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doi: 10.3934/dcdsb.2022099
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Delay-dependent flocking dynamics of a two-group coupling system

College of Liberal Arts and Sciences, National University of Defense Technology, Changsha, 410073, China

*Corresponding author: Yicheng Liu

Received  September 2021 Revised  April 2022 Early access June 2022

Fund Project: The first author is supported by the National Natural Science Foundation of China grant 11671011, Postgraduate Scientific Research Innovation Project of Hunan Province grant CX20200011

A group coupling model for a system with large-scale nodes is investigated. The model is formulated as a system of functional differential equations. It incorporates two additional factors that exist in the evolution of flocking behavior, but are often ignored in modeling: (ⅰ) the diversity of interactions, including inter-group and intra-group interactions and (ⅱ) the delayed response of particles to signals from the environment or neighbors, including transmission and processing delays. Theoretically, using the divide-and-conquer method and under different delay factors, sufficient conditions for self-organizing flocking are derived by constructing a dissipative differential inequalities with continuous parameters respectively, which involve some analytical expressions of the upper bound of the delay that the system can tolerate. Results of systematic numerical simulations are presented. They not only validate the analytical results, but hint at a somehow surprising behavior of system, that is, weak flocking behavior occurs when two types of delays coexist.

Citation: Maoli Chen, Yicheng Liu, Xiao Wang. Delay-dependent flocking dynamics of a two-group coupling system. Discrete and Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2022099
References:
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J. Haskovec and I. Markou, Asymptotic flocking in the Cucker-Smale model with reaction-type delays in the non-oscillatory regime, Kinet. Relat. Mod., 13 (2020), 795-813.  doi: 10.3934/krm.2020027.

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J. Juang and Y.-H. Liang, Avoiding collisions in Cucker-Smale flocking models under group-hierarchical multileadership, SIAM J. Appl. Math., 78 (2018), 531-550.  doi: 10.1137/16M1098401.

[30]

W. KinzelA. EnglertG. ReentsM. Zigzag and I. Kanter, Synchronization of networks of chaotic units with time-delayed couplings, Phys. Rev. E, 79 (2009), 056207.  doi: 10.1103/PhysRevE.79.056207.

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[32]

H. LiuX. WangY. Huang and Y. Liu, A new class of fixed-time bipartite flocking protocols for multi-agent systems, Appl. Math. Model., 84 (2020), 501-521.  doi: 10.1016/j.apm.2020.04.016.

[33]

H. LiuX. WangX. Li and Y. Liu, Finite-time flocking and collision avoidance for second-order multi-agent systems, Int. J. Syst. Sci., 51 (2020), 102-115.  doi: 10.1080/00207721.2019.1701133.

[34]

Y. Liu and J. Wu, Flocking and asymptotic velocity of the Cucker-Smale model with processing delay, J. Math. Anal. Appl., 415 (2014), 53-61.  doi: 10.1016/j.jmaa.2014.01.036.

[35]

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S. Motsch and E. Tadmor, A new model for self-organized dynamics and its flocking behavior, J. Stat. Phys., 144 (2011), 923-947.  doi: 10.1007/s10955-011-0285-9.

[37]

X. Mu and Y. He, Hierarchical Cucker-Smale flocking under random interactions with time-varying failure probabilities, J. Franklin. I., 355 (2018), 8723-8742.  doi: 10.1016/j.jfranklin.2018.09.014.

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[43]

W. Ren, On consensus algorithms for double-integrator dynamics, IEEE Trans. Autom. Control, 53 (2008), 1503-1509.  doi: 10.1109/TAC.2008.924961.

[44]

C. W. Reynolds, Flocks, herds and schools: A distributed behavioral model, SIGGRAPH Computer Graphics, 21 (1987), 25-34.  doi: 10.1145/37402.37406.

[45]

L. RuY. Liu and X. Wang, New conditions to avoid collisions in the discrete Cucker-Smale model with singular interactions, Appl. Math. Lett., 114 (2021), 106906.  doi: 10.1016/j.aml.2020.106906.

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X. YinD. Yue and Z. Chen, Asymptotic behavior and collision avoidance in the Cucker-Smale model, IEEE Trans. Autom. Control, 65 (2020), 3112-3119.  doi: 10.1109/TAC.2019.2948473.

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show all references

References:
[1]

S. M. Ahn and S.-Y. Ha, Stochastic flocking dynamics of the Cucker-Smale model with multiplicative white noises, J. Math. Phys., 51 (2010), 103301.  doi: 10.1063/1.3496895.

[2]

M. CaoA. S. Morse and B. D. O. Anderson, Reaching a consensus in a dynamically changing environment: A graphical approach, SIAM J. Control Optim., 47 (2008), 575-600.  doi: 10.1137/060657005.

[3]

J. A. CarrilloY.-P. ChoiP. B. Mucha and J. Peszek, Sharp conditions to avoid collisions in singular Cucker-Smale interactions, Nonlinear Anal. Real World Appl., 37 (2017), 317-328.  doi: 10.1016/j.nonrwa.2017.02.017.

[4]

M. Chen and X. Wang, Flocking dynamics for multi-agent system with measurement delay, Math. Comput. Simulat., 171 (2020), 187-200.  doi: 10.1016/j.matcom.2019.09.015.

[5]

Y.-P. Choi and J. Haskovec, Cucker-Smale model with normalized communication weights and time delay, Kinet. Relat. Mod., 10 (2017), 1011-1033.  doi: 10.3934/krm.2017040.

[6]

Y.-P. ChoiD. KaliseJ. Peszek and A. A. Peters, A collisionless singular Cucker-Smale model with decentralized formation control, SIAM J. Appl. Dyn. Syst., 18 (2019), 1954-1981.  doi: 10.1137/19M1241799.

[7]

Y.-P. Choi and Z. Li, Emergent behavior of Cucker-Smale flocking particles with heterogeneous time delays, Appl. Math. Lett., 86 (2018), 49-56.  doi: 10.1016/j.aml.2018.06.018.

[8]

Y.-P. Choi and S. Salem, Cucker-Smale flocking particles with multiplicative noises: Stochastic mean-field limit and phase transition, Kinet. Relat. Mod., 12 (2019), 573-592.  doi: 10.3934/krm.2019023.

[9]

F. Cucker and J.-G. Dong, On flocks under switching directed interaction topologies, SIAM J. Appl. Math., 79 (2019), 95-110.  doi: 10.1137/18M116976X.

[10]

F. Cucker and J.-G. Dong, A general collision-avoiding flocking framework, IEEE Trans. Autom. Control, 56 (2011), 1124-1129.  doi: 10.1109/TAC.2011.2107113.

[11]

F. Cucker and J.-G. Dong, A conditional, collision-avoiding, model for swarming, Discrete Cont. Dyn.-A, 34 (2014), 1009-1020.  doi: 10.3934/dcds.2014.34.1009.

[12]

F. Cucker and S. Smale, Emergent behavior in flocks, IEEE Trans. Autom. Control, 52 (2007), 852-862.  doi: 10.1109/TAC.2007.895842.

[13]

F. Cucker and S. Smale, On the mathematics of emergence, Jap. J. Math., 2 (2007), 197-227.  doi: 10.1007/s11537-007-0647-x.

[14]

J.-G. DongS.-Y. HaJ. Jung and D. Kim, On the stochastic flocking of the Cucker-Smale flock with randomly switching topologies, SIAM J. Control Optim., 58 (2020), 2332-2353.  doi: 10.1137/19M1279150.

[15]

J.-G. DongS.-Y. HaD. Kim and J. Kim, Time-delay effect on the flocking in an ensemble of thermomechanical Cucker-Smale particles, J. Differ. Equations., 266 (2019), 2373-2407.  doi: 10.1016/j.jde.2018.08.034.

[16]

R. ErbanJ. Haskovec and Y. Z. Sun, A Cucker-Smale model with noise and delay, SIAM J. Appl. Math., 76 (2016), 1535-1557.  doi: 10.1137/15M1030467.

[17]

E. FerranteA. E. TurgutC. HuepeA. StranieriC. Pinciroli and M. Dorigo, Self-organized flocking with a mobile robot swarm: A novel motion control method, Adapt. Behav., 20 (2012), 460-477.  doi: 10.1177/1059712312462248.

[18]

R. C. Fetecau and A. Guo, A mathematical model for flight guidance in honeybee swarms, Bull. Math. Biol., 74 (2012), 2600-2621.  doi: 10.1007/s11538-012-9769-2.

[19]

S.-Y. HaJ. JeongS. E. NohQ. Xiao and X. Zhang, Emergent dynamics of Cucker-Smale flocking particles in a random environment, J. Differ. Equations., 262 (2017), 2554-2591.  doi: 10.1016/j.jde.2016.11.017.

[20]

S.-Y. HaJ. Jung and M. Röckner, Collective stochastic dynamics of the Cucker-Smale ensemble under uncertain communication, J. Differ. Equations., 284 (2021), 39-82.  doi: 10.1016/j.jde.2021.02.046.

[21]

S.-Y. HaJ. Kim and X. Zhang, Uniform stability of the Cucker-Smale model and its application to the mean-field limit, Kinet. Relat. Mod., 11 (2018), 1157-1181.  doi: 10.3934/krm.2018045.

[22]

S.-Y. Ha and J.-G. Liu, A simple proof of the Cucker-Smale flocking dynamics and mean-field limit, Commun. Math. Sci., 7 (2009), 297-325.  doi: 10.4310/CMS.2009.v7.n2.a2.

[23]

S.-Y. Ha and E. Tadmor, From particle to kinetic and hydrodynamic descriptions of flocking, Kinet. Relat. Mod., 1 (2008), 415-435.  doi: 10.3934/krm.2008.1.415.

[24]

S.-Y. HaQ. Xiao and X. Zhang, Emergent dynamics of Cucker-Smale particles under the effects of random communication and incompressible fluids, J. Differ. Equations., 264 (2018), 4669-4706.  doi: 10.1016/j.jde.2017.12.020.

[25]

J. Haskovec, A simple proof of asymptotic consensus in the Hegselmann-Krause and Cucker-Smale models with normalization and delay, SIAM J. Appl. Dyn. Syst., 20 (2021), 130-148.  doi: 10.1137/20M1341350.

[26]

J. Haskovec and I. Markou, Asymptotic flocking in the Cucker-Smale model with reaction-type delays in the non-oscillatory regime, Kinet. Relat. Mod., 13 (2020), 795-813.  doi: 10.3934/krm.2020027.

[27]

J. Haskovec and I. Markou, Exponential asymptotic flocking in the Cucker-Smale model with distributed reaction delays, Math. Biosci. Eng., 17 (2020), 5651-5671.  doi: 10.3934/mbe.2020304.

[28]

Y. HongG. Chen and L. Bushnell, Distributed observers design for leader-following control of multi-agent networks, Automatica, 44 (2008), 846-850.  doi: 10.1016/j.automatica.2007.07.004.

[29]

J. Juang and Y.-H. Liang, Avoiding collisions in Cucker-Smale flocking models under group-hierarchical multileadership, SIAM J. Appl. Math., 78 (2018), 531-550.  doi: 10.1137/16M1098401.

[30]

W. KinzelA. EnglertG. ReentsM. Zigzag and I. Kanter, Synchronization of networks of chaotic units with time-delayed couplings, Phys. Rev. E, 79 (2009), 056207.  doi: 10.1103/PhysRevE.79.056207.

[31]

Z. Li and X. Xue, Cucker-Smale flocking under rooted leadership with fixed and switching topologies, SIAM J. Appl. Math., 70 (2010), 3156-3174.  doi: 10.1137/100791774.

[32]

H. LiuX. WangY. Huang and Y. Liu, A new class of fixed-time bipartite flocking protocols for multi-agent systems, Appl. Math. Model., 84 (2020), 501-521.  doi: 10.1016/j.apm.2020.04.016.

[33]

H. LiuX. WangX. Li and Y. Liu, Finite-time flocking and collision avoidance for second-order multi-agent systems, Int. J. Syst. Sci., 51 (2020), 102-115.  doi: 10.1080/00207721.2019.1701133.

[34]

Y. Liu and J. Wu, Flocking and asymptotic velocity of the Cucker-Smale model with processing delay, J. Math. Anal. Appl., 415 (2014), 53-61.  doi: 10.1016/j.jmaa.2014.01.036.

[35]

I. Markou, Collision-avoiding in the singular Cucker-Smale model with nonlinear velocity couplings, Discrete Cont. Dyn.-A, 38 (2018), 5245-5260.  doi: 10.3934/dcds.2018232.

[36]

S. Motsch and E. Tadmor, A new model for self-organized dynamics and its flocking behavior, J. Stat. Phys., 144 (2011), 923-947.  doi: 10.1007/s10955-011-0285-9.

[37]

X. Mu and Y. He, Hierarchical Cucker-Smale flocking under random interactions with time-varying failure probabilities, J. Franklin. I., 355 (2018), 8723-8742.  doi: 10.1016/j.jfranklin.2018.09.014.

[38]

K.-K. OhM.-C. Park and H.-S. Ahn, A survey of multi-agent formation control, Automatica, 53 (2015), 424-440.  doi: 10.1016/j.automatica.2014.10.022.

[39]

R. Olfati-Saber, Flocking for multi-agent dynamic systems: Algorithms and theory, IEEE Trans. Autom. Control, 51 (2006), 401-420.  doi: 10.1109/TAC.2005.864190.

[40]

P.-Y. Oudeyer, Self-organization: Complex dynamical systems in the evolution of speech, The Language Phenomenon Springer(Ed), 2013,191–216. doi: 10.1007/978-3-642-36086-2_9.

[41]

L. M. Pecora and T. L. Carroll, Synchronization in chaotic systems, Phys. Rev. Lett., 64 (1990), 821-824.  doi: 10.1103/PhysRevLett.64.821.

[42]

C. Pignotti and E. Trélat, Convergence to consensus of the general finite-dimensional Cucker-Smale model with time-varying delays, Commun. Math. Sci., 16 (2018), 2053-2076.  doi: 10.4310/CMS.2018.v16.n8.a1.

[43]

W. Ren, On consensus algorithms for double-integrator dynamics, IEEE Trans. Autom. Control, 53 (2008), 1503-1509.  doi: 10.1109/TAC.2008.924961.

[44]

C. W. Reynolds, Flocks, herds and schools: A distributed behavioral model, SIGGRAPH Computer Graphics, 21 (1987), 25-34.  doi: 10.1145/37402.37406.

[45]

L. RuY. Liu and X. Wang, New conditions to avoid collisions in the discrete Cucker-Smale model with singular interactions, Appl. Math. Lett., 114 (2021), 106906.  doi: 10.1016/j.aml.2020.106906.

[46]

J. Shen, Cucker-Smale flocking under hierarchical leadership, SIAM J. Appl. Math., 68 (2007), 694-719.  doi: 10.1137/060673254.

[47]

Y. SunY. Wang and D. Zhao, Flocking of multi-agent systems with multiplicative and independent measurement noises, Physica A, 440 (2015), 81-89.  doi: 10.1016/j.physa.2015.08.005.

[48]

T. VicsekA. CzirókE. Ben-JacobI. Cohen and O. Shochet, Novel type of phase transition in a system of self-driven particles, Phys. Rev. Lett., 75 (1995), 1226-1229.  doi: 10.1103/PhysRevLett.75.1226.

[49]

X. WangL. Wang and J. Wu, Impacts of time delay on flocking dynamics of a two-agent flock model, Commun. Nonlinear. Sci., 70 (2019), 80-88.  doi: 10.1016/j.cnsns.2018.10.017.

[50]

Q. XiaoH. LiuX. Wang and Y. Huang, A note on the fixed-time bipartite flocking for nonlinear multi-agent systems, Appl. Math. Lett., 99 (2020), 105973.  doi: 10.1016/j.aml.2019.07.004.

[51]

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Figure 1.  Schematic diagram of a two-group coupling system subject to processing delay $ \tau_1 $ and transmission delay $ \tau_2 $
Figure 2.  Schematic diagram of flocking results under Scenario 1
Figure 3.  Schematic diagram of particle velocity evolution in Scenario 1 with large time delay
Figure 4.  Schematic diagram of flocking results under Scenario 2
Figure 5.  Schematic diagram of particle velocity evolution in Scenario 2 with large time delay
Figure 6.  Schematic diagram of the relationship between the upper bounds of the two types of time delay and the system parameters. To ensure the flocking behavior, the upper bound of the transmission delay allowed by the system decreases with the increase of the position diameter margin $ d $, and the upper bound of the processing delay allowed by the system decreases with the increase of the total number of members $ N $
Figure 7.  When the transmission delay and the processing delay coexist, a weak flocking phenomenon occurs in system (2)-(3)
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