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Complete Cluster Predictability of the Cucker–Smale Flocking Model on the Real Line

  • Seung-Yeal Ha
  • Jeongho Kim
  • Jinyeong Park
  • Xiongtao Zhang
Article
  • 55 Downloads

Abstract

We present the complete predictability of clustering for the Cucker–Smale (C–S) model on the line. Emergence of multi-cluster flocking is often observed in numerical simulations for the C–S model with short-range interactions. However, the explicit computation of the number of emergent multi-clusters a priori is a challenging problem for the Cucker–Smale flocking model. In this paper, we present an explicit criterion and algorithm to calculate the number of clusters and their bulk velocities in terms of initial configuration, coupling strength and communication weight function in a one-dimensional setting. We present a finite increasing sequence of coupling strengths in which the number of asymptotic clusters has a jump. We also provide several numerical examples and compare them with analytical results.

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Acknowledgements

The authors would like to thank the anonymous referee for his/her careful reading and constructive comments on the original manuscript.

References

  1. 1.
    Ahn S., Choi H., Ha S.-Y., Lee H.: On the collision avoiding initial-configurations to the Cucker–Smale type flocking models. Commun. Math. Sci. 10, 625–643 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Ahn S., Ha S.-Y.: Stochastic flocking dynamics of the Cucker–Smale model with multiplicative white noises. J. Math. Phys. 51, 103301 (2010)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bellomo N., Ha S.-Y.: A quest toward a mathematical theory of the dynamics of swarms. Math. Models Methods Appl. Sci. 27, 745–770 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bolley F., Canizo J.A., Carrillo J.A.: Stochastic mean-field limit: non-Lipschitz forces and swarming. Math. Models Methods Appl. Sci. 21, 2179–2210 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Canizo J.A., Carrillo J.A., Rosado J.: A well-posedness theory in measures for some kinetic models of collective motion. Math. Models Methods Appl. Sci. 21, 515–539 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Carrillo J.A., D’ Orsogna M.R., Panferov V.: Double milling in self-propelled swarms from kinetic theory. Kinet. Relat. Models 2, 363–378 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Carrillo J.A., Fornasier M., Rosado J., Toscani G.: Asymptotic flocking dynamics for the kinetic Cucker–Smale model. SIAM J. Math. Anal. 42, 218–236 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Carrillo J.A., Klar A., Martin S., Tiwari S.: Self-propelled interacting particle systems with roosting force. Math. Models Methods Appl. Sci. 20, 1533–1552 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Cho J., Ha S.-Y., Huang F., Jin C., Ko D.: Emergence of bi-cluster flocking for the Cucker–Smale model. Math. Models Methods Appl. Sci. 26, 1191–1218 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Cho J., Ha S.-Y., Huang F., Jin C., Ko D.: Emergence of bi-cluster flocking for agent-based models with unit speed constraint. Anal. Appl. 14, 1–35 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Cucker F., Dong J.-G.: Avoiding collisions in flocks. IEEE Trans. Autom. Control 55, 1238–1243 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Cucker F., Mordecki E.: Flocking in noisy environments. J. Math. Pure Appl. 89, 278–296 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Cucker F., Smale S.: Emergent behavior in flocks. IEEE Trans. Automat. Contr. 52, 852–862 (2007)CrossRefzbMATHGoogle Scholar
  14. 14.
    Degond P., Motsch S.: Macroscopic limit of self-driven particles with orientation interaction. C. R. Math. Acad. Sci. Paris 345, 555–560 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Degond P., Motsch S.: Large-scale dynamics of the persistent Turing Walker model of fish behavior. J. Stat. Phys. 131, 989–1022 (2008)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Degond P., Motsch S.: Continuum limit of self-driven particles with orientation interaction. Math. Models Methods Appl. Sci. 18, 1193–1215 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Duan R., Fornasier M., Toscani G.: A kinetic flocking model with diffusion. Commun. Math. Phys. 300, 95–145 (2010)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Fornasier M., Haskovec J., Toscani G.: Fluid dynamic description of flocking via Povzner–Boltzmann equation. Phys. D 240, 21–31 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Ha S.-Y., Ha T., Kim J.: Asymptotic flocking dynamics for the Cucker–Smale model with the Rayleigh friction. J. Phys. A Math. Theor. 43, 315201 (2010)CrossRefzbMATHGoogle Scholar
  20. 20.
    Ha S.-Y., Ko D., Zhang Y.: Critical coupling strength of the Cucker–Smale model for flocking. Math. Models Methods Appl. Sci. 27, 1051–1087 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Ha S.-Y., Lee K., Levy D.: Emergence of time-asymptotic flocking in a stochastic Cucker–Smale system. Commun. Math. Sci. 7, 453–469 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Ha S.-Y., Liu J.-G.: A simple proof of Cucker–Smale flocking dynamics and mean field limit. Commun. Math. Sci. 7, 297–325 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Ha, S.-Y., Park, J., Zhang, X.: On the first-order reduction of the Cucker–Smale model and its clustering dynamics (submitted)Google Scholar
  24. 24.
    Ha S.-Y., Slemrod M.: Flocking dynamics of a singularly perturbed oscillator chain and the Cucker–Smale system. J. Dyn. Differ. Equ. 22, 325–330 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Ha S.-Y., Tadmor E.: From particle to kinetic and hydrodynamic description of flocking. Kinet. Relat. Models 1, 415–435 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Hendrickx M., Tsitsiklis J.N.: Convergence of type-symmetric and cut-balanced consensus seeking systems. IEEE Trans. Autom. Control 58, 214–218 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Kuramoto Y.: International symposium on mathematical problems in mathematical physics. Lect. Notes Theor. Phys. 30, 420 (1975)ADSCrossRefGoogle Scholar
  28. 28.
    Leonard N.E., Paley D.A., Lekien F., Sepulchre R., Fratantoni D.M., Davis R.E.: Collective motion, sensor networks and ocean sampling. Proc. IEEE 95, 48–74 (2007)CrossRefGoogle Scholar
  29. 29.
    Li Z., Xue X.: Cucker–Smale flocking under rooted leadership with fixed and switching topologies. SIAM J. Appl. Math. 70, 3156–3174 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Motsch S., Tadmor E.: Heterophilious dynamics enhances consensus. SIAM Rev. 56, 577–621 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Motsch S., Tadmor E.: A new model for self-organized dynamics and its flocking behavior. J. Stat. Phys. 144, 923–947 (2011)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Perea L., Elosegui P., Gómez G.: Extension of the Cucker–Smale control law to space flight formation. J. Guid. Control Dyn. 32, 526–536 (2009)ADSCrossRefGoogle Scholar
  33. 33.
    Paley D.A., Leonard N.E., Sepulchre R., Grunbaum D., Parrish J.K.: Oscillator models and collective motion. IEEE Control Syst. 27, 89–105 (2007)CrossRefGoogle Scholar
  34. 34.
    Park J., Kim H., Ha S.-Y.: Cucker–Smale flocking with inter-particle bonding forces. IEEE Tran. Autom. Control 55, 2617–2623 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Poyato D., Soler J.: Euler-type equations and commutators in singular and hyperbolic limits of kinetic Cucker–Smale models. Math. Models Methods Appl. Sci., 6, 1089–1152 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Shen J.: Cucker–Smale flocking under hierarchical leadership. SIAM J. Appl. Math. 68, 694–719 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Toner J., Tu Y.: Flocks, herds, and schools: A quantitative theory of flocking. Phys. Rev. E 58, 4828–4858 (1998)ADSMathSciNetCrossRefGoogle Scholar
  38. 38.
    Topaz C.M., Bertozzi A.L.: Swarming patterns in a two-dimensional kinematic model for biological groups. SIAM J. Appl. Math. 65, 152–174 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Vicsek T., Czirók A., Ben-Jacob E., Cohen I., Schochet O.: Novel type of phase transition in a system of self-driven particles. Phys. Rev. Lett. 75, 1226–1229 (1995)ADSMathSciNetCrossRefGoogle Scholar
  40. 40.
    Winfree A. T.: Biological rhythms and the behavior of populations of coupled oscillators. J. Theor. Biol. 16, 15–42 (1967)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Mathematical Sciences and Research Institute of MathematicsSeoul National UniversitySeoulRepublic of Korea
  2. 2.Korea Institute for Advanced StudySeoulRepublic of Korea
  3. 3.Department of Mathematical SciencesSeoul National UniversitySeoulRepublic of Korea
  4. 4.Department of Mathematics and Research Institute of Natural SciencesHanyang UniversitySeoulRepublic of Korea
  5. 5.Center for Mathematical SciencesHuazhong University of Science and TechnologyWuhanChina

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