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Emergent Dynamics of the Cucker–Smale Flocking Model and Its Variants

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Active Particles, Volume 1

Abstract

In this chapter, we present the Cucker–Smale-type flocking models and discuss their mathematical structures and flocking theorems in terms of coupling strength, interaction topologies, and initial data. In 2007, two mathematicians Felipe Cucker and Steve Smale introduced a second-order particle model which resembles Newton’s equations in N-body system and present how their simple model can exhibit emergent flocking behavior under sufficient conditions expressed only in terms of parameters and initial data. After Cucker–Smale’s seminal works in [31, 32], their model has received lots of attention from applied math and control engineering communities. We discuss the state of the art for the flocking theorems to Cucker–Smale-type flocking models.

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References

  1. Ahn, S., Choi, H., Ha, S.-Y., and Lee, H.: On the collision avoiding initial-congurations to the Cucker-Smale type flocking models, Comm. Math. Sci. 10, 625–643 (2012).

    Article  MATH  Google Scholar 

  2. Bae, H.-O., Choi, Y.-P., Ha, S.-Y., and Kang, M.-J.: Time-asymptotic interacton of flocking particles and incompressible viscous fluid, Nonlinearity 25, 1155–1177 (2012).

    Article  MathSciNet  MATH  Google Scholar 

  3. Bae, H.-O., Choi, Y.-P., Ha, S.-Y., and Kang, M.-J.: Asymptotic flocking dynamics of Cucker-Smale particles immersed in compressible fluids, Disc. and Cont. Dyn. Sys. 34, 4419–4458 (2014).

    Article  MathSciNet  MATH  Google Scholar 

  4. Bae, H.-O., Choi, Y.-P., Ha, S.-Y., and Kang, M.-J.: Global existence of strong solution for the Cucker-Smale-Navier-Stokes system, J. Diff. Eqns. 257, 2225–2255 (2014).

    Article  MathSciNet  MATH  Google Scholar 

  5. Bae, H.-O., Choi, Y.-P., Ha, S.-Y., and Kang, M.-J.: Global existence of strong solutions to the Cucker-Smale-Stokes system, J. Math. Fluid Mech., 18, 381–396 (2016).

    Google Scholar 

  6. Ballerini, M., Cabibbo, N., Candelier, R., et al.: Interaction ruling animal collective behavior depends on topological rather than metric distance: Evidence from a field study, Proc. Nat. Acad. Sci. 105, 1232–1237 (2008).

    Article  Google Scholar 

  7. Boudin, L., Desvillettes, L., and Motte, R.: A modelling of compressible droplets in a fluid, Comm. Math. Sci. 1, 657–669 (2003).

    Article  MATH  Google Scholar 

  8. Cañizo, J. A., Carrillo, J. A., and Rosado, J.: A well-posedness theory in measures for some kinetic models of collective motion, Math. Mod. Meth. Appl. Sci. 21, 515–539 (2011).

    Article  MathSciNet  MATH  Google Scholar 

  9. Carrillo, J. A., Choi, Y.-P., and Hauray, M.: Local well-posedness of the generalized Cucker-Smale model, ESAIM: Proc. 47, 17–35 (2014).

    Google Scholar 

  10. Carrillo, J. A., Choi, Y.-P., Hauray, M., and Salem, S.: Mean-field limit for collective behavior models with sharp sensitivity regions, to appear in J. Eur. Math. Soc.

    Google Scholar 

  11. Carrillo, J. A., Choi, Y.-P., and Karper, T.: On the analysis of a coupled kinetic-fluid model with local alignment forces, Annales de I’IHP-ANL, 33, (2016), 273–307.

    Google Scholar 

  12. Carrillo, J. A., Choi, Y.-P., and Peszek, J.: Sharp Conditions to avoid collisions in singular Cucker-Smale interactions, Preprint.

    Google Scholar 

  13. Carrillo, J. A., Choi, Y.-P., Tadmor, E., and Tan, C.: Critical thresholds in 1D Euler equations with nonlocal interaction forces, Math. Mod. Meth. Appl. Sci. 26, 185–206 (2016).

    Article  MATH  Google Scholar 

  14. Carrillo, J. A., Fornasier, M., Toscani, G. and Vecil, F.: Particle, kinetic and hydrodynamic models of swarming. In Mathematical modeling of collective behavior in Socio-Economic and Life Sciences, 297–336 (2010).

    Google Scholar 

  15. Carrillo, J. A., Fornasier, M. Rosado, J., and Toscani, G.: Asymptotic flocking dynamics for the kinetic Cucker-Smale model, SIAM J. Math. Anal. 42, 218–236 (2010).

    Article  MathSciNet  MATH  Google Scholar 

  16. Cho, J., Ha, S.-Y., Huang, F., Jin, C. and D. Ko: Emergence of bi-cluster flocking for the Cucker-Smale model, Math. Mod. Meth. Appl. Sci. 14, (2016) doi:10.1142/S0219530515400023.

  17. Cho, J., Ha, S.-Y., Huang, F., Jin, C. and D. Ko: Emergence of bi-cluster flocking for agent-based models with unit speed constraint. Analysis and Applications 14, 39–73 (2016).

    Article  MathSciNet  MATH  Google Scholar 

  18. Choi, Y.-P.: Global classical solutions of the Vlasov-Fokker-Planck equation with local alignment forces, Nonlinearity, 29, (2016), 1887–1916.

    Google Scholar 

  19. Choi, Y.-P.: Large-time behavior for the Vlasov/compressible Navier-Stokes equations, J. Math. Phys., 57, 071501, (2016).

    Google Scholar 

  20. Choi, Y.-P.: Compressible Euler equations interacting with incompressible flow, Kinetic and Related Models 8, 335–358 (2015).

    Article  MathSciNet  MATH  Google Scholar 

  21. Choi, Y.-P.: Global classical solutions and large-time behavior of the two-phase fluid model, SIAM J. Math. Anal., 48, (2016), 3090–3122.

    Google Scholar 

  22. Choi, Y.-P.: Finite-time blow-up phenomena of Vlasov/Navier-Stokes equations and related systems, preprint.

    Google Scholar 

  23. Choi, Y.-P. and Lee, J.: Global existence of weak and strong solutions to Cucker-Smale-Navier-Stokes equations in \({\mathbb{R}}^{2}\), Nonlinear Anal.-Real. 27, 158–182 (2016).

    Google Scholar 

  24. Choi, Y.-P. and Kwon, B.: Two-species flocking particles immersed in a fluid, Comm. Info. Sys. 13, 123–149 (2013).

    MathSciNet  MATH  Google Scholar 

  25. Choi, Y.-P. and Kwon, B.: Global well-posedness and large-time behavior for the inhomogeneous Vlasov-Navier-Stokes equations, Nonlinearity 28, 3309–3336 (2015).

    Article  MathSciNet  MATH  Google Scholar 

  26. Choi, Y.-P. and Kwon, B.: The Cauchy problem for the pressureless Euler/isentropic Navier-Stokes equations, J. Diff. Eqns., 261, 654–711 (2016).

    Google Scholar 

  27. Cucker, F., and Dong J.G.: On the critical exponent for flocks under hierarchical leadership, Math. Mod. Meth. Appl. Sci. 19, 1391–1404 (2009).

    Article  MathSciNet  MATH  Google Scholar 

  28. Cucker. F., and Huepe, C.: Flocking with informed agents, Maths. in Action 1, 1–25 (2008).

    Article  MathSciNet  Google Scholar 

  29. Cao, M., Morse, A. S., and Anderson, B. D. O.: Reaching a consensus in a dynamically changing environment: A graphic approach, SIAM J. Control Optim. 47, 575–600 (2008).

    Article  MathSciNet  MATH  Google Scholar 

  30. Cao, M., Morse, A. S., and Anderson, B. D. O.: Reaching a consensus in a dynamically changing environment: Vonvergence rates, meansurement delays, and asynchronous events, SIAM J. Control Optim. 47, 601–623 (2008).

    Article  MathSciNet  MATH  Google Scholar 

  31. Cucker, F., and Smale S.: Emergent behavior in flocks, IEEE Trans. Autom. Control 52, 852–862 (2007).

    Article  MathSciNet  Google Scholar 

  32. Cucker, F., and Smale S.: On the mathematics of emergence, Japan. J. Math. 2, 197–227 (2007).

    Article  MathSciNet  MATH  Google Scholar 

  33. Dalmao, F., and Mordecki, E.: Cucker-Smale flocking under hierarchical leadership and random interactions, SIAM J. Appl. Math. 71, 1307–1316 (2010).

    Article  MathSciNet  MATH  Google Scholar 

  34. Diestel, R.: Graph Theory, Graduate Texts in Mathematics New York, U.S.A.: Springer-Verlag, (1997).

    Google Scholar 

  35. Dobrushin, R.: Vlasov equations, Funct. Anal. Appl. 13, 115–123, (1979).

    Article  MATH  Google Scholar 

  36. Duan, R., Fornasier, M., and Toscani, G.: A kinetic flocking model with diffusion, Commun. Math. Phys. 300, 95–145 (2010).

    Article  MathSciNet  MATH  Google Scholar 

  37. Ha, S.-Y., Kang, M.-J., and Kwon, B.: A hydrodynamic model for the interaction of Cucker-Smale particles and incompressible fluid, Math. Mod. Meth. Appl. Sci. 24, 2311–2359 (2014).

    Article  MathSciNet  MATH  Google Scholar 

  38. Ha, S.-Y., Kang, M.-J., and Kwon, B.: Emergent dynamics for the hydrodynamic Cucker-Smale system in a moving domain, SIAM. Math. Anal. 47, 3813–3831 (2015).

    Article  MathSciNet  MATH  Google Scholar 

  39. Ha, S.-Y., Ko, D., Zhang, Y. and Zhang, X.: Emergent dynamics in the interactions of Cucker-Smale ensembles, to appear in Kinetic and Related Models.

    Google Scholar 

  40. Ha, S.-Y., Ko, D. and Zhang, Y.: A criterion for non-flocking and emergence of multi-cluster flocking for the Cucker-Smale model, to appear in Math. Mod. Meth. Appl. Sci.

    Google Scholar 

  41. Ha, S.-Y. and Liu, J.-G.: A simple proof of Cucker-Smale flocking dynamics and mean field limit. Comm. Math. Sci. 7, 297–325 (2009).

    Article  MathSciNet  MATH  Google Scholar 

  42. Ha, S.-Y. and Tadmor, E.: From particle to kinetic and hydrodynamic description of flocking, Kinetic and Related Models 1, 415–435 (2008).

    Article  MathSciNet  MATH  Google Scholar 

  43. Haskovec, J.: Flocking dynamics and mean-field limit in the Cucker-Smale-type model with topological interactions, Physica D 261, 42–51 (2013).

    Article  MathSciNet  MATH  Google Scholar 

  44. Jadbabaie, A., Lin, J., and Morse, A.: Coordination of groups of mobile autonomous agents using nearest neighbor rules, IEEE Trans. Autom. Control 48 988-1001 (2003).

    Article  MathSciNet  Google Scholar 

  45. Karper, T. K., Mellet, A., and Trivisa, K.: Hydrodynamic limit of the kinetic Cucker-Smale flocking model 25, 131–163 (2015).

    Google Scholar 

  46. Li, Z.: Effectual leadership in flocks with hierarchy and individual preference, Disc. Cont. Dyn. Syst. A 34, 3683–3702 (2014).

    Article  MathSciNet  MATH  Google Scholar 

  47. Li, Z., and Ha, S.-Y.: On the Cucker-Smale flocking with alternating leaders, Quart. Appl. Math. 73, 693–709 (2015).

    Article  MathSciNet  MATH  Google Scholar 

  48. Li, Z., Ha, S.-Y., and Xue, X.: Emergent phenomena in an ensemble of Cucker-Smale particles under joint rooted leadership, Math. Mod. Meth. Appl. Sci. 24, 1389–1419 (2014).

    Article  MathSciNet  MATH  Google Scholar 

  49. Leonard, N. E., Paley, D. A., Lekien, F., Sepulchre, R., Fratantoni, D.M. and Davis, R. E.: Collective motion, sensor networks and ocean sampling, Proc. IEEE 95, 48–74 (2007).

    Article  Google Scholar 

  50. Li, Z. and Xue, X.: Cucker-Smale flocking under rooted leadership with fixed and switching topologies, SIAM J. Appl. Math. 70, 3156–3174 (2010).

    Article  MathSciNet  MATH  Google Scholar 

  51. Motsch, S. and Tadmor, E.: Heterophilious dynamics enhances consensus, SIAM Rev. 56, 577–621 (2014).

    Article  MathSciNet  MATH  Google Scholar 

  52. Motsch, S. and Tadmor, E.: A new model for self-organized dynamics and its flocking behavior, J. Stat. Phys. 144, 923–947 (2011).

    Article  MathSciNet  MATH  Google Scholar 

  53. Nagy, M., Ákos, Z., Biro, D., and Vicsek, T.: Hierarchical group dynamics in pigeon flocks, Nature 464, 890–893 (2010).

    Article  Google Scholar 

  54. O’Rourke, P.: Collective drop effects on vaporising liquid sprays, Ph. D. Thesis, Princeton University, Princeton, NJ, 1981.

    Google Scholar 

  55. Paley, D.A., Leonard, N. E., Sepulchre, R., Grunbaum, D. and Parrish, J. K.: Oscillator models and collective motion, IEEE Control Systems 27, 89–105 (2007).

    Article  Google Scholar 

  56. Park, J., Kim, H., and Ha, S.-Y.: Cucker-Smale flocking with inter-particle bonding forces, IEEE Tran. Automatic Control 55, 2617–2623 (2010).

    Article  MathSciNet  Google Scholar 

  57. Perea, L., Gómez, G., and Elosegui, P.: Extension of the Cucker-Smale control law to space flight formation, J. Guidance, Control and Dynamics 32, 526–536 (2009).

    Article  Google Scholar 

  58. Reynolds, C. W.: Flocks, herds and schools: A distributed behavioral model, Proceeding SIGGRAPH 87 Proceedings of the 14th annual conference on Computer graphics and interactive techniques 25–34 (1987).

    Google Scholar 

  59. Ranz, W. and Marshall, W.: Evaporization from drops, Chem. Eng. Prog. 48, 141–180 (1952).

    Google Scholar 

  60. Shen, J.: Cucker-Smale Flocking under Hierarchical Leadership, SIAM J. Appl. Math. 68, 694–719 (2007).

    Article  MathSciNet  MATH  Google Scholar 

  61. Spannenberg, A. and Galvin, K. P.: Continuous differential sedimentation of a binary suspension, Chem. Eng. Aust. 21, 7–11 (1996).

    Google Scholar 

  62. Tadmor, E. and Tan, C.: Critical thresholds in flocking hydrodynamics with nonlocal alignment, Proc. Royal Soc. A, 372:20130401 (2014).

    MathSciNet  MATH  Google Scholar 

  63. Toner, J. and Tu, Y.: Flocks, herds, and Schools: A quantitative theory of flocking, Physical Review E. 58, 4828–4858 (1998).

    Article  MathSciNet  Google Scholar 

  64. Vicsek, T., Czirók, A., Ben-Jacob, E., Cohen I., and Shochet O.: Novel type of phase transition in a system of self-driven particles, Phys. Rev. Lett. 75, 1226–1229 (1995).

    Article  MathSciNet  Google Scholar 

  65. Vinkovic, I., Aguirre, C., Simoëns S., and Gorokhovski, M.: Large eddy simulation of droplet dispersion for inhomogeneous turbulent wall flow, Int. J. Multiph. Flow 32, 344–364 (2006).

    Article  MATH  Google Scholar 

  66. Williams, F. A.: Spray combustion and atomization, Phys. fluids 1, 541–555 (1958).

    Article  MATH  Google Scholar 

  67. Xue, X., and Guo, L.: A kind of nonnegative matrices and its application on the stability of discrete dynamical systems, J. Math. Anal. Appl. 331, 1113–1121 (2007).

    Article  MathSciNet  MATH  Google Scholar 

  68. Xue, X., and Li, Z.: Asymptotic stability analysis of a kind of switched positive linear discrete systems, IEEE Trans. Autom. Control 55, 2198–2203 (2010).

    Article  MathSciNet  Google Scholar 

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Acknowledgements

The work of S.-Y. Ha was supported by the Samsung Science and Technology Foundation under Project Number SSTF-BA1401-03. The work of Y.-P. Choi was supported by Engineering and Physical Sciences Research Council (EP/K008404/1). The work of Z. Li was supported by the National Natural Science Foundation of China Grant No.11401135, and Fundamental Research Funds for the Central Universities (HIT.BRETIII.201501 and HIT.PIRS.201610).

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Authors and Affiliations

  1. Fakultät für Mathematik, Technische Universität München, Boltzmannstraße 3, 85748, Garching bei München, Germany

    Young-Pil Choi

  2. Department of Mathematical Sciences and Research Institute of Mathematics, Seoul National University, Seoul, 151-747, Republic of Korea

    Seung-Yeal Ha

  3. Department of Mathematics, Harbin Institute of Technology, Harbin, 150001, People’s Republic of China

    Zhuchun Li

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  1. Young-Pil Choi
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  2. Seung-Yeal Ha
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  3. Zhuchun Li
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Correspondence to Seung-Yeal Ha .

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Editors and Affiliations

  1. Department of Mathematical Sciences, King Abdulaziz University, Jeddah, Saudi Arabia

    Nicola Bellomo

  2. Department of Mathematics, Imperial College London, London, United Kingdom

    Pierre Degond

  3. CSCAMM, University of Maryland, College Park, USA

    Eitan Tadmor

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Choi, YP., Ha, SY., Li, Z. (2017). Emergent Dynamics of the Cucker–Smale Flocking Model and Its Variants. In: Bellomo, N., Degond, P., Tadmor, E. (eds) Active Particles, Volume 1 . Modeling and Simulation in Science, Engineering and Technology. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-49996-3_8

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