Flocking and synchronization of particle models

Quarterly of Applied Mathematics

Volumn 69, Issue 1, 2011, Pages 91-103

  Ha, Seung Yeal ^a   Lattanzio, Corrado ^b   Rubino, Bruno ^b   Slemrod, Marshall ^c  

^a Seoul National University   (South Korea)

^b UNIVERSITY OF L'AQUILA   (Italy)

^c UNIVERSITY OF WISCONSIN MADISON   (United States)

Author keywords

Flocking; Mechanical Model; Particles; Synchronization

Indexed keywords

ELEMENTARY PARTICLES; STATISTICAL MECHANICS;

FLOCKING; FREQUENCY SYNCHRONIZATION; KURAMOTO MODELS; MECHANICAL MODEL; MULTI DIMENSIONAL; NON-LINEAR OSCILLATORS; OSCILLATORY CHAINS; SYNCHRONIZATION MODELS;

SYNCHRONIZATION;

EID: 79952583203     PISSN: 0033569X     EISSN: 15524485     Source Type: Journal    
DOI: 10.1090/s0033-569x-2010-01200-7     Document Type: Article

References (14)

1. Acebron, J.A. and Bonilla, L.L.: The Kuramoto model: A simple paradigm for synchronization phenomena. Rev. Mod. Phys. 77, 137-185 (2005).
2. Artstein, Z. and Vigodner, A.: Singularly perturbed ordinary differential equations with dynamic limits. Proc. Roy. Soc. Edinburgh Sec. A. 126 (1996), 541-569.
3. Carrillo, J.A., Fornasier, M., Rosado, J. and Toscani, G.: Asymptotic flocking dynamics for the kinetic Cucker-Smale model. Preprint.
4. Cucker, F. and Smale, S.: On the mathematics of emergence. Japan. J. Math. 2, 197-227 (2007).
5. Cucker, F. and Smale, S.: Emergent behavior in flocks. IEEE Trans. Automat. Control 52, 852-862 (2007).
6. Erdmann, U., Ebeling, W. and Mikhailov, A.: Noise-induced transition from translational to rotational motion of swarms. Phys. Review E 71, 051904 (2005).
7. Ha, S.-Y. and Liu, J.-G.: A simple proof of the Cucker-Smale flocking dynamics and mean-field limit. Commun. Math. Sci. 7, 297-325 (2009).
8. Ha, S.-Y. and Tadmor, E.: From particle to kinetic and hydrodynamic description of flocking. Kinetic and Related Models. 1, 415-435 (2008).
9. Ha, S.-Y. and Slemrod, M.: Flocking dynamics of singularly perturbed oscillator chain and the Cucker-Smale system. J. Dyna. Differential Equations. 21, 371-376 (2009).
10. Kuramoto, Y.: Chemical oscillations, waves and turbulence. Springer-Verlag, Berlin, 1984.
11. O'Malley, R.E.: Singular perturbation methods for ordinary differential equations. Applied Mathematical Sciences, 89. Springer-Verlag, New York, 1991.
12. Shen, J.: Cucker-Smale flocking under hierarchical leadership. SIAM J. Appl. Math. 68, 694-719 (2007/08).
13. Tikhonov, A.N.: Systems of differential equations containing a small parameter multiplying the derivative. Mat. Sb. N.S. 27, 147-156 (1950).
14. Verhulst, F.: Methods and applications of singular perturbations. Boundary layers and multiple timescale dynamics. Texts in Applied Mathematics, 50, Springer, New York, 2005.