DOI QR코드

DOI QR Code

COLLECTIVE BEHAVIORS OF SECOND-ORDER NONLINEAR CONSENSUS MODELS WITH A BONDING FORCE

  • Hyunjin Ahn (Department of Mathematics Myongji University) ;
  • Junhyeok Byeon (Research Institute of Basic Sciences Seoul National University) ;
  • Seung-Yeal Ha (Department of Mathematical Sciences, and Research Institute of Mathematics Seoul National University) ;
  • Jaeyoung Yoon (Department of Mathematical Sciences Seoul National University)
  • Received : 2023.08.16
  • Accepted : 2024.01.17
  • Published : 2024.05.01

Abstract

We study the collective behaviors of two second-order nonlinear consensus models with a bonding force, namely the Kuramoto model and the Cucker-Smale model with inter-particle bonding force. The proposed models contain feedback control terms which induce collision avoidance and emergent consensus dynamics in a suitable framework. Through the cooperative interplays between feedback controls, initial state configuration tends to an ordered configuration asymptotically under suitable frameworks which are formulated in terms of system parameters and initial configurations. For a two-particle system on the real line, we show that the relative state tends to the preassigned value asymptotically, and we also provide several numerical examples to analyze the possible nonlinear dynamics of the proposed models, and compare them with analytical results.

Keywords

Acknowledgement

The work of H. Ahn was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MIST)(2022R1C12007321), and the work of J. Byeon was supported by the National Research Foundation of Korea (NRF) grant funded by the Korean government (MEST): No.2019R1A6A1A10073437. The work of S.-Y. Ha was supported by NRF-2020R1A2C3A01003881, and the work of J. Yoon was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIP): NRF-2016K2A9A2A13003815.

References

  1. J. A. Acebron, L. L. Bonilla, C. J. P. Perez Vicente, F. Ritort, and R. Spigler, The Kuramoto model: A simple paradigm for synchronization phenomena, Rev. Mod. Phys. 77 (2005), 137-185. 
  2. H. Ahn, J. Byeon, S.-Y. Ha, and J. Yoon, Collective behaviors of second-order nonlinear consensus models with a bonding force, arXiv:2112.14875. 
  3. 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. https://doi.org/10.1142/S0218202519500374 
  4. I. Barbalat, Systemes d'equations differentielles d'oscillations non lineaires, Rev. Math. Pures Appl. 4 (1959), 267-270. 
  5. D. Benedetto, E. Caglioti, and U. Montemagno, On the complete phase synchronization for the Kuramoto model in the mean-field limit, Commun. Math. Sci. 13 (2015), no. 7, 1775-1786. https://doi.org/10.4310/CMS.2015.v13.n7.a6 
  6. D. Benedetto, E. Caglioti, and U. Montemagno, Exponential dephasing of oscillators in the kinetic Kuramoto model, J. Stat. Phys. 162 (2016), no. 4, 813-823. https://doi.org/10.1007/s10955-015-1426-3 
  7. J. C. Bronski, L. DeVille, and M. J. Park, Fully synchronous solutions and the synchronization phase transition for the finite-N Kuramoto model, Chaos 22 (2012), no. 3, 033133, 17 pp. https://doi.org/10.1063/1.4745197 
  8. J. Buck and E. Buck, Biology of synchronous flashing of fireflies, Nature 211 (1966), 562-564. 
  9. J. A. Carrillo, Y.-P. Choi, S.-Y. Ha, M.-J. Kang, and Y. Kim, Contractivity of transport distances for the kinetic Kuramoto equation, J. Stat. Phys. 156 (2014), no. 2, 395-415. https://doi.org/10.1007/s10955-014-1005-z 
  10. J. A. Carrillo, M. Fornasier, J. Rosado, and G. Toscani, Asymptotic flocking dynamics for the kinetic Cucker-Smale model, SIAM J. Math. Anal. 42 (2010), no. 1, 218-236. https://doi.org/10.1137/090757290 
  11. H. Cho, J.-G. Dong, and S.-Y. Ha, Emergent behaviors of a thermodynamic Cucker-Smale flock with a time-delay on a general digraph, Math. Methods Appl. Sci. 45 (2022), no. 1, 164-196. https://doi.org/10.1002/mma.7771 
  12. Y.-P. Choi, S.-Y. Ha, S. Jung, and Y. Kim, Asymptotic formation and orbital stability of phase-locked states for the Kuramoto model, Phys. D 241 (2012), no. 7, 735-754. https://doi.org/10.1016/j.physd.2011.11.011 
  13. Y.-P. Choi, S.-Y. Ha, and Z. Li, Emergent dynamics of the Cucker-Smale flocking model and its variants, In N. Bellomo, P. Degond, and E. Tadmor (Eds.), Active Particles Vol.I - Theory, Models, Applications (tentative title), Series: Modeling and Simulation in Science and Technology, Birkhauser, Springer, 2019. 
  14. N. Chopra and M. W. Spong, On exponential synchronization of Kuramoto oscillators, IEEE Trans. Automat. Control 54 (2009), no. 2, 353-357. https://doi.org/10.1109/TAC.2008.2007884 
  15. J. Cort'es, Discontinuous dynamical systems: a tutorial on solutions, nonsmooth analysis, and stability, IEEE Control Syst. Mag. 28 (2008), no. 3, 36-73. https://doi.org/10.1109/MCS.2008.919306 
  16. F. Cucker and S. Smale, Emergent behavior in flocks, IEEE Trans. Automat. Control 52 (2007), no. 5, 852-862. https://doi.org/10.1109/TAC.2007.895842 
  17. P. Degond and S. Motsch, Macroscopic limit of self-driven particles with orientation interaction, C. R. Math. Acad. Sci. Paris 345 (2007), no. 10, 555-560. https://doi.org/10.1016/j.crma.2007.10.024 
  18. P. Degond and S. Motsch, Large scale dynamics of the persistent turning walker model of fish behavior, J. Stat. Phys. 131 (2008), no. 6, 989-1021. https://doi.org/10.1007/s10955-008-9529-8 
  19. P. Degond and S. Motsch, Continuum limit of self-driven particles with orientation interaction, Math. Models Methods Appl. Sci. 18 (2008), suppl., 1193-1215. https://doi.org/10.1142/S0218202508003005 
  20. J.-G. Dong and X. P. Xue, Synchronization analysis of Kuramoto oscillators, Commun. Math. Sci. 11 (2013), no. 2, 465-480. https://doi.org/10.4310/CMS.2013.v11.n2.a7 
  21. F. Dorfler and F. Bullo, On the critical coupling for Kuramoto oscillators, SIAM J. Appl. Dyn. Syst. 10 (2011), no. 3, 1070-1099. https://doi.org/10.1137/10081530X 
  22. A. F. Filippov, Differential equations with multi-valued discontinuous right-hand side, Dokl. Akad. Nauk SSSR 151 (1963), 65-68. 
  23. A. F. Filippov, Classical solutions of differential equations with multi-valued right-hand side, SIAM J. Control 5 (1967), 609-621. 
  24. A. F. Filippov, Differential equations with discontinuous righthand sides, translated from the Russian, Mathematics and its Applications (Soviet Series), 18, Kluwer Acad. Publ., Dordrecht, 1988. https://doi.org/10.1007/978-94-015-7793-9 
  25. S.-Y. Ha, J. Kim, J. Park, and X. Zhang, Uniform stability and mean-field limit for the augmented Kuramoto model, Netw. Heterog. Media 13 (2018), no. 2, 297-322. https://doi.org/10.3934/nhm.2018013 
  26. S.-Y. Ha, J. Kim, J. Park, and X. Zhang, Complete cluster predictability of the Cucker-Smale flocking model on the real line, Arch. Ration. Mech. Anal. 231 (2019), no. 1, 319-365. https://doi.org/10.1007/s00205-018-1281-x 
  27. S.-Y. Ha, J. Kim, and T. Ruggeri, From the relativistic mixture of gases to the relativistic Cucker-Smale flocking, Arch. Ration. Mech. Anal. 235 (2020), no. 3, 1661-1706. https://doi.org/10.1007/s00205-019-01452-y 
  28. S.-Y. Ha, H. K. Kim, and S.-Y. Ryoo, Emergence of phase-locked states for the Kuramoto model in a large coupling regime, Commun. Math. Sci. 14 (2016), no. 4, 1073-1091. https://doi.org/10.4310/CMS.2016.v14.n4.a10 
  29. S.-Y. Ha, D. Kim, and F. W. Schloder, Emergent behaviors of Cucker-Smale flocks on Riemannian manifolds, IEEE Trans. Automat. Control 66 (2021), no. 7, 3020-3035. 
  30. S.-Y. Ha, C. Lattanzio, B. Rubino, and M. Slemrod, Flocking and synchronization of particle models, Quart. Appl. Math. 69 (2011), no. 1, 91-103. https://doi.org/10.1090/S0033-569X-2010-01200-7 
  31. S.-Y. Ha and J.-G. Liu, A simple proof of the Cucker-Smale flocking dynamics and mean-field limit, Commun. Math. Sci. 7 (2009), no. 2, 297-325. http://projecteuclid.org/euclid.cms/1243443982 
  32. S.-Y. Ha, J. Park, and X. Zhang, A first-order reduction of the Cucker-Smale model on the real line and its clustering dynamics, Commun. Math. Sci. 16 (2018), no. 7, 1907-1931. https://doi.org/10.4310/CMS.2018.v16.n7.a8 
  33. S.-Y. Ha, W. Shim, and J. Yoon, An energy preserving discretization method for the thermodynamic Kuramoto model and collective behaviors, Commun. Math. Sci. 20 (2022), no. 2, 495-521. https://doi.org/10.4310/CMS.2022.v20.n2.a9 
  34. S.-Y. Ha and E. Tadmor, From particle to kinetic and hydrodynamic descriptions of flocking, Kinet. Relat. Models 1 (2008), no. 3, 415-435. https://doi.org/10.3934/krm.2008.1.415 
  35. Y. Kuramoto, International symposium on mathematical problems in mathematical physics, Lecture Notes Theor. Phys. 30 (1975), 420. 
  36. C. Lancellotti, On the Vlasov limit for systems of nonlinearly coupled oscillators without noise, Transport Theory Statist. Phys. 34 (2005), no. 7, 523-535. https://doi.org/10.1080/00411450508951152 
  37. Y. C. Liu and J. Wu, Flocking and asymptotic velocity of the Cucker-Smale model with processing delay, J. Math. Anal. Appl. 415 (2014), no. 1, 53-61. https://doi.org/10.1016/j.jmaa.2014.01.036 
  38. Y. C. Liu and J. Wu, Local phase synchronization and clustering for the delayed phase-coupled oscillators with plastic coupling, J. Math. Anal. Appl. 444 (2016), no. 2, 947-956. https://doi.org/10.1016/j.jmaa.2016.06.049 
  39. S. Motsch and E. Tadmor, Heterophilious dynamics enhances consensus, SIAM Rev. 56 (2014), no. 4, 577-621. https://doi.org/10.1137/120901866 
  40. J. Park, H. J. Kim, and S.-Y. Ha, Cucker-Smale flocking with inter-particle bonding forces, IEEE Trans. Automat. Control 55 (2010), no. 11, 2617-2623. https://doi.org/10.1109/TAC.2010.2061070 
  41. C. S. Peskin, Mathematical aspects of heart physiology, Courant Inst. Math. Sci., New York University, New York, 1975. 
  42. A. Pikovsky, M. Rosenblum, and J. Kurths, Synchronization, Cambridge Nonlinear Science Series, 12, Cambridge Univ. Press, Cambridge, 2001. https://doi.org/10.1017/CBO9780511755743 
  43. L. Ru, Y. C. Liu, and X. Wang, New conditions to avoid collisions in the discrete Cucker-Smale model with singular interactions, Appl. Math. Lett. 114 (2021), Paper No. 106906, 6 pp. https://doi.org/10.1016/j.aml.2020.106906 
  44. W. Shim, On the generic complete synchronization of the discrete Kuramoto model, Kinet. Relat. Models 13 (2020), no. 5, 979-1005. https://doi.org/10.3934/krm.2020034 
  45. R. Shvydkoy, Dynamics and analysis of alignment models of collective behavior, Necas Center Series, Birkhauser/Springer, Cham, 2021. https://doi.org/10.1007/978-3-030-68147-0 
  46. S. 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. https://doi.org/10.1016/S0167-2789(00)00094-4 
  47. J. Toner and Y. Tu, Flocks, herds, and schools: a quantitative theory of flocking, Phys. Rev. E (3) 58 (1998), no. 4, 4828-4858. https://doi.org/10.1103/PhysRevE.58.4828 
  48. C. M. Topaz and A. L. Bertozzi, Swarming patterns in a two-dimensional kinematic model for biological groups, SIAM J. Appl. Math. 65 (2004), no. 1, 152-174. https://doi.org/10.1137/S0036139903437424 
  49. J. L. van Hemmen and W. F. Wreszinski, Lyapunov function for the Kuramoto model of nonlinearly coupled oscillators, J. Stat. Phys. 72 (1993), 145-166. 
  50. T. Vicsek and A. Zefeiris, Collective motion, Phys. Rep. 517 (2012), 71-140. 
  51. A. T. Winfree, The geometry of biological time, Biomathematics, 8, Springer, Berlin, 1980.