Cucker-smale flocking under hierarchical leadership

SIAM Journal on Applied Mathematics

Volumn 68, Issue 3, 2007, Pages 694-719

  Shen, Jackie ^a  

^a NONE   (United States)

Author keywords

Bio flocking; Convergence; Cucker smale model; Dynamic graphs; Fiedler number; Free will; Graph laplacian; Leaders; Leadership; Perturbation

Indexed keywords

APPLIED MATHEMATICS; CONVERGENCE (MATHEMATICS); EMERGENT BEHAVIORS; EVOLUTION (CO); MATHEMATICAL THEORIES; MULTI AGENTS; RATES OF CONVERGENCE; SOCIAL BEHAVIORS;

SENSORS;

SENSOR NETWORKS;

EID: 47149118340     PISSN: 00361399     EISSN: None     Source Type: Journal    
DOI: 10.1137/060673254     Document Type: Article

References (27)

1. D. CHANDLER, Introduction to Modern Statistical Mechanics, Oxford University Press, New York, Oxford, 1987.
2. Y.-L. CHUANG, M. R. D'ORSOGNA, D. MARTHALER, A. L. BERTOZZI, AND L. CHAYES, State transitions and the continuum limit for a 2D interacting, self-propelled particle system, Phys. D, submitted.
3. F. R. K. CHUNG, Spectral Graph Theory, AMS, Providence, RI, 1997.
4. I. D. COUZIN, J. KRAUSE, N. R. FRANKS, AND S. LEVIN, Effective leadership and decision making in animal groups on the move, Nature, 433 (2005), pp. 513-516.
5. T. M. COVER AND J. A. THOMAS, Elements of Information Theory, John Wiley & Sons, New York, 1991.
6. F. CUCKER AND S. SMALE, On the mathematical foundations of learning, Bull. Amer. Math. Soc. (N.S.), 39 (2001), pp. 1-49.
7. F. CUCKER AND S. SMALE, Emergent behavior in flocks, IEEE Trans. Automat. Control, 52 (2007), pp. 852-862.
8. F. CUCKER AND S. SMALE, Lectures on emergence, Japan J. Math., 2 (2007), pp. 197-227.
9. F. CUCKER, S. SMALE, AND D. ZHOU, Modeling language evolution, Found. Comput. Math., 4 (2004), pp. 315-343.
10. I. DAUBECHIES AND J. C. LACARIAS, Sets of matrices all infinite products of which converge, Linear Algebra Appl., 161 (1992), pp. 227-263.
11. M. R. D'ORSOGNA, Y.-L. CHUANG, A. L. BERTOZZI, AND L. CHAYES, Self-propelled particles with soft-core interactions: Patterns, stability, and collapse, Phys. Rev. Lett., 96 (2006), 104302.
12. G. FLIERL, D. GRÜNBAUM, S. LEVIN, AND D. OLSON, From individuals to aggregations: The interplay between behavior and physics, J. Theoret. Biol., 196 (1999), pp. 397-454.
13. M. W. HIRSCH, S. SMALE, AND R. DEVANEY, Differential Equations, Dynamical Systems, and an Introduction to Chaos, 2nd ed., Academic Press, New York, 2003.
14. A. JADBABAIE, J. LIN, AND A. S. MORSE, Coordination of groups of mobile autonomous agents using nearest neighbor rules, IEEE Trans. Automat. Control, 48 (2003), pp. 988-1001.
15. H. LEVINE AND W.-J. RAPPEL, Self-organization in systems of self-propelled particles, Phys. Rev. E, 63 (2000), article 017101.
16. E. H. LIEB AND M. LOSS, Analysis, 2nd ed., AMS, Providence, RI, 2001.
17. Y. LIU AND K. PASSINO, Stable social foraging swarms in a noisy environment, IEEE Trans. Automat. Control, 49 (2004), pp. 30-44.
18. J. K. PARRISH, S. V. VISCIDO, AND D. GRÜNBAUM, Self-organized fish schools: An examination of emergent properties, Biol. Bull., 202 (2002), pp. 296-305.
19. G.-C. ROTA AND G. STRANG, A note on the joint spectral radius, Indag. Math., 22 (1960), pp. 379-381.
20. J. SHEN, Compactification of a set of matrices with convergent infinite products, Linear Algebra Appl., 311 (2000), pp. 177-186.
21. J. SHEN, A geometric approach to ergodic non-homogeneous Markov chains, in Wavelet Analysis and Multiresolution Methods, Lecture Notes in Pure and Appl. Math. 212, Marcel Dekker, New York, 2000, pp. 341-366.
22. G. STRANG AND T. NGUYEN, Wavelets and Filter Banks, Wellesley-Cambridge Press, Wellesley, MA, 1996.
23. C. M. TOPAZ AND A. L. BERTOZZI, Swarming patterns in a two-dimensional kinematic model for biological groups, SIAM J. Appl. Math., 65 (2004), pp. 152-174.
24. C. M. TOPAZ, A. L. BERTOZZI, AND M. A. LEWIS, A nonlocal continuum model for biological aggregation, Bull. Math. Bio., 68 (2006), pp. 1601-1623.
25. J. N. TSITSIKLIS, D. P. BERTSEKAS, AND M. ATHANS, Distributed asynchronous deterministic and stochastic gradient optimization algorithms, IEEE Trans. Automat. Control, AC-31 (1986), pp. 803-812.
26. V. VAPNIK, The Nature of Statistical Learning Theory, Springer-Verlag, New York, 1995.
27. T. VICSEK, A. CZIRÓK, E. BEN- JACOB, I. COHEN, AND O. SCHOCHET, Novel type of phase transition transition in a system of self-driven particles, Phys. Rev. Lett., 75 (1995), pp. 1226-1229.