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February  2021, 26(2): 1111-1127. doi: 10.3934/dcdsb.2020155

Flocking of non-identical Cucker-Smale models on general coupling network

1. 

Department of Applied Mathematics, National Chiao Tung University, Hsinchu, Taiwan, ROC

2. 

Department of Applied Mathematics, National University of Kaohsiung, Kaohsiung 700, Taiwan, ROC

* Corresponding author: Jonq Juang

Received  July 2019 Revised  December 2019 Published  May 2020

Fund Project: This work is partially supported by the Ministry of Science and Technology of the Republic of China under grant No.\ MOST 107-2115-M-009-011-MY2 and No. MOST 107-2115-M-390-006-MY2

The purpose of the paper is to investigate the flocking behavior of the discrete-time Cucker-Smale(C-S) model under general interaction network topologies with agents having their free-will accelerations. We prove theoretically that if the free-will accelerations of agents are summable, then, for any given initial conditions, the solution achieves flocking with a finite moving speed by suitably choosing the time step as well as the communication rate of the system or the strength of the interaction between agents. In particular, if the communication rate $ \beta $ of the system is subcritical, i.e., $ \beta $ is less than a critical value $ \beta_c $, then flocking holds for any initial conditions regardless of the strength of the interaction between agents. While, if the communication rate is critical ($ \beta = \beta_c $) or supercritical ($ \beta > \beta_c $), then flocking can only be achieved by making the strength of the interaction large enough. We also present some numerical simulations to support our obtained theoretical results.

Citation: Yu-Jhe Huang, Zhong-Fu Huang, Jonq Juang, Yu-Hao Liang. Flocking of non-identical Cucker-Smale models on general coupling network. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 1111-1127. doi: 10.3934/dcdsb.2020155
References:
[1]

S. M. Ahn, H. Choi, S.-Y. Ha and H. Lee, Stochastic flocking dynamics of the Cucker-Smale model with multiplicative white noises, J. Math. Phys., 51 (2010), 103301, 17pp. doi: 10.1063/1.3496895.  Google Scholar

[2]

S. M. AhnH. ChoiS.-Y. Ha and H. Lee, On collision-avoiding initial configurations to Cucker-Smale type flocking models, Commun. Math. Sci., 10 (2012), 625-643.  doi: 10.4310/CMS.2012.v10.n2.a10.  Google Scholar

[3]

S. M. AhnH.-O. BaeS.-Y. HaY. Kim and H. Lim, Application of flocking mechanism to the modeling of stochastic volatility, Math. Models Methods in Appli. Sci., 23 (2013), 1603-1628.  doi: 10.1142/S0218202513500176.  Google Scholar

[4]

P. Antoniou, A. Pitsillides, T. Blackwell and A. Engelbrecht, Employing the flocking behavior of birds for controlling congestion in autonomous decentralized networks, IEEE Congr. Evol. Comp., (2009), 1753–1761. doi: 10.1109/CEC.2009.4983153.  Google Scholar

[5]

J. A. CañizoJ. A. Carrillo and J. Rosado, A well-posedness theory in measures for some kinetic models of collective motion, Math. Models Methods Appl. Sci., 21 (2011), 515-539.  doi: 10.1142/S0218202511005131.  Google Scholar

[6]

J. A. CarrilloM. FornasierJ. 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

[7]

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

[8]

F. Cucker and J.-G. Dong, Avoiding collisions in flocks, IEEE Trans. Automat. Control, 55 (2010), 1238-1243.  doi: 10.1109/TAC.2010.2042355.  Google Scholar

[9]

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

[10]

F. Cucker and J.-G. Dong, On flocks influenced by closest neighbors, Math. Models Methods Appl. Sci., 26 (2016), 2685-2708.  doi: 10.1142/S0218202516500639.  Google Scholar

[11]

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

[12]

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

[13]

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

[14]

P. Degond and S. Motsch, Macroscopic limit of self-driven particles with orientation interaction, C. R. Math. Acad. Sci. Paris, 345 (2007), 555-560.  doi: 10.1016/j.crma.2007.10.024.  Google Scholar

[15]

P. Degond and S. Motsch, Continuum limit of self-driven particles with orientation interaction, Math. Models Methods Appl. Sci., 18 (2008), 1193-1215.  doi: 10.1142/S0218202508003005.  Google Scholar

[16]

P. Degond and S. Motsch, Large scale dynamics of the persistent turning walker model of fish behavior, J. Stat. Phys., 131 (2008), 989-1021.  doi: 10.1007/s10955-008-9529-8.  Google Scholar

[17]

P. Degond and T. Yang, Diffusion in a continuum model of self-propelled particles with alignment interaction, Math. Models Methods Appl. Sci., 20 (2010), 1459-1490.  doi: 10.1142/S0218202510004659.  Google Scholar

[18]

J.-G. Dong and L. Qiu, Flocking of the Cucker-Smale model on general digraphs, IEEE Trans. Automat. Control, 62 (2017), 5234-5239.  doi: 10.1109/TAC.2016.2631608.  Google Scholar

[19]

S.-Y. HaK. 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

[20]

S.-Y. HaZ. LiM. Slemrod and X. Xue, Flocking behavior of the Cucker-Smale model under rooted leadership in a large coupling limit, Quart. Appl. Math., 72 (2014), 689-701.  doi: 10.1090/S0033-569X-2014-01350-5.  Google Scholar

[21]

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.  Google Scholar

[22]

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

[23]

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

[24]

Z. Li and S.-Y. Ha, On the Cucker-Smale flocking with alternating leaders, Quart. Appl. Math., 73 (2015), 693-709.  doi: 10.1090/qam/1401.  Google Scholar

[25]

Z. LiS.-Y. Ha and and X. Xue, Emergent phenomena in an ensemble of Cucker-Smale particles under joint rooted leadership, Math. Models Methods Appl. Sci., 24 (2014), 1389-1419.  doi: 10.1142/S0218202514500043.  Google Scholar

[26]

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

[27]

Z. Li and X. Xue, Cucker-Smale flocking under rooted leadership with free-will agent, Physica A, 410 (2014), 205-217.  doi: 10.1016/j.physa.2014.05.008.  Google Scholar

[28]

C.-H. Li and S.-Y. Yang, A new discrete Cucker-Smale flocking model under hierarchical leadership, Discrete Contin. Dyn. Syst. Ser. B, 21 (2016), 2587-2599.  doi: 10.3934/dcdsb.2016062.  Google Scholar

[29]

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

[30]

J. ParkH. Kim and S.-Y. Ha, Cucker-Smale flocking with inter-particle bonding forces, IEEE Trans. Automat. Control, 55 (2010), 2617-2623.  doi: 10.1109/TAC.2010.2061070.  Google Scholar

[31]

L. PereaP. Elosegui and G. Gómez, Extension of the Cucker-Smale control law to space flight formations, J. Guid., Control, and Dyn., 32 (2009), 527-537.  doi: 10.2514/1.36269.  Google Scholar

[32]

B. PiccoliF. Rossi and E. Trélat, Control to flocking of the kinetic Cucker-Smale model, SIAM J. Math. Anal., 47 (2015), 4685-4719.  doi: 10.1137/140996501.  Google Scholar

[33]

Z. Qu, Cooperative Control Of Dynamical Systems, Springer-Verlag London, 2009. doi: 10.1007/978-1-84882-325-9.  Google Scholar

[34]

J. Shen, Cucker-Smale flocking under hierarchical leadership, SIAM J. Appl. Math., 68 (2008), 694-719.  doi: 10.1137/060673254.  Google Scholar

[35]

J. Toner and Y. Tu, Flocks, herds, and schools: A quantitative theory of flocking, Phys. Rev. E, 58 (1998), 4828-4858.  doi: 10.1103/PhysRevE.58.4828.  Google Scholar

[36]

C. M. Topaz and A. L. Bertozzi, Swarming patterns in a two-dimensional kinematic model for biological groups, SIAM J. Appl. Math., 65 (2004), 152-174.  doi: 10.1137/S0036139903437424.  Google Scholar

[37]

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.  Google Scholar

[38]

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

[39]

C.-W. Wu, Synchronization in Complex Networks of Nonlinear Dynamical Systems, Singapore: World Scientific, 2007, 88–89. doi: 10.1142/6570.  Google Scholar

show all references

References:
[1]

S. M. Ahn, H. Choi, S.-Y. Ha and H. Lee, Stochastic flocking dynamics of the Cucker-Smale model with multiplicative white noises, J. Math. Phys., 51 (2010), 103301, 17pp. doi: 10.1063/1.3496895.  Google Scholar

[2]

S. M. AhnH. ChoiS.-Y. Ha and H. Lee, On collision-avoiding initial configurations to Cucker-Smale type flocking models, Commun. Math. Sci., 10 (2012), 625-643.  doi: 10.4310/CMS.2012.v10.n2.a10.  Google Scholar

[3]

S. M. AhnH.-O. BaeS.-Y. HaY. Kim and H. Lim, Application of flocking mechanism to the modeling of stochastic volatility, Math. Models Methods in Appli. Sci., 23 (2013), 1603-1628.  doi: 10.1142/S0218202513500176.  Google Scholar

[4]

P. Antoniou, A. Pitsillides, T. Blackwell and A. Engelbrecht, Employing the flocking behavior of birds for controlling congestion in autonomous decentralized networks, IEEE Congr. Evol. Comp., (2009), 1753–1761. doi: 10.1109/CEC.2009.4983153.  Google Scholar

[5]

J. A. CañizoJ. A. Carrillo and J. Rosado, A well-posedness theory in measures for some kinetic models of collective motion, Math. Models Methods Appl. Sci., 21 (2011), 515-539.  doi: 10.1142/S0218202511005131.  Google Scholar

[6]

J. A. CarrilloM. FornasierJ. 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

[7]

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

[8]

F. Cucker and J.-G. Dong, Avoiding collisions in flocks, IEEE Trans. Automat. Control, 55 (2010), 1238-1243.  doi: 10.1109/TAC.2010.2042355.  Google Scholar

[9]

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

[10]

F. Cucker and J.-G. Dong, On flocks influenced by closest neighbors, Math. Models Methods Appl. Sci., 26 (2016), 2685-2708.  doi: 10.1142/S0218202516500639.  Google Scholar

[11]

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

[12]

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

[13]

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

[14]

P. Degond and S. Motsch, Macroscopic limit of self-driven particles with orientation interaction, C. R. Math. Acad. Sci. Paris, 345 (2007), 555-560.  doi: 10.1016/j.crma.2007.10.024.  Google Scholar

[15]

P. Degond and S. Motsch, Continuum limit of self-driven particles with orientation interaction, Math. Models Methods Appl. Sci., 18 (2008), 1193-1215.  doi: 10.1142/S0218202508003005.  Google Scholar

[16]

P. Degond and S. Motsch, Large scale dynamics of the persistent turning walker model of fish behavior, J. Stat. Phys., 131 (2008), 989-1021.  doi: 10.1007/s10955-008-9529-8.  Google Scholar

[17]

P. Degond and T. Yang, Diffusion in a continuum model of self-propelled particles with alignment interaction, Math. Models Methods Appl. Sci., 20 (2010), 1459-1490.  doi: 10.1142/S0218202510004659.  Google Scholar

[18]

J.-G. Dong and L. Qiu, Flocking of the Cucker-Smale model on general digraphs, IEEE Trans. Automat. Control, 62 (2017), 5234-5239.  doi: 10.1109/TAC.2016.2631608.  Google Scholar

[19]

S.-Y. HaK. 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

[20]

S.-Y. HaZ. LiM. Slemrod and X. Xue, Flocking behavior of the Cucker-Smale model under rooted leadership in a large coupling limit, Quart. Appl. Math., 72 (2014), 689-701.  doi: 10.1090/S0033-569X-2014-01350-5.  Google Scholar

[21]

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.  Google Scholar

[22]

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

[23]

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

[24]

Z. Li and S.-Y. Ha, On the Cucker-Smale flocking with alternating leaders, Quart. Appl. Math., 73 (2015), 693-709.  doi: 10.1090/qam/1401.  Google Scholar

[25]

Z. LiS.-Y. Ha and and X. Xue, Emergent phenomena in an ensemble of Cucker-Smale particles under joint rooted leadership, Math. Models Methods Appl. Sci., 24 (2014), 1389-1419.  doi: 10.1142/S0218202514500043.  Google Scholar

[26]

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

[27]

Z. Li and X. Xue, Cucker-Smale flocking under rooted leadership with free-will agent, Physica A, 410 (2014), 205-217.  doi: 10.1016/j.physa.2014.05.008.  Google Scholar

[28]

C.-H. Li and S.-Y. Yang, A new discrete Cucker-Smale flocking model under hierarchical leadership, Discrete Contin. Dyn. Syst. Ser. B, 21 (2016), 2587-2599.  doi: 10.3934/dcdsb.2016062.  Google Scholar

[29]

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

[30]

J. ParkH. Kim and S.-Y. Ha, Cucker-Smale flocking with inter-particle bonding forces, IEEE Trans. Automat. Control, 55 (2010), 2617-2623.  doi: 10.1109/TAC.2010.2061070.  Google Scholar

[31]

L. PereaP. Elosegui and G. Gómez, Extension of the Cucker-Smale control law to space flight formations, J. Guid., Control, and Dyn., 32 (2009), 527-537.  doi: 10.2514/1.36269.  Google Scholar

[32]

B. PiccoliF. Rossi and E. Trélat, Control to flocking of the kinetic Cucker-Smale model, SIAM J. Math. Anal., 47 (2015), 4685-4719.  doi: 10.1137/140996501.  Google Scholar

[33]

Z. Qu, Cooperative Control Of Dynamical Systems, Springer-Verlag London, 2009. doi: 10.1007/978-1-84882-325-9.  Google Scholar

[34]

J. Shen, Cucker-Smale flocking under hierarchical leadership, SIAM J. Appl. Math., 68 (2008), 694-719.  doi: 10.1137/060673254.  Google Scholar

[35]

J. Toner and Y. Tu, Flocks, herds, and schools: A quantitative theory of flocking, Phys. Rev. E, 58 (1998), 4828-4858.  doi: 10.1103/PhysRevE.58.4828.  Google Scholar

[36]

C. M. Topaz and A. L. Bertozzi, Swarming patterns in a two-dimensional kinematic model for biological groups, SIAM J. Appl. Math., 65 (2004), 152-174.  doi: 10.1137/S0036139903437424.  Google Scholar

[37]

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.  Google Scholar

[38]

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

[39]

C.-W. Wu, Synchronization in Complex Networks of Nonlinear Dynamical Systems, Singapore: World Scientific, 2007, 88–89. doi: 10.1142/6570.  Google Scholar

Figure 1.  The network consisting of $ 9 $ vertices has a spanning tree. For this network, $ \mathcal{R} = \{1,2,3\} $, $ n = 9 $, $ \ell = 3 $, $ h = 3 $ and $ r = 2 $
Figure 2.  The graph of $ G $
Figure 3.  (Subcritical) Numerical simulation for model Eq. (4) with the network given in Fig. 1. This simulation result shows the solution achieves flocking. Here parameters in Eq. (4) are chosen as $ \beta = 1/10 $, $ \kappa = 10 $ and $ \varepsilon = 0.001 $. The initial conditions are randomly chosen and they satisfy $ a\approx 0.8932 $, $ b\approx 0.8104 $. Here $ a,b $ are defined in (20a)
Figure 4.  (Critical) Numerical simulation for model Eq. (4) with the network given in Fig. 1. This simulation result shows the solution achieves flocking. Here parameters in Eq. (4) are chosen as $ \beta = 1/6 $, $ \kappa = 120 $ and $ \varepsilon = 0.001 $. The initial conditions are randomly chosen and they satisfy $ a\approx 0.8932 $, $ b\approx 0.8104 $. Here $ a,b $ are defined in (20a)
Figure 5.  (Supercritical) Numerical simulation for model Eq. (4) with the network given in Fig. 1. This simulation result shows the solution achieves flocking. Here parameters in Eq. (4) are chosen as $ \beta = 1/3 $, $ \kappa = 200 $ and $ \varepsilon = 0.001 $. The initial conditions are randomly chosen and they satisfy $ a\approx 0.8932 $, $ b\approx 0.8104 $. Here $ a,b $ are defined in (20a)
Figure 6.  Numerical simulation for model Eq. (4) under the network provided in Fig. 1. This simulation result shows the solution does not achieve flocking. Here parameters in Eq. (4) are chosen as $ \beta = 1 $ (supercritical), $ \kappa = 0.1 $ and $ \varepsilon = 0.001 $. The initial conditions are randomly chosen and they satisfy $ a\approx 0.2872 $, $ b\approx 0.2850 $. Here $ a,b $ are defined in (20a). Such set of parameters and initial conditions do not satisfy the sufficient condition (23)
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Ka Luen Cheung, Man Chun Leung. Asymptotic behavior of positive solutions of the equation $ \Delta u + K u^{\frac{n+2}{n-2}} = 0$ in $IR^n$ and positive scalar curvature. Conference Publications, 2001, 2001 (Special) : 109-120. doi: 10.3934/proc.2001.2001.109

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