For a collective system, the connectedness of the adjacency matrix plays a key role in making the system achieve its emergent feature, such as flocking or multi-clustering. In this paper, we study a nonsymmetric multi-particle system with a constant and local cut-off weight. A distributed communication delay is also introduced into both the velocity adjoint term and the cut-off weight. As a new observation, we show that the desired multi-particle system undergoes both flocking and clustering behaviors when the eigenvalue 1 of the adjacency matrix is semi-simple. In this case, the adjacency matrix may lose the connectedness. In particular, the number of clusters is discussed by using subspace analysis. In terms of results, for both the non-critical and general neighbourhood situations, some criteria of flocking and clustering emergence with an exponential convergent rate are established by the standard matrix analysis for when the delay is free. As a distributed delay is involved, the corresponding criteria are also found, and these small time lags do not change the emergent properties qualitatively, but alter the final value in a nonlinear way. Consequently, some previous works [14] are extended.
Yicheng LIU
. EXPONENTIAL STABILITY OF A MULTI-PARTICLE SYSTEM WITH LOCAL INTERACTION AND DISTRIBUTED DELAY[J]. Acta mathematica scientia, Series B, 2022
, 42(5)
: 2165
-2187
.
DOI: 10.1007/s10473-022-0525-x
[1] Atay F M. The consensus problem in networks with transmission delays. Phil Trans Roy Soc A, 2013, 371: 20120460
[2] Bellomo N, Degond P, Tadmor E, eds. Active Particles Vol I: Advances in Theory, Models, and Applications. Modelling and Simulation in Science and Technology. Basel: Birkhäuser, 2017
[3] Choi Y, Haskovec J. Cucker-Smale model with normalized communication weights and time delay. Kinet Relat Models, 2017, 10: 1011–1033
[4] Choi Y, Li Z. Emergent behavior of Cucker-Smale flocking particles with heterogeneous time delays. Appl Math Lett, 2018, 86: 49–56
[5] Cucker F, Smale S. Emergent behavior in flocks. IEEE Trans Automat Control, 2007, 52: 852–862
[6] Cucker F, Smale S. On the mathematics of emergence. Jpn J Math, 2007, 2 197–227
[7] Dong J, Ha S-Y, Kim D, Kim J. Time-delay effect on the flocking in an ensemble of thermomechanical Cucker-Smale particles. J Differential Equations, 2019, 266: 2373–2407
[8] Erban R, Haskovec J, Sun Y. A Cucker-Smale model with noise and delay. SIAM J Appl Math, 2016, 76: 1535–1557
[9] Ha S-Y, Lee K, Levy D. Emergence of time-asymptotic flocking in a stochastic Cucker-Smale system. Commun Math Sci, 2009, 7: 453–469
[10] Hale J K, Lunel S M V. Introduction to Functional Differential Equations. Berlin: Springer-Verlag, 1993
[11] Haskovec J, Markou I. Delayed Cucker-Smale model with and without noise revisited. arXiv:1810.01084v2
[12] Jabin P-E, Motsch S. Clustering and asymptotic behavior in opinion formation. J Differ Equ, 2014, 257: 4165–4187
[13] Jadbabaie A, Jie L, Morse A S. Coordination of groups of mobile autonomous agents using nearest neighbour rules. IEEE Trans Automat Control, 2003, 48: 988–1001
[14] Jin C. Flocking of the motsch-tadmor model with a cut-off interaction function. J Stat Phys, 2018, 171: 345–360
[15] Ke X, Yuan B, Xiao Y. A Stability problem for the 3D magnetohydrodynamic equations near equilibrium. Acta Math Sci, 2021, 41B: 1107–1118
[16] Kumar R, Sharma A K, Sahu G P. Dynamical behavior of an innovation diffusion model with intra-specific competition between competing adopters. Acta Math Sci, 2022, 42B: 364–386
[17] Li X, Liu Y, Wu J. Flocking and pattern motion in a modified Cucker-Smale model. Bull Korean Math Soc, 2016, 53: 1327–1339
[18] Li Z, Xue X. Cucker-Smale flocking under rooted leadership with fixed and switching topologies. SIAM J Appl Math, 2010, 70: 3156–3174
[19] Liu Y, Wu J. Flocking and asymptotic velocity of the Cucker-Smale model with processing delay. J Math Anal Appl, 2014, 415: 53–61
[20] Liu Y, Wu J. Local phase synchronization and clustering for the delayed phase-coupled oscillators with plastic coupling. J Math Anal Appl, 2016, 444: 947–956
[21] Liu Y, Wu J. Opinion consensus with delay when the zero eigenvalue of the connection matrix is semi-simple. Journal of Dynamics and Differential Equations, 2017, 29: 1539–1551
[22] Motsch S, Tadmor E. A new model for self-organized dynamics and its flocking behavior. J Stat Phys, 2011, 144: 923–947
[23] Pignotti C, Trelat E. Convergence to consensus of the general finite-dimensional Cucker-Smale model with time-varying delays. Commun Math Sci, 2017, 16: 2053–2076
[24] Pignotti C, Vallejo I. Flocking estimates for the Cucker-Smale model with time lag and hierarchical leadership. J Math Anal Appl, 2018, 464: 1313–1332
[25] Ru L, Xue X. Multi-cluster flocking behavior of the hierarchical Cucker-Smale model. J Franklin Inst, 2017, 354: 2371–2392
[26] Serre D. Matrices. Graduate Texts in Mathematics 216. Springer, 2010
[27] Shi L, Cheng Y, Huang J, Shao J. Cucker-Smale flocking under rooted leadership and time-varying heterogeneous delays. Appl Math Lett, 2019, 98: 453–460
[28] Vicsek T, Zafeiris A. Collective motion. Physics Reports, 2012, 517: 71–140
[29] Yang J, Wang L. Dynamics analysis of a delayed HIV infection model with CTL immune response and antibody immune response. Acta Math Sci, 2021, 41B: 991–1016
