Journal of Statistical Physics

, Volume 171, Issue 2, pp 345–360 | Cite as

Flocking of the Motsch–Tadmor Model with a Cut-Off Interaction Function

Article
  • 70 Downloads

Abstract

In this paper, we study the flocking behavior of the Motsch–Tadmor model with a cut-off interaction function. Our analysis shows that connectedness is important for flocking of this kind of model. Fortunately, we get a sufficient condition imposed only on the model parameters and initial data to guarantee the connectedness of the neighbor graph associated with the system. Then we present a theoretical analysis for flocking, and show that the system achieves consensus at an exponential rate.

Keywords

Motsch–Tadmor model Cucker–Smale model Flocking Stochastic matrix Neighbor graph 

Mathematics Subject Classification

34A36 34D06 34F15 34K25 70E55 

Notes

Acknowledgements

The author would like to thank the referee for detailed comments, which benefit him a lot and improve the presentation of this paper significantly.

References

  1. 1.
    Bae, H.-O., Choi, Y.-P., Ha, S.-Y., Kang, M.-J.: Asymptotic flocking dynamics of Cucker–Smale particles immersed in compressible fluids. Discrete Contin. Dyn. Syst. 34(11), 4419–4458 (2014)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    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(03), 515–539 (2011)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Carrillo, J.A., Choi, Y.-P., Hauray, M., Salem, S.: Mean-field limit for collective behavior models with sharp sensitivity regions, J. Eur. Math. Soc (to appear)Google Scholar
  4. 4.
    Carrillo, J.A., Fornasier, M., Rosado, J., Toscani, G.: Asymptotic flocking dynamics for the kinetic Cucker–Smale model. SIAM J. Math. Anal. 42(1), 218–236 (2010)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Carrillo, J.A., Fornasier, M., Toscani, G., Vecil, F.: Particle, kinetic, and hydrodynamic models of swarming. In: Mathematical Modeling of Collective Behavior in Socio-Economic and Life Sciences, pp. 297–336. Springer (2010)Google Scholar
  6. 6.
    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(01), 39–73 (2016)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Cho, J., Ha, S.-Y., Huang, F., Jin, C., Ko, Dongnam: Emergence of bi-cluster flocking for the Cucker–Smale model. Math. Models Methods Appl. Sci. 26(06), 1191–1218 (2016)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Cucker, F., Smale, S.: Emergent behavior in flocks. IEEE Trans. Autom. Control 52(5), 852–862 (2007)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Ha, S.-Y., Huang, F., Wang, Y.: A global unique solvability of entropic weak solution to the one-dimensional pressureless Euler system with a flocking dissipation. J. Differ. Equ. 257(5), 1333–1371 (2014)ADSMathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Ha, S.-Y., Kang, M.-J., Kwon, B.: A hydrodynamic model for the interaction of Cucker–Smale particles and incompressible fluid. Math. Models Methods Appl. Sci. 24(11), 2311–2359 (2014)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Ha, S.-Y., Liu, J.-G.: A simple proof of the Cucker–Smale flocking dynamics and mean-field limit. Commun. Math. Sci. 7(2), 297–325 (2009)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Ha, S.-Y., Tadmor, E.: From particle to kinetic and hydrodynamic descriptions of flocking. Kinet. Relat. Models 1(3), 415–435 (2008)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Haskovec, J.: Flocking dynamics and mean-field limit in the Cucker–Smale type model with topological interactions. Phys. D 261(15), 42–51 (2013)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Jabin, P.-E., Motsch, S.: Clustering and asymptotic behavior in opinion formation. J. Differ. Equ. 257(11), 4165–4187 (2014)ADSMathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Jadbabaie, A., Lin, J.: Coordination of groups of mobile autonomous agents using nearest neighbor rules. IEEE Trans. Automat. Control 48(6), 988–1001 (2003)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Jin, C.: Well posedness for pressureless Euler system with a flocking dissipation in Wasserstein space. Nonlinear Anal. 128, 412–422 (2015)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Karper, T.K., Mellet, A., Trivisa, K.: Existence of weak solutions to kinetic flocking models. SIAM J. Math. Anal. 45(1), 215–243 (2013)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Liu, Z., Guo, L.: Connectivity and synchronization of Vicsek model. Sci. China Ser. F 51(7), 848–858 (2008)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Liu, Z., Han, J., Hu, X.: The number of leaders needed for consensus. In: Decision and Control, 2009 Held Jointly with the 2009 28th Chinese Control Conference. CDC/CCC 2009. Proceedings of the 48th IEEE Conference on, pp. 3745–3750. IEEE (2009)Google Scholar
  20. 20.
    Motsch, S., Tadmor, E.: A new model for self-organized dynamics and its flocking behavior. J. Stat. Phys. 144(5), 923–947 (2011)ADSMathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Serre, D.: Matrices. Graduate Texts in Mathematics 216. Springer (2010)Google Scholar
  22. 22.
    Tang, G., Guo, L.: Convergence of a class of multi-agent systems in probabilistic framework. J. Syst. Sci. Complex. 20(2), 173–197 (2007)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Vicsek, T., Czirók, A., Ben-Jacob, E., Cohen, I., Shochet, O.: Novel type of phase transition in a system of self-driven particles. Phys. Rev. Lett. 75(6), 1226–1229 (1995)ADSMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Beijing International Center for Mathematical ResearchPeking UniversityBeijingPeople’s Republic of China

Personalised recommendations