Emergent behaviors in group ring flocks
Authors:
Seung-Yeal Ha and Hansol Park
Journal:
Quart. Appl. Math. 79 (2021), 617-640
MSC (2020):
Primary 70G60, 34D06, 70F10
DOI:
https://doi.org/10.1090/qam/1595
Published electronically:
May 17, 2021
MathSciNet review:
4328141
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Additional Information
Abstract: In this paper, we present a first-order aggregation model on a group ring and investigate its asymptotic dynamics. In a positive coupling strength regime, we show that the flow generated by the proposed model approaches to an equilibrium manifold asymptotically using a nonlinear functional and LaSalle invariance principle. Moreover, we also verify that equilibrium manifold’s structure is strongly dependent on the structure of an underlying group structure.
References
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- Sun-Ho Choi and Seung-Yeal Ha, Quantum synchronization of the Schrödinger-Lohe model, J. Phys. A 47 (2014), no. 35, 355104, 16. MR 3254872, DOI 10.1088/1751-8113/47/35/355104
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- Seung-Yeal Ha, Dongnam Ko, and Sang Woo Ryoo, Emergent dynamics of a generalized Lohe model on some class of Lie groups, J. Stat. Phys. 168 (2017), no. 1, 171–207. MR 3659983, DOI 10.1007/s10955-017-1797-8
- Seung-Yeal Ha, Dongnam Ko, Jinyeong Park, and Xiongtao Zhang, Collective synchronization of classical and quantum oscillators, EMS Surv. Math. Sci. 3 (2016), no. 2, 209–267. MR 3576533, DOI 10.4171/EMSS/17
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- S.-Y. Ha and H. Park, Emergent behaviors of the generalized Lohe matrix model. To appear in Discrete and Continuous Dynamical Systems Series B.
- Seung-Yeal Ha and Hansol Park, Complete aggregation of the Lohe tensor model with the same free flow, J. Math. Phys. 61 (2020), no. 10, 102702, 27. MR 4160274, DOI 10.1063/5.0007292
- Seung-Yeal Ha and Hansol Park, Emergent behaviors of Lohe tensor flocks, J. Stat. Phys. 178 (2020), no. 5, 1268–1292. MR 4081228, DOI 10.1007/s10955-020-02505-3
- Seung-Yeal Ha and Sang Woo Ryoo, On the emergence and orbital stability of phase-locked states for the Lohe model, J. Stat. Phys. 163 (2016), no. 2, 411–439. MR 3478317, DOI 10.1007/s10955-016-1481-4
- Seung-Yeal Ha and Marshall Slemrod, A fast-slow dynamical systems theory for the Kuramoto type phase model, J. Differential Equations 251 (2011), no. 10, 2685–2695. MR 2831709, DOI 10.1016/j.jde.2011.04.004
- Seung-Yeal Ha and Marshall Slemrod, Flocking dynamics of singularly perturbed oscillator chain and the Cucker-Smale system, J. Dynam. Differential Equations 22 (2010), no. 2, 325–330. MR 2665438, DOI 10.1007/s10884-009-9142-9
- Vladimir Jaćimović and Aladin Crnkić, Low-dimensional dynamics in non-Abelian Kuramoto model on the 3-sphere, Chaos 28 (2018), no. 8, 083105, 8. MR 3840744, DOI 10.1063/1.5029485
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- J. P. LaSalle, Stability theory for ordinary differential equations, J. Differential Equations 4 (1968), 57–65. MR 222402, DOI 10.1016/0022-0396(68)90048-X
- M. A. Lohe, Quantum synchronization over quantum networks, J. Phys. A 43 (2010), no. 46, 465301, 20. MR 2735225, DOI 10.1088/1751-8113/43/46/465301
- M. A. Lohe, Non-abelian Kuramoto models and synchronization, J. Phys. A 42 (2009), no. 39, 395101, 25. MR 2539317, DOI 10.1088/1751-8113/42/39/395101
- J. Markdahl, Synchronization on Riemannian manifolds: Multiply connected implies multistable. Available in https://arxiv.org/abs/1906.07452
- Johan Markdahl, Johan Thunberg, and Jorge Gonçalves, Almost global consensus on the $n$-sphere, IEEE Trans. Automat. Control 63 (2018), no. 6, 1664–1675. MR 3807655, DOI 10.1109/tac.2017.2752799
- Reza Olfati-Saber, Flocking for multi-agent dynamic systems: algorithms and theory, IEEE Trans. Automat. Control 51 (2006), no. 3, 401–420. MR 2205679, DOI 10.1109/TAC.2005.864190
- D. S. Passman, What is a group ring?, Amer. Math. Monthly 83 (1976), no. 3, 173–185. MR 390033, DOI 10.2307/2977018
- Donald S. Passman, Infinite group rings, Pure and Applied Mathematics, vol. 6, Marcel Dekker, Inc., New York, 1971. MR 0314951
- Charles S. Peskin, Mathematical aspects of heart physiology, Courant Institute of Mathematical Sciences, New York University, New York, 1975. Notes based on a course given at New York University during the year 1973/74. MR 0414135
- Arkady Pikovsky, Michael Rosenblum, and Jürgen Kurths, Synchronization, Cambridge Nonlinear Science Series, vol. 12, Cambridge University Press, Cambridge, 2001. A universal concept in nonlinear sciences. MR 1869044, DOI 10.1017/CBO9780511755743
- Alain Sarlette and Rodolphe Sepulchre, Consensus optimization on manifolds, SIAM J. Control Optim. 48 (2009), no. 1, 56–76. MR 2480126, DOI 10.1137/060673400
- Steven H. Strogatz, From Kuramoto to Crawford: exploring the onset of synchronization in populations of coupled oscillators, Phys. D 143 (2000), no. 1-4, 1–20. Bifurcations, patterns and symmetry. MR 1783382, DOI 10.1016/S0167-2789(00)00094-4
- Johan Thunberg, Johan Markdahl, Florian Bernard, and Jorge Goncalves, A lifting method for analyzing distributed synchronization on the unit sphere, Automatica J. IFAC 96 (2018), 253–258. MR 3844968, DOI 10.1016/j.automatica.2018.07.007
- John Toner and Yuhai Tu, Flocks, herds, and schools: a quantitative theory of flocking, Phys. Rev. E (3) 58 (1998), no. 4, 4828–4858. MR 1651324, DOI 10.1103/PhysRevE.58.4828
- Chad M. Topaz and Andrea L. Bertozzi, Swarming patterns in a two-dimensional kinematic model for biological groups, SIAM J. Appl. Math. 65 (2004), no. 1, 152–174. MR 2111591, DOI 10.1137/S0036139903437424
- Roberto Tron, Bijan Afsari, and René Vidal, Riemannian consensus for manifolds with bounded curvature, IEEE Trans. Automat. Control 58 (2013), no. 4, 921–934. MR 3038794, DOI 10.1109/TAC.2012.2225533
- Tamás Vicsek, András Czirók, Eshel Ben-Jacob, Inon Cohen, and Ofer Shochet, Novel type of phase transition in a system of self-driven particles, Phys. Rev. Lett. 75 (1995), no. 6, 1226–1229. MR 3363421, DOI 10.1103/PhysRevLett.75.1226
- T. Vicsek and A. Zefeiris, Collective motion, Phys. Rep. 517 (2012), 71-140.
- A. T. Winfree, Biological rhythms and the behavior of populations of coupled oscillators, J. Theor. Biol. 16 (1967), 15-42.
- Arthur T. Winfree, The geometry of biological time, Biomathematics, vol. 8, Springer-Verlag, Berlin-New York, 1980. MR 572965
- Jiandong Zhu, Synchronization of Kuramoto model in a high-dimensional linear space, Phys. Lett. A 377 (2013), no. 41, 2939–2943. MR 3108680, DOI 10.1016/j.physleta.2013.09.010
References
- J. A. Acebron, L. L. Bonilla, C. J. P. Pérez Vicente, F. Ritort, and R. Spigler, The Kuramoto model: A simple paradigm for synchronization phenomena. Rev. Mod. Phys. 77 (2005), 137-185.
- G. Albi, N. Bellomo, L. Fermo, S.-Y. Ha, J. Kim, L. Pareschi, D. 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), no. 10, 1901–2005. MR 4014449, DOI 10.1142/S0218202519500374
- Aylin Aydoğdu, Sean T. McQuade, and Nastassia Pouradier Duteil, Opinion dynamics on a general compact Riemannian manifold, Netw. Heterog. Media 12 (2017), no. 3, 489–523. MR 3714980, DOI 10.3934/nhm.2017021
- Dario Benedetto, Emanuele Caglioti, and Umberto Montemagno, On the complete phase synchronization for the Kuramoto model in the mean-field limit, Commun. Math. Sci. 13 (2015), no. 7, 1775–1786. MR 3393174, DOI 10.4310/CMS.2015.v13.n7.a6
- Jared C. Bronski, Thomas E. Carty, and Sarah E. Simpson, A matrix-valued Kuramoto model, J. Stat. Phys. 178 (2020), no. 2, 595–624. MR 4055252, DOI 10.1007/s10955-019-02442-w
- J. Buck and E. Buck, Biology of synchronous flashing of fireflies, Nature 211 (1966), 562.
- Dongpyo Chi, Sun-Ho Choi, and Seung-Yeal Ha, Emergent behaviors of a holonomic particle system on a sphere, J. Math. Phys. 55 (2014), no. 5, 052703, 18. MR 3390625, DOI 10.1063/1.4878117
- Sun-Ho Choi and Seung-Yeal Ha, Quantum synchronization of the Schrödinger-Lohe model, J. Phys. A 47 (2014), no. 35, 355104, 16. MR 3254872, DOI 10.1088/1751-8113/47/35/355104
- Sun-Ho Choi and Seung-Yeal Ha, Complete entrainment of Lohe oscillators under attractive and repulsive couplings, SIAM J. Appl. Dyn. Syst. 13 (2014), no. 4, 1417–1441. MR 3267148, DOI 10.1137/140961699
- Young-Pil Choi, Seung-Yeal Ha, Sungeun Jung, and Yongduck Kim, Asymptotic formation and orbital stability of phase-locked states for the Kuramoto model, Phys. D 241 (2012), no. 7, 735–754. MR 2897541, DOI 10.1016/j.physd.2011.11.011
- Nikhil Chopra and Mark W. Spong, On exponential synchronization of Kuramoto oscillators, IEEE Trans. Automat. Control 54 (2009), no. 2, 353–357. MR 2491964, DOI 10.1109/TAC.2008.2007884
- Felipe Cucker and Steve Smale, Emergent behavior in flocks, IEEE Trans. Automat. Control 52 (2007), no. 5, 852–862. MR 2324245, DOI 10.1109/TAC.2007.895842
- Pierre Degond, Amic Frouvelle, Sara Merino-Aceituno, and Ariane Trescases, Quaternions in collective dynamics, Multiscale Model. Simul. 16 (2018), no. 1, 28–77. MR 3743738, DOI 10.1137/17M1135207
- Pierre Degond, Amic Frouvelle, and Sara Merino-Aceituno, A new flocking model through body attitude coordination, Math. Models Methods Appl. Sci. 27 (2017), no. 6, 1005–1049. MR 3659045, DOI 10.1142/S0218202517400085
- Pierre Degond and Sébastien Motsch, Large scale dynamics of the persistent turning walker model of fish behavior, J. Stat. Phys. 131 (2008), no. 6, 989–1021. MR 2407377, DOI 10.1007/s10955-008-9529-8
- Lee DeVille, Synchronization and stability for quantum Kuramoto, J. Stat. Phys. 174 (2019), no. 1, 160–187. MR 3904514, DOI 10.1007/s10955-018-2168-9
- Jiu-Gang Dong and Xiaoping Xue, Synchronization analysis of Kuramoto oscillators, Commun. Math. Sci. 11 (2013), no. 2, 465–480. MR 3002560, DOI 10.4310/CMS.2013.v11.n2.a7
- Florian Dörfler and Francesco Bullo, Synchronization in complex networks of phase oscillators: a survey, Automatica J. IFAC 50 (2014), no. 6, 1539–1564. MR 3214901, DOI 10.1016/j.automatica.2014.04.012
- E. Ferrante, A. E. Turgut, A. Stranieri, C. Pinciroli, and M. Dorigo, Self-organized flocking with a mobile robot swarm: a novel motion control method, Adapt. Behav. 20 (2012), 460-477.
- R. C. Fetecau, H. Park, and F. S. Patacchini, Well-posedness and asymptotic behaviour of an aggregation model with intrinsic interactions on sphere and other manifolds. To appear in Analysis and Applications.
- Razvan C. Fetecau and Beril Zhang, Self-organization on Riemannian manifolds, J. Geom. Mech. 11 (2019), no. 3, 397–426. MR 4026014, DOI 10.3934/jgm.2019020
- Seung-Yeal Ha, Hwa Kil Kim, and Sang Woo Ryoo, Emergence of phase-locked states for the Kuramoto model in a large coupling regime, Commun. Math. Sci. 14 (2016), no. 4, 1073–1091. MR 3491817, DOI 10.4310/CMS.2016.v14.n4.a10
- Seung-Yeal Ha, Dongnam Ko, and Sang Woo Ryoo, Emergent dynamics of a generalized Lohe model on some class of Lie groups, J. Stat. Phys. 168 (2017), no. 1, 171–207. MR 3659983, DOI 10.1007/s10955-017-1797-8
- Seung-Yeal Ha, Dongnam Ko, Jinyeong Park, and Xiongtao Zhang, Collective synchronization of classical and quantum oscillators, EMS Surv. Math. Sci. 3 (2016), no. 2, 209–267. MR 3576533, DOI 10.4171/EMSS/17
- Seung-Yeal Ha and Hansol Park, On the Schrödinger-Lohe hierarchy for aggregation and its emergent dynamics, J. Stat. Phys. 181 (2020), no. 6, 2150–2190. MR 4179802, DOI 10.1007/s10955-020-02659-0
- S.-Y. Ha and H. Park, Emergent behaviors of the generalized Lohe matrix model. To appear in Discrete and Continuous Dynamical Systems Series B.
- Seung-Yeal Ha and Hansol Park, Complete aggregation of the Lohe tensor model with the same free flow, J. Math. Phys. 61 (2020), no. 10, 102702, 27. MR 4160274, DOI 10.1063/5.0007292
- Seung-Yeal Ha and Hansol Park, Emergent behaviors of Lohe tensor flocks, J. Stat. Phys. 178 (2020), no. 5, 1268–1292. MR 4081228, DOI 10.1007/s10955-020-02505-3
- Seung-Yeal Ha and Sang Woo Ryoo, On the emergence and orbital stability of phase-locked states for the Lohe model, J. Stat. Phys. 163 (2016), no. 2, 411–439. MR 3478317, DOI 10.1007/s10955-016-1481-4
- Seung-Yeal Ha and Marshall Slemrod, A fast-slow dynamical systems theory for the Kuramoto type phase model, J. Differential Equations 251 (2011), no. 10, 2685–2695. MR 2831709, DOI 10.1016/j.jde.2011.04.004
- Seung-Yeal Ha and Marshall Slemrod, Flocking dynamics of singularly perturbed oscillator chain and the Cucker-Smale system, J. Dynam. Differential Equations 22 (2010), no. 2, 325–330. MR 2665438, DOI 10.1007/s10884-009-9142-9
- Vladimir Jaćimović and Aladin Crnkić, Low-dimensional dynamics in non-Abelian Kuramoto model on the 3-sphere, Chaos 28 (2018), no. 8, 083105, 8. MR 3840744, DOI 10.1063/1.5029485
- Y. Kuramoto, International symposium on mathematical problems in mathematical physics, Lecture Notes Theor. Phys. 30 (1975), 420.
- J. P. LaSalle, Stability theory for ordinary differential equations, J. Differential Equations 4 (1968), 57–65. MR 222402, DOI 10.1016/0022-0396(68)90048-X
- M. A. Lohe, Quantum synchronization over quantum networks, J. Phys. A 43 (2010), no. 46, 465301, 20. MR 2735225, DOI 10.1088/1751-8113/43/46/465301
- M. A. Lohe, Non-abelian Kuramoto models and synchronization, J. Phys. A 42 (2009), no. 39, 395101, 25. MR 2539317, DOI 10.1088/1751-8113/42/39/395101
- J. Markdahl, Synchronization on Riemannian manifolds: Multiply connected implies multistable. Available in https://arxiv.org/abs/1906.07452
- Johan Markdahl, Johan Thunberg, and Jorge Gonçalves, Almost global consensus on the $n$-sphere, IEEE Trans. Automat. Control 63 (2018), no. 6, 1664–1675. MR 3807655, DOI 10.1109/tac.2017.2752799
- Reza Olfati-Saber, Flocking for multi-agent dynamic systems: algorithms and theory, IEEE Trans. Automat. Control 51 (2006), no. 3, 401–420. MR 2205679, DOI 10.1109/TAC.2005.864190
- D. S. Passman, What is a group ring?, Amer. Math. Monthly 83 (1976), no. 3, 173–185. MR 390033, DOI 10.2307/2977018
- Donald S. Passman, Infinite group rings, Marcel Dekker, Inc., New York, 1971. Pure and Applied Mathematics, 6. MR 0314951
- Charles S. Peskin, Mathematical aspects of heart physiology, Notes based on a course given at New York University during the year 1973/74, Courant Institute of Mathematical Sciences, New York University, New York, 1975. MR 0414135
- Arkady Pikovsky, Michael Rosenblum, and Jürgen Kurths, Synchronization: A universal concept in nonlinear sciences, Cambridge Nonlinear Science Series, vol. 12, Cambridge University Press, Cambridge, 2001. MR 1869044, DOI 10.1017/CBO9780511755743
- Alain Sarlette and Rodolphe Sepulchre, Consensus optimization on manifolds, SIAM J. Control Optim. 48 (2009), no. 1, 56–76. MR 2480126, DOI 10.1137/060673400
- Steven H. Strogatz, From Kuramoto to Crawford: exploring the onset of synchronization in populations of coupled oscillators: Bifurcations, patterns and symmetry, Phys. D 143 (2000), no. 1-4, 1–20. MR 1783382, DOI 10.1016/S0167-2789(00)00094-4
- Johan Thunberg, Johan Markdahl, Florian Bernard, and Jorge Goncalves, A lifting method for analyzing distributed synchronization on the unit sphere, Automatica J. IFAC 96 (2018), 253–258. MR 3844968, DOI 10.1016/j.automatica.2018.07.007
- John Toner and Yuhai Tu, Flocks, herds, and schools: a quantitative theory of flocking, Phys. Rev. E (3) 58 (1998), no. 4, 4828–4858. MR 1651324, DOI 10.1103/PhysRevE.58.4828
- Chad M. Topaz and Andrea L. Bertozzi, Swarming patterns in a two-dimensional kinematic model for biological groups, SIAM J. Appl. Math. 65 (2004), no. 1, 152–174. MR 2111591, DOI 10.1137/S0036139903437424
- Roberto Tron, Bijan Afsari, and René Vidal, Riemannian consensus for manifolds with bounded curvature, IEEE Trans. Automat. Control 58 (2013), no. 4, 921–934. MR 3038794, DOI 10.1109/TAC.2012.2225533
- Tamás Vicsek, András Czirók, Eshel Ben-Jacob, Inon Cohen, and Ofer Shochet, Novel type of phase transition in a system of self-driven particles, Phys. Rev. Lett. 75 (1995), no. 6, 1226–1229. MR 3363421, DOI 10.1103/PhysRevLett.75.1226
- T. Vicsek and A. Zefeiris, Collective motion, Phys. Rep. 517 (2012), 71-140.
- A. T. Winfree, Biological rhythms and the behavior of populations of coupled oscillators, J. Theor. Biol. 16 (1967), 15-42.
- Arthur T. Winfree, The geometry of biological time, Biomathematics, vol. 8, Springer-Verlag, Berlin-New York, 1980. MR 572965
- Jiandong Zhu, Synchronization of Kuramoto model in a high-dimensional linear space, Phys. Lett. A 377 (2013), no. 41, 2939–2943. MR 3108680, DOI 10.1016/j.physleta.2013.09.010
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Additional Information
Seung-Yeal Ha
Affiliation:
Department of Mathematical Sciences and Research Institute of Mathematics, Seoul National University, Seoul 08826; and Korea Institute for Advanced Study, Hoegiro 85, Seoul 02455, Republic of Korea
MR Author ID:
684438
Email:
syha@snu.ac.kr
Hansol Park
Affiliation:
Department of Mathematical Sciences, Seoul National University, Seoul 08826, Republic of Korea
MR Author ID:
1356829
ORCID:
0000-0002-1075-6472
Email:
hansol960612@snu.ac.kr
Received by editor(s):
March 8, 2021
Received by editor(s) in revised form:
March 28, 2021
Published electronically:
May 17, 2021
Additional Notes:
The work of the first author was supported by National Research Foundation of Korea (NRF-2020R1A2C3A01003881) and the work of the second author was supported by Basic Science Research Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Education (2019R1I1A1A01059585)
Article copyright:
© Copyright 2021
Brown University
