 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.
[26]	S.-Y. Ha, D. Kim, J. 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.
[27]	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.
[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.
[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.
[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.
[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.
[32]	M. A. Lohe, Non-Abelian Kuramoto model and synchronization, J. Phys. A, 42 (2009), 395101. doi: 10.1088/1751-8113/42/39/395101.
[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.
[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.
[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.
[36]	J. Markdahl, J. 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.
[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.
[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.
[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.
[40]	L. Perea, G. 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.
[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.
[42]	M. Rubenstein, A. Cornejo and R. Nagapal, Programmable self-assembly in a thousand-robot swarm, Science, 345 (2014), 795-799. doi: 10.1126/science.1254295.
[43]	T. Vicsek, A. Czirók, E. Ben-Jacob, I. 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.
[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.
[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.
[46]	J. Zhu, J. 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.