Flocking and synchronization of particle models
Authors:
Seung-Yeal Ha, Corrado Lattanzio, Bruno Rubino and Marshall Slemrod
Journal:
Quart. Appl. Math. 69 (2011), 91-103
MSC (2000):
Primary 92D25, 74A25, 76N10
DOI:
https://doi.org/10.1090/S0033-569X-2010-01200-7
Published electronically:
December 9, 2010
MathSciNet review:
2807979
Full-text PDF Free Access
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Additional Information
Abstract: In this note, we present a multi-dimensional flocking model rigorously derived from a vector oscillatory chain model and study the connection between the Cucker-Smale flocking model and the Kuramoto synchronization model appearing in the statistical mechanics of nonlinear oscillators. We provide an alternative direct approach for frequency synchronization to the Kuramoto model as an application of the flocking estimate for the Cucker-Smale model.
References
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- Zvi Artstein and Alexander Vigodner, Singularly perturbed ordinary differential equations with dynamic limits, Proc. Roy. Soc. Edinburgh Sect. A 126 (1996), no. 3, 541–569. MR 1396278, DOI https://doi.org/10.1017/S0308210500022903
- Carrillo, J. A., Fornasier, M., Rosado, J. and Toscani, G.: Asymptotic flocking dynamics for the kinetic Cucker-Smale model. Preprint.
- Felipe Cucker and Steve Smale, On the mathematics of emergence, Jpn. J. Math. 2 (2007), no. 1, 197–227. MR 2295620, DOI https://doi.org/10.1007/s11537-007-0647-x
- Felipe Cucker and Steve Smale, Emergent behavior in flocks, IEEE Trans. Automat. Control 52 (2007), no. 5, 852–862. MR 2324245, DOI https://doi.org/10.1109/TAC.2007.895842
- Erdmann, U., Ebeling, W. and Mikhailov, A.: Noise-induced transition from translational to rotational motion of swarms. Phys. Review E 71, 051904 (2005).
- Seung-Yeal Ha and Jian-Guo Liu, A simple proof of the Cucker-Smale flocking dynamics and mean-field limit, Commun. Math. Sci. 7 (2009), no. 2, 297–325. MR 2536440
- Seung-Yeal Ha and Eitan Tadmor, From particle to kinetic and hydrodynamic descriptions of flocking, Kinet. Relat. Models 1 (2008), no. 3, 415–435. MR 2425606, DOI https://doi.org/10.3934/krm.2008.1.415
- 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).
- Y. Kuramoto, Chemical oscillations, waves, and turbulence, Springer Series in Synergetics, vol. 19, Springer-Verlag, Berlin, 1984. MR 762432
- Robert E. O’Malley Jr., Singular perturbation methods for ordinary differential equations, Applied Mathematical Sciences, vol. 89, Springer-Verlag, New York, 1991. MR 1123483
- Jackie Shen, Cucker-Smale flocking under hierarchical leadership, SIAM J. Appl. Math. 68 (2007/08), no. 3, 694–719. MR 2375291, DOI https://doi.org/10.1137/060673254
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- Ferdinand Verhulst, Methods and applications of singular perturbations, Texts in Applied Mathematics, vol. 50, Springer, New York, 2005. Boundary layers and multiple timescale dynamics. MR 2148856
References
- Acebron, J. A. and Bonilla, L. L.: The Kuramoto model: A simple paradigm for synchronization phenomena. Rev. Mod. Phys. 77, 137-185 (2005).
- Artstein, Z. and Vigodner, A.: Singularly perturbed ordinary differential equations with dynamic limits. Proc. Roy. Soc. Edinburgh Sec. A. 126 (1996), 541-569. MR 1396278 (97g:34073)
- Carrillo, J. A., Fornasier, M., Rosado, J. and Toscani, G.: Asymptotic flocking dynamics for the kinetic Cucker-Smale model. Preprint.
- Cucker, F. and Smale, S.: On the mathematics of emergence. Japan. J. Math. 2, 197-227 (2007). MR 2295620 (2007m:91126)
- Cucker, F. and Smale, S.: Emergent behavior in flocks. IEEE Trans. Automat. Control 52, 852-862 (2007). MR 2324245 (2008h:91132)
- Erdmann, U., Ebeling, W. and Mikhailov, A.: Noise-induced transition from translational to rotational motion of swarms. Phys. Review E 71, 051904 (2005).
- 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). MR 2536440
- Ha, S.-Y. and Tadmor, E.: From particle to kinetic and hydrodynamic description of flocking. Kinetic and Related Models. 1, 415-435 (2008). MR 2425606
- 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).
- Kuramoto, Y.: Chemical oscillations, waves and turbulence. Springer-Verlag, Berlin, 1984. MR 762432 (87e:92054)
- O’Malley, R. E.: Singular perturbation methods for ordinary differential equations. Applied Mathematical Sciences, 89. Springer-Verlag, New York, 1991. MR 1123483 (92i:34071)
- Shen, J.: Cucker-Smale flocking under hierarchical leadership. SIAM J. Appl. Math. 68, 694-719 (2007/08). MR 2375291 (2008k:92066)
- Tikhonov, A. N.: Systems of differential equations containing a small parameter multiplying the derivative. Mat. Sb. N.S. 27, 147-156 (1950). MR 0036902 (12:181d)
- Verhulst, F.: Methods and applications of singular perturbations. Boundary layers and multiple timescale dynamics. Texts in Applied Mathematics, 50, Springer, New York, 2005. MR 2148856 (2006k:34001)
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Additional Information
Seung-Yeal Ha
Affiliation:
Department of Mathematical Sciences, Seoul National University, Seoul 151-747, Korea
MR Author ID:
684438
Email:
syha@snu.ac.kr
Corrado Lattanzio
Affiliation:
Department of Pure and Applied Mathematics, University of L’Aquila, loc. Coppito, 67010 L’Aquila, Italy
Email:
corrado.lattanzio@univaq.it
Bruno Rubino
Affiliation:
Department of Pure and Applied Mathematics, University of L’Aquila, loc. Coppito, 67010 L’Aquila, Italy
Email:
bruno.rubino@univaq.it
Marshall Slemrod
Affiliation:
Department of Mathematics, University of Wisconsin-Madison, Wisconsin 53706-1388
MR Author ID:
163635
Email:
slemrod@math.wisc.edu
Keywords:
Flocking,
particles,
mechanical model,
synchronization
Received by editor(s):
July 17, 2009
Published electronically:
December 9, 2010
Article copyright:
© Copyright 2010
Brown University
