This paper studies the emergent dynamics of a flock with nonlinear inherent dynamics under closest neighbors model. We establish sufficient frameworks for convergence to flocking in terms of initial state and system parameters. When the number of closest neighbors is at least half of the population, it is shown that convergence to flocking occurs regardless of the initial state provided that the Lipschitz constant of nonlinear dynamics is smaller than the coupling strength. In contrast, when this number of closest neighbors is less than half of the population, we need to impose some restrictive conditions on the initial state to ensure the emergence of flocking based on the disturbed graphs approach. Our results are applicable to both continuous and discrete time cases. Finally, the validity of our theoretical analysis is tested by numerical simulations.
Yuan LIANG
,
Chen WU
,
Jiugang DONG
. EMERGENT BEHAVIOR OF CLOSEST NEIGHBORS MODEL WITH NONLINEAR INHERENT DYNAMICS[J]. Acta mathematica scientia, Series B, 2025
, 45(4)
: 1640
-1658
.
DOI: 10.1007/s10473-025-0421-2
[1] Bellomo N, Gibelli L, Quaini A, Reali A. Towards a mathematical theory of behavioral human crowds. Math Models Methods Appl Sci, 2022, 32(2): 321-358
[2] Beaver L E, Malikopoulos A A. An overview on optimal flocking. Annu Rev Control, 2021, 51: 88-99
[3] Couzin I D, Krause J, Franks N R, Levin S A. Effective leadership and decision-making in animal groups on the move. Nature, 2005, 433: 513-516
[4] Leonard N E, Paley D A, Lekien F, et al. Collective motion, sensor networks,ocean sampling. Proc IEEE, 2007, 95(1): 48-74
[5] Park J, Choi H H, Lee J R. Flocking-inspired transmission power control for fair resource allocation in vehicle-mounted mobile relay networks. IEEE Trans Veh Technol, 2019, 68(1): 754-764
[6] Perea L, Gómez G, Elosegui P. Extension of the Cucker-Smale control law to space flight formations. J Guidance Control Dynam, 2009, 32(2): 526-536
[7] Reynolds C W. Flocks, herds,schools: A distributed behavioral model. Comput Graphics, 1987, 21(4): 25-34
[8] Paley D A, Leonard N E, Sepulchre R, et al. Oscillator models and collective motion. IEEE Control Syst Mag, 2007, 27(4): 89-105
[9] Toner J, Tu Y. Flocks, herds,schools: A quantitative theory of flocking. Phys Rev E, 1998, 58(4): 4828-4858
[10] Li C H, Yang S Y. A new discrete Cucker-Smale flocking model under hierarchical leadership. Discrete Contin Dyn Syst Ser B, 2016, 21(8): 2587-2599
[11] Yang T, Meng Z, Shi G, et al. Network synchronization with nonlinear dynamics and switching interactions. IEEE Trans Automat Control, 2016, 61(10): 3103-3108
[12] Ballerini M, Cabibbo N, Candelier R, et al. Interaction ruling animal collective behaviour depends on topological rather than metric distance: Evidence from a field study. Proc Natl Acad Sci, 2008, 105(4): 1232-1237
[13] Martin S. Multi-agent flocking under topological interactions. Syst Control Lett, 2014, 69: 53-61
[14] Cucker F, Dong J G. On flocks influenced by closest neighbors. Math Models Methods Appl Sci, 2016, 26(14): 2685-2708
[15] Dong J G, Ha S Y, Kim D. On the Cucker-Smale ensemble with $q$-closest neighbors under time-delayed communications. Kinet Relat Models, 2020, 13(4): 653-676
[16] Dong J G, Ha S Y, Kim D. On the Cucker-Smale ensemble with the $q$-closest neighbors in a self-consistent temperature field. SIAM J Control Optim, 2020, 58(1): 368-392
[17] Mucha P B, Peszek J. A fuzzy $q$-closest alignment model. Nonlinearity, 2024, 37(8): Article 085007
[18] Cao Y, Ren W, Casbeer D W, Schumacher C. Finite-time connectivity-preserving consensus of networked nonlinear agents with unknown Lipschitz terms. IEEE Trans Automat Control, 2016, 61(6): 1700-1705
[19] Su H, Chen G, Wang X, Lin Z. Adaptive second-order consensus of networked mobile agents with nonlinear dynamics. Automatica, 2011, 47(2): 368-375
[20] Yu W, Chen G, Cao M. Consensus in directed networks of agents with nonlinear dynamics. IEEE Trans Automat Control, 2011, 56(6): 1436-1441
[21] Yu W, Chen G, Cao M, Kurths J. Second-order consensus for multiagent systems with directed topologies and nonlinear dynamics. IEEE Trans Syst Man Cybern B Cybern, 2010, 40(3): 881-891
[22] Wu C W.Synchronization in Complex Networks of Nonlinear Dynamical Systems. Singapore: World Scientific, 2007
[23] Dong J G, Qiu L. Flocking of the Cucker-Smale model on general digraphs. IEEE Trans Automat Control, 2017, 62(10): 5234-5239
[24] Lin Z, Francis B, Maggiore M. State agreement for continuous-time coupled nonlinear systems. SIAM J Control Optim, 2007, 46(1): 288-307
[25] Dong J G. Flocks with nonlinear inherent dynamics under fixed and switching digraphs. SIAM J Appl Dyn Syst, 2024, 23(2): 1242-1271
