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Flocking with General Local Interaction and Large Population

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Abstract

This paper studies a flocking model in which the interaction between agents is described by a general local nonlinear function depending on the distance between agents. The existing analysis provided sufficient conditions for flocking under an assumption imposed on the system’s closed-loop states; however this assumption is hard to verify. To avoid this kind of assumption the authors introduce some new methods including large deviations theory and estimation of spectral radius of random geometric graphs. For uniformly and independently distributed initial states, the authors establish sufficient conditions and necessary conditions for flocking with large population. The results reveal that under some conditions, the critical interaction radius for flocking is almost the same as the critical radius for connectivity of the initial neighbor graph.

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References

  1. Toner J and Tu Y, Flocks, herds, and schools: A quantitative theory of flocking, Phys. Rev. E, 1998, 58(4): 4828–4858.

    Article  MathSciNet  Google Scholar 

  2. Reynolds C, Flocks, herds, and schools: A distributed behavioral model, Computer Graphics, 1987, 21: 25–34.

    Article  Google Scholar 

  3. Vicsek T, Czirók A, Jacob E B, et al., Novel type of phase transition in a system of self-driven particles, Phys. Rev. Lett., 1995, 75: 1226–1229.

    Article  MathSciNet  Google Scholar 

  4. Buhl J, Sumpter D J T, Couzin I D, et al., From disorder to order in marching locusts, Science, 2006, 312(5778): 1402–1406.

    Article  Google Scholar 

  5. Chazelle B, The convergence of bird flocking, Journal of the ACM, 2014, 61(4): 1–35.

    Article  MathSciNet  Google Scholar 

  6. Olfati-Saber R, Flocking for multi-agent dynamic systems: Algorithms and theory, IEEE Trans. Autom. Control, 2006, 51(3): 401–420.

    Article  MathSciNet  Google Scholar 

  7. Jadbabaie A, Lin J, and Morse A S, Coordination of groups of mobile autonomous agents using nearest neighbor rules, IEEE Trans. Autom. Control, 2003, 48(9): 988–1001.

    Article  MathSciNet  Google Scholar 

  8. Savkin A V, Coordinated collective motion of groups of autonomous mobile robots: Analysis of Vicsek’s model, IEEE Trans. Autom. Control, 2004, 39: 981–983.

    Article  MathSciNet  Google Scholar 

  9. Li Q and Jiang Z P, Global analysis of multi-agent systems based on Vicsek’s model, IEEE Trans. Auto. Control, 2009, 54(12): 2876–2881.

    Article  MathSciNet  Google Scholar 

  10. Chen G, Small noise may diversify collective motion in Vicsek model, IEEE Trans. Auto. Control, 2017, 62(2): 636–651.

    Article  MathSciNet  Google Scholar 

  11. Cucker F and Smale S, Emergent behavior in flocks, IEEE Trans. Autom. Control, 2007, 52(5): 852–862.

    Article  MathSciNet  Google Scholar 

  12. Cucker F and Mordecki E, Flocking in noisy environments, J. Math. Pures Appl., 2008, 89: 278–296.

    Article  MathSciNet  Google Scholar 

  13. Cucker F and Dong J G, A general collision-avoiding flocking framework, IEEE Trans. Autom. Control, 2011, 56: 1124–1129.

    Article  MathSciNet  Google Scholar 

  14. Peszek J, Existence of piecewise weak solutions of a discrete Cucker-Smale’s flocking model with a singular communication weight, Journal of Differential Equations, 2014, 257(8): 2900–2925.

    Article  MathSciNet  Google Scholar 

  15. Carrillo J A, Fornasier M, Rosado J, et al., Asymptotic flocking dynamics for the kinetic Cucker- Smale model, SIAM J. Math. Anal., 2010, 42(1): 218–236.

    Article  MathSciNet  Google Scholar 

  16. Ahn S M and Ha S, Stochastic flocking dynamics of the Cucker-Smale model with multiplicative white noises, Journal of Mathematical Physics, 2010, 51(10): 1634–1642.

    Article  MathSciNet  Google Scholar 

  17. Shen J, Cucker-Smale flocking under hierarchical leadership, SIAM J. Appl. Math., 2007, 68(3): 694–719.

    Article  MathSciNet  Google Scholar 

  18. Park J, Kim H J, and Ha S Y, Cucker-Smale flocking with inter-particle bonding forces, IEEE Trans. Auto. Control, 2010, 55(11): 2617–2623.

    Article  MathSciNet  Google Scholar 

  19. Ha S Y, Ha T, and Kim J H, Emergent behavior of a Cucker-Smale type particle model with nonlinear velocity couplings, IEEE Trans. Autom. Control, 2010, 55(7): 1679–1683.

    Article  MathSciNet  Google Scholar 

  20. Ha S Y, Jeong J, Noh S E, et al., Emergent dynamics of Cucker-Smale flocking particles in a random environment, Journal of Differential Equations, 2017, 262(3): 2554–2591.

    Article  MathSciNet  Google Scholar 

  21. Rosenthal S B, Twomey C R, Hartnett A T, et al., Revealing the hidden networks of interaction in mobile animal groups allows prediction of complex behavioral contagion, P. Natl. Acad. Sci. USA, 2015, 112(15): 4690–4695.

    Article  Google Scholar 

  22. Rieu J P, Upadhyaya A, Glazier J A, et al., Diffusion and deformations of single Hydra cells in cellular aggregates, Biophys. J., 2000, 79(4): 1903–1914.

    Article  Google Scholar 

  23. Rieu J P, Kataoka N, and Sawada Y, Quantitative analysis of cell motion during sorting in two-dimensional aggregates of dissociated Hydra cells, Phys. Rev. E, 1998, 57(1): 924–931.

    Article  Google Scholar 

  24. Martin S, Girard A, Fazeli A, et al., Multiagent flocking under general communication rule, IEEE Transactions on Control of Network Systems, 2014, 1(2): 155–166.

    Article  MathSciNet  Google Scholar 

  25. Tang G G and Guo L, Convergence of a class of multi-agent systems in probabilistic framework, Journal of Systems Science and Complexity, 2007, 20(2): 173–197.

    Article  MathSciNet  Google Scholar 

  26. Liu Z X and Guo L, Synchronization of multi-agent systems without connectivity assumption, Automatica, 2009, 45: 2744–2753.

    Article  MathSciNet  Google Scholar 

  27. Chen G, Liu Z X, and Guo L, The smallest possible interaction radius for synchronization of self-propelled particles, SIAM Rev., 2014, 56(3): 499–521.

    Article  MathSciNet  Google Scholar 

  28. Penrose M D, Random Geometric Graphs, Oxford University Press, Oxford, UK, 2003.

    Book  Google Scholar 

  29. Gupta P and Kumar P R, The capacity of wireless networks, IEEE Trans. Inform. Theory, 2000, 46: 388–404.

    Article  MathSciNet  Google Scholar 

  30. Dembo A and Zeitouni O, Large Deviations Techniques and Applications, 2nd Edition, Springer, New York, 1998.

    Book  Google Scholar 

  31. Gupta P and Kumar P R, Critical power for asymptotic connectivity in wireless networks, Stochastic Analysis, Control, Optimization and Applications, Birkhäuser Boston, Boston, MA, 1999, 547–566.

    Chapter  Google Scholar 

  32. Penrose M D, The longest edge of the random minimal spanning tree, Ann. Appl. Probab., 1997, 7(2): 340–361.

    Article  MathSciNet  Google Scholar 

  33. Diaconis P and Strook D, Geometric bounds for eigenvalues ofMarkov chains, Ann. Appl. Probab., 1991, 1: 36–61.

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Key Laboratory of Systems and Control & National Center for Mathematics and Interdisciplinary Sciences, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, China

    Ge Chen & Zhixin Liu

Authors
  1. Ge Chen
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  2. Zhixin Liu
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Corresponding author

Correspondence to Ge Chen.

Additional information

The research was supported by the National Natural Science Foundation of China under Grant No. 11688101, 91634203, 91427304, and 61673373, and the National Key Basic Research Program of China (973 Program) under Grant No. 2016YFB0800404. Part of this paper was presented in the 10th World Congress on Intelligent Control and Automation, pp. 3515–3519, July 6–8, 2012, Beijing, China.

This paper was recommended for publication by Editor CHEN Jie.

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Chen, G., Liu, Z. Flocking with General Local Interaction and Large Population. J Syst Sci Complex 32, 1498–1525 (2019). https://doi.org/10.1007/s11424-019-7407-x

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  • DOI: https://doi.org/10.1007/s11424-019-7407-x

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