No Access

EMERGENCE OF MULTI-CLUSTER CONFIGURATIONS FROM ATTRACTIVE AND REPULSIVE INTERACTIONS

SEUNG-YEAL HA

Department of Mathematical Sciences, Seoul National University, Seoul 151-747, Korea

Search for more papers by this author

EUNHEE JEONG

Department of Mathematical Sciences, Seoul National University, Seoul 151-747, Korea

Search for more papers by this author

JEONG-HAN KANG

Department of Sociology, Yeonsei University, Seoul 120-749, Korea

Search for more papers by this author

, and

KYUNGKEUN KANG

Department of Mathematics, Yeonsei University, Seoul 120-749, Korea

Corresponding author.

Search for more papers by this author

https://doi.org/10.1142/S0218202512500133Cited by:20 (Source: Crossref)

Abstract

We discuss a first-order Cucker–Smale-type consensus model with attractive and repulsive interactions and present upper and lower bound estimates on the number of asymptotic point-clusters depending on the relative ranges of interactions and coupling strength. When the number of agents approaches infinity, we introduce a scalar conservation law with a non-local flux for a macroscopic description. We show that the corresponding conservation law admits a classical solution for sufficiently smooth initial data, which illustrates the shock avoidance effect due to the non-locality of the interactions. We also study the dynamics of special Dirac-Comb-type solutions consisting of two and three point-clusters.

Keywords:

AMSC: 22E46, 53C35, 57S20

References

M. Agueh, R. Illner and A. Richardson, Kinetic Relat. Models 4, 1 (2011). Crossref, Web of Science, Google Scholar
G. Aletti, G. Naldi and G. Toscani, SIAM J. Appl. Math. 67, 837 (2007). Crossref, Web of Science, Google Scholar
I. Aoki, Bull. Japan Soc. Sci. Fisheries 48, 1081 (1982). Crossref, Web of Science, Google Scholar
R. Axelrod, J. Conflict Resolution 41, 203 (1997). Crossref, Web of Science, Google Scholar
M. Balleriniet al., Proc. Natl. Acad. Sci. USA 105, 1232 (2008). Crossref, Web of Science, Google Scholar
A. B. T. Barbaroet al., Math. Comput. Simulat. 79, 3397 (2009). Crossref, Web of Science, Google Scholar
N. Bellomo , Modeling Complex Living Systems: A Kinetic Theory and Stochastic Game Approach ( Birkhäuser , 2008 ) . Crossref, Google Scholar
N. Bellomoet al., Math. Models Methods Appl. Sci. 20, 1179 (2010). Link, Web of Science, Google Scholar
N. Bellomoet al., Math. Models Methods Appl. Sci. 20, 1391 (2010). Link, Web of Science, Google Scholar
N. Bellomo and C. Dogbe, SIAM Rev. 53, 409 (2011). Crossref, Web of Science, Google Scholar
E. Ben-Naim, Europhys. Lett. 69, 671 (2005). Crossref, Google Scholar
M. L. Bertotti and M. Delitala, Nonlinear Anal. Real World Appl. 9, 183 (2008). Crossref, Web of Science, Google Scholar
M. L. Bertotti and M. Delitala, Z. Angew. Math. Phys. 61, 583 (2010). Crossref, Web of Science, Google Scholar
V. Blondel, J. M. Hendrickx and J. Tsitsiklis, IEEE Trans. Automat. Control 54, 2506 (2009). Web of Science, Google Scholar
V. Blondel, J. M. Hendrickx and J. Tsitsiklis, SIAM J. Control Optim. 48, 5214 (2010). Crossref, Web of Science, Google Scholar
J. A. Carrilloet al., SIAM J. Math. Anal. 42, 218 (2010). Crossref, Web of Science, Google Scholar
A. Cavagnaet al., Math. Models Methods Appl. Sci. 20, 1491 (2010). Link, Web of Science, Google Scholar
F. Cucker and S. Smale, Japan J. Math. 2, 197 (2007). Crossref, Web of Science, Google Scholar
F. Cucker and S. Smale, IEEE Trans. Automat. Control 52, 852 (2007). Crossref, Web of Science, Google Scholar
C. M. Dafermos , Hyperbolic Conservation Laws in Continuum Physics , Grundlehren der mathematishen Wissenschaften 325 ( Springer , 2000 ) . Crossref, Google Scholar
G. Deffusant, F. Amblard and G. Weisbuch, Physica A 353, 555 (2005). Web of Science, Google Scholar
P. Degond and S. Motsch, J. Stat. Phys. 131, 989 (2008). Crossref, Web of Science, Google Scholar
R. C. Fetecau and R. Eftimie, J. Math. Biol. 61, 545 (2010). Crossref, Web of Science, Google Scholar
A. Flache and M. W. Macy, Local convergence and global diversity: From interpersonal to social influence, to appear in J. Conflict Resolution . Google Scholar
S. Galam, Y. Gefen and Y. Shapir, J. Math. Sociology 9, 1 (1982). Crossref, Web of Science, Google Scholar
S. Galam and J.-D. Zucker, Physica A 287, 644 (2000). Crossref, Web of Science, Google Scholar
S.-Y. Ha, T. Ha and J.-H. Kim, J. Phys. A: Math. Theor. 43, 315201 (2010). Crossref, Web of Science, Google Scholar
S.-Y. Ha, T. Ha and J.-H. Kim, IEEE Trans. Automat. Control 55, 1679 (2010). Web of Science, Google Scholar
S.-Y. Ha and J.-G. Liu, Commun. Math. Sci. 7, 297 (2009). Crossref, Web of Science, Google Scholar
S.-Y. Ha and M. Slemrod, J. Dynam. Differential Equations 22, 325 (2010). Crossref, Web of Science, Google Scholar
S.-Y. Ha and E. Tadmor, Kinetic Relat. Models 1, 415 (2008). Crossref, Web of Science, Google Scholar
R. Hegselmann and U. Krause, J. Artif. Soc. Social Simulation 5, (2002), http://jasss.soc.surrey.ac.uk/5/3/2.html. Google Scholar
J.-H. Kang, S.-Y. Ha, K.-K. Kang and E. Jeong, How do cultural class emerge from assimilation and distinction? An extension of the Cucker–Smale model, to appear in J. Math. Sociology (available at http://www.math.snu.ac.kr/syha/Distinction-JMS.pdf) . Google Scholar
Y.-X. Li, R. Lukeman and L. Edelstein-Keshet, Physica D 237, 699 (2008). Crossref, Web of Science, Google Scholar
N. P. Mark, Amer. Soc. Rev. 68, 319 (2003). Crossref, Web of Science, Google Scholar
R. Ocrombel, Int. J. Mod. Phys. C 12, 1091 (2010). Link, Web of Science, Google Scholar
J. Shen, SIAM J. Appl. Math. 68, 694 (2008). Crossref, Web of Science, Google Scholar
S. Strogatz , Sync: The Emerging Science of Spontaneous Order ( Hyperion , 2003 ) . Google Scholar
D. J. T. Sumpteret al., Current Biol. 105, 6948 (2008). Google Scholar
K. Sznajd-Weron and J. Sznajd, Int. J. Mod. Phys. C 11, 1157 (2000). Link, Web of Science, Google Scholar
C. M. Topaz and A. L. Bertozzi, SIAM J. Appl. Math. 65, 152 (2004). Crossref, Web of Science, Google Scholar
G. Toscani, Commun. Math. Sci. 4, 481 (2006). Crossref, Web of Science, Google Scholar
A. J. W. Wardet al., PNAS 108, 2312 (2011). Crossref, Web of Science, Google Scholar
W. Weidlich , Sociodynamics: A Systematic Approach to Mathematical Modeling in the Social Sciences ( Harwood Academic , 2000 ) . Google Scholar