American Institute of Mathematical Sciences

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July  2021, 26(7): 3693-3716. doi: 10.3934/dcdsb.2020253

Flocking and line-shaped spatial configuration to delayed Cucker-Smale models

Zhisu Liu , Yicheng Liu , and  Xiang Li

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

* Corresponding author: Yicheng Liu

Received  July 2019 Revised  February 2020 Published  August 2020

Fund Project: This work was supported by the NSFC (11701267;11671011;11801562) and Hunan Natural Science Excellent Youth Fund (2020JJ3029)

As we known, it is popular for a designed system to achieve a prescribed performance, which have remarkable capability to regulate the flow of information from distinct and independent components. Also, it is not well understand, in both theories and applications, how self propelled agents use only limited environmental information and simple rules to organize into an ordered motion. In this paper, we focus on analysis the flocking behaviour and the line-shaped pattern for collective motion involving time delay effects. Firstly, we work on a delayed Cucker-Smale-type system involving a general communication weight and a constant delay $ \tau>0 $ for modelling collective motion. In a result, by constructing a new Lyapunov functional approach, combining with two delayed differential inequalities established by $ L^2 $-analysis, we show that the flocking occurs for the general communication weight when $ \tau $ is sufficiently small. Furthermore, to achieve the prescribed performance, we introduce the line-shaped inner force term into the delayed collective system, and analytically show that there is a flocking pattern with an asymptotic flocking velocity and line-shaped pattern. All results are novel and can be illustrated by numerical simulations using some concrete influence functions. Also, our results significantly extend some known theorems in the literature.

Keywords: Cucker-Smale system, flocking, time delay, differential inequality.
Mathematics Subject Classification: 34H15, 34K20, 34K35.
Citation: Zhisu Liu, Yicheng Liu, Xiang Li. Flocking and line-shaped spatial configuration to delayed Cucker-Smale models. Discrete & Continuous Dynamical Systems - B, 2021, 26 (7) : 3693-3716. doi: 10.3934/dcdsb.2020253
References:
[1]

N. Bellomo, P. Degond and E. Tadmor, eds., Active Particles. Vol. 1, Advances in Theory, Models, and Applications, Birkhäuser/Springer, Cham, 2017.  Google Scholar

[2]

N. Bellomo, P. Degond and E. Tadmor, eds., Active Particles. Vol. 2, Advances in Theory, Models, and Applications, Birkhäuser/Springer, Cham, 2019. doi: 10.1007/978-3-030-20297-2.  Google Scholar

[3]

J. A. Carrillo, Y.-P. Choi, P. 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.  Google Scholar

[4]

J. A. Carrillo, M. Fornasier, J. Rosado and G. Toscani, Asymptotic flocking dynamics for the kinetic Cucker-Smale model, SIAM J. Math. Anal., 42 (2010), 218-236.  doi: 10.1137/090757290.  Google Scholar

[5]

J. Cho, S.-Y. Ha, F. Huang, C. Jin and D. Ko, Emergence of bi-cluster flocking for the Cucker-Smale model, Math. Models Methods Appl. Sci., 26 (2016), 1191-1218.  doi: 10.1142/S0218202516500287.  Google Scholar

[6]

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

[7]

Y.-P. Choi, S. Ha and Z. Li, Emergent dynamics of the Cucker-Smale flocking model and its variants, In Active Particles, Vol. 1, Advances in Theory, Models, and Applications, Birkhäuser/Springer, Cham, (2017), 299–331.  Google Scholar

[8]

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.  Google Scholar

[9]

Y.-P. Choi and C. Pignotti, Emergent behavior of Cucker-Smale models with normalized weights and distributed time delays, Netw. Heterog. Media, 14 (2019), 789-804.  doi: 10.3934/nhm.2019032.  Google Scholar

[10]

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

[11]

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

[12]

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

[13]

F. Cucker and J.-G. Dong, On the critical exponent for flocks under hierarchical leadership, Math. Models Methods Appl. Sci., 19 (2009), 1391-1404.  doi: 10.1142/S0218202509003851.  Google Scholar

[14]

J. Dong, S. -Y. Ha, D. Kim and J. Kim, Time-delay effect on the flocking in an ensemble of thermomechanical Cucker-Smale particles, J. Differential Equations, 266 (2019), 2373-2407.  doi: 10.1016/j.jde.2018.08.034.  Google Scholar

[15]

J. Dong, S.-Y. Ha, D. Kim and J. Kim, Interplay of time-delay and velocity alignment in the Cucker-Smale model on a general digraph, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 5569-5596.  doi: 10.3934/dcdsb.2019072.  Google Scholar

[16]

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

[17]

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

[18]

S.-Y. Ha, K. Lee and D. Levy, Emergence of time-asymptotic flocking in a stochastic Cucker-Smale system, Commun. Math. Sci., 7 (2009), 453-469.  doi: 10.4310/CMS.2009.v7.n2.a9.  Google Scholar

[19]

S.-Y. Ha and M. Slemrod, Flocking dynamics of a singularly perturbed oscillator chain and the Cucker-Smale system, J. Dynam. Differential Equations, 22 (2010), 325-330.  doi: 10.1007/s10884-009-9142-9.  Google Scholar

[20]

S. -Y. Ha, J. Kim, J. Park and X. Zhang, Complete cluster predictability of the Cucker-Smale flocking model on the real line, Arch. Ration. Mech. Anal., 231 (2019), 319-365.  doi: 10.1007/s00205-018-1281-x.  Google Scholar

[21]

J. Haskovec and I. Markou, Delay Cucker-Smale model with and without noise revised, preprint, arXiv: 1810.01084v2. Google Scholar

[22]

L. Li, L. Huang and J. Wu, Cascade flocking with free-will, Discrete Contin. Dyn. Syst. Ser.B, 21 (2016), 497-522.  doi: 10.3934/dcdsb.2016.21.497.  Google Scholar

[23]

L. Li, W. Wang, L. Huang and J. Wu, Some weak flocking models and its application to target tracking, J. Math. Anal. Appl., 480 (2019), 123404. doi: 10.1016/j.jmaa.2019.123404.  Google Scholar

[24]

X. Li, Y. Liu and J. Wu, Flocking and pattern motion in a modified Cucker-Smale model, Bull. Korean. Math. Soc., 53 (2016), 1327-1339.  doi: 10.4134/BKMS.b150629.  Google Scholar

[25]

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.  Google Scholar

[26]

Z. Li, Effectual leadership in flocks with hierarchy and individual preference, Discrete Contin. Dyn. Syst., 34 (2014), 3683-3702.  doi: 10.3934/dcds.2014.34.3683.  Google Scholar

[27]

H. Liu, X. Wang, Y. Liu and X. Li, On non-collision flocking and line-shaped spatial configuration for a modified singular Cucker-Smale model, Commun. Nonlinear. Sci. Numer. Simul., 75 (2019), 280-301.  doi: 10.1016/j.cnsns.2019.04.006.  Google Scholar

[28]

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.  Google Scholar

[29]

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.  Google Scholar

[30]

P. Mucha and J. Peszek, The Cucker-Smale equation: Singular communication weight, measure-valued solutions and weakatomic uniqueness, Arch. Ration. Mech. Anal., 227 (2018), 273-308.  doi: 10.1007/s00205-017-1160-x.  Google Scholar

[31]

C. Pignotti and E. Trelat, 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.  Google Scholar

[32]

C. Pignotti and I. Vallejo, Flocking estimates for the Cucker-Smale model with time lag and hierarchical leadership, J. Math. Anal. Appl., 464 (2018), 1313-1332.  doi: 10.1016/j.jmaa.2018.04.070.  Google Scholar

[33]

C. Pignotti and I. Vallejo, Asymptotic analysis of a Cucker-Smale system with leadership and distributed delay, in Trends in Control Theory and Partial Differential Equations, Springer INdAM Ser. 32, Springer, Cham, 2019.  Google Scholar

[34]

L. Ru and X. Xue, Multi-cluster flocking behavior of the hierarchical Cucker-Smale model, J. Franklin Inst., 354 (2017), 2371-2392.  doi: 10.1016/j.jfranklin.2016.12.018.  Google Scholar

[35]

J. Shen, Cucker-Smale flocking under hierarchical leadership, SIAM J. Appl. Math., 68 (2007/08), 694-719.  doi: 10.1137/060673254.  Google Scholar

[36]

T. Vicsek and A. Zafeiris, Collective motion, Physics Reports, 517 (2012), 71-140.  doi: 10.1016/j.physrep.2012.03.004.  Google Scholar

[37]

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

show all references

References:
[1]

N. Bellomo, P. Degond and E. Tadmor, eds., Active Particles. Vol. 1, Advances in Theory, Models, and Applications, Birkhäuser/Springer, Cham, 2017.  Google Scholar

[2]

N. Bellomo, P. Degond and E. Tadmor, eds., Active Particles. Vol. 2, Advances in Theory, Models, and Applications, Birkhäuser/Springer, Cham, 2019. doi: 10.1007/978-3-030-20297-2.  Google Scholar

[3]

J. A. Carrillo, Y.-P. Choi, P. 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.  Google Scholar

[4]

J. A. Carrillo, M. Fornasier, J. Rosado and G. Toscani, Asymptotic flocking dynamics for the kinetic Cucker-Smale model, SIAM J. Math. Anal., 42 (2010), 218-236.  doi: 10.1137/090757290.  Google Scholar

[5]

J. Cho, S.-Y. Ha, F. Huang, C. Jin and D. Ko, Emergence of bi-cluster flocking for the Cucker-Smale model, Math. Models Methods Appl. Sci., 26 (2016), 1191-1218.  doi: 10.1142/S0218202516500287.  Google Scholar

[6]

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

[7]

Y.-P. Choi, S. Ha and Z. Li, Emergent dynamics of the Cucker-Smale flocking model and its variants, In Active Particles, Vol. 1, Advances in Theory, Models, and Applications, Birkhäuser/Springer, Cham, (2017), 299–331.  Google Scholar

[8]

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.  Google Scholar

[9]

Y.-P. Choi and C. Pignotti, Emergent behavior of Cucker-Smale models with normalized weights and distributed time delays, Netw. Heterog. Media, 14 (2019), 789-804.  doi: 10.3934/nhm.2019032.  Google Scholar

[10]

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

[11]

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

[12]

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

[13]

F. Cucker and J.-G. Dong, On the critical exponent for flocks under hierarchical leadership, Math. Models Methods Appl. Sci., 19 (2009), 1391-1404.  doi: 10.1142/S0218202509003851.  Google Scholar

[14]

J. Dong, S. -Y. Ha, D. Kim and J. Kim, Time-delay effect on the flocking in an ensemble of thermomechanical Cucker-Smale particles, J. Differential Equations, 266 (2019), 2373-2407.  doi: 10.1016/j.jde.2018.08.034.  Google Scholar

[15]

J. Dong, S.-Y. Ha, D. Kim and J. Kim, Interplay of time-delay and velocity alignment in the Cucker-Smale model on a general digraph, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 5569-5596.  doi: 10.3934/dcdsb.2019072.  Google Scholar

[16]

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

[17]

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

[18]

S.-Y. Ha, K. Lee and D. Levy, Emergence of time-asymptotic flocking in a stochastic Cucker-Smale system, Commun. Math. Sci., 7 (2009), 453-469.  doi: 10.4310/CMS.2009.v7.n2.a9.  Google Scholar

[19]

S.-Y. Ha and M. Slemrod, Flocking dynamics of a singularly perturbed oscillator chain and the Cucker-Smale system, J. Dynam. Differential Equations, 22 (2010), 325-330.  doi: 10.1007/s10884-009-9142-9.  Google Scholar

[20]

S. -Y. Ha, J. Kim, J. Park and X. Zhang, Complete cluster predictability of the Cucker-Smale flocking model on the real line, Arch. Ration. Mech. Anal., 231 (2019), 319-365.  doi: 10.1007/s00205-018-1281-x.  Google Scholar

[21]

J. Haskovec and I. Markou, Delay Cucker-Smale model with and without noise revised, preprint, arXiv: 1810.01084v2. Google Scholar

[22]

L. Li, L. Huang and J. Wu, Cascade flocking with free-will, Discrete Contin. Dyn. Syst. Ser.B, 21 (2016), 497-522.  doi: 10.3934/dcdsb.2016.21.497.  Google Scholar

[23]

L. Li, W. Wang, L. Huang and J. Wu, Some weak flocking models and its application to target tracking, J. Math. Anal. Appl., 480 (2019), 123404. doi: 10.1016/j.jmaa.2019.123404.  Google Scholar

[24]

X. Li, Y. Liu and J. Wu, Flocking and pattern motion in a modified Cucker-Smale model, Bull. Korean. Math. Soc., 53 (2016), 1327-1339.  doi: 10.4134/BKMS.b150629.  Google Scholar

[25]

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.  Google Scholar

[26]

Z. Li, Effectual leadership in flocks with hierarchy and individual preference, Discrete Contin. Dyn. Syst., 34 (2014), 3683-3702.  doi: 10.3934/dcds.2014.34.3683.  Google Scholar

[27]

H. Liu, X. Wang, Y. Liu and X. Li, On non-collision flocking and line-shaped spatial configuration for a modified singular Cucker-Smale model, Commun. Nonlinear. Sci. Numer. Simul., 75 (2019), 280-301.  doi: 10.1016/j.cnsns.2019.04.006.  Google Scholar

[28]

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.  Google Scholar

[29]

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.  Google Scholar

[30]

P. Mucha and J. Peszek, The Cucker-Smale equation: Singular communication weight, measure-valued solutions and weakatomic uniqueness, Arch. Ration. Mech. Anal., 227 (2018), 273-308.  doi: 10.1007/s00205-017-1160-x.  Google Scholar

[31]

C. Pignotti and E. Trelat, 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.  Google Scholar

[32]

C. Pignotti and I. Vallejo, Flocking estimates for the Cucker-Smale model with time lag and hierarchical leadership, J. Math. Anal. Appl., 464 (2018), 1313-1332.  doi: 10.1016/j.jmaa.2018.04.070.  Google Scholar

[33]

C. Pignotti and I. Vallejo, Asymptotic analysis of a Cucker-Smale system with leadership and distributed delay, in Trends in Control Theory and Partial Differential Equations, Springer INdAM Ser. 32, Springer, Cham, 2019.  Google Scholar

[34]

L. Ru and X. Xue, Multi-cluster flocking behavior of the hierarchical Cucker-Smale model, J. Franklin Inst., 354 (2017), 2371-2392.  doi: 10.1016/j.jfranklin.2016.12.018.  Google Scholar

[35]

J. Shen, Cucker-Smale flocking under hierarchical leadership, SIAM J. Appl. Math., 68 (2007/08), 694-719.  doi: 10.1137/060673254.  Google Scholar

[36]

T. Vicsek and A. Zafeiris, Collective motion, Physics Reports, 517 (2012), 71-140.  doi: 10.1016/j.physrep.2012.03.004.  Google Scholar

[37]

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

Figure 1.  Time-domain behaviors of the state variables $ x_{1}(t) $, $ x_{2}(t) $, $ v_{1}(t) $, $ v_{2}(t) $ in Example 4.1
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Figure 2.  Time-domain behaviors of the state variables $ x_{1}(t) $, $ x_{2}(t) $, $ v_{1}(t) $, $ v_{2}(t) $ for the case $ \beta = \frac{1}{2} $ in Example 4.2
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Figure 3.  Time-domain behaviors of the state variables $ x_{1}(t) $, $ x_{2}(t) $, $ v_{1}(t) $, $ v_{2}(t) $ for the case $ \beta = 1 $ in Example 4.2
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Figure 4.  The first is the initial position of each particle in the system; the second is the population distribution after iteration $ 200 $ ($ 2s $). the third is the population distribution after iteration $ 2000 $ ($ 20s $). The value of each parameter is given as: $ N = 100, \lambda = 2, \tau = 0.2, \psi = \frac{1}{(1+r^2)^{1/3}} $, $ \gamma = 1 $
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Figure 5.  The first is the initial position of each particle in the system; the second is the population distribution after iteration $ 2000 $ ($ 20s $); the third is the population distribution after iteration $ 5000 $ ($ 50s $). The value of each parameter is given as: $ N = 100, \lambda = 20, \tau = 0.2, \psi = \frac{1}{(1+r^2)^{1/2}} $, $ \gamma = 1 $
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