• Previous Article
    The two dimensional Vlasov-Poisson system with steady spatial asymptotics
  • KRM Home
  • This Issue
  • Next Article
    Global strong solutions to the planar compressible magnetohydrodynamic equations with large initial data and vacuum
December  2017, 10(4): 1011-1033. doi: 10.3934/krm.2017040

Cucker-Smale model with normalized communication weights and time delay

1. 

Department of Mathematics, Inha University, Incheon, 402-751, Republic of Korea

2. 

Computer, Electrical and Mathematical Sciences & Engineering, King Abdullah University of Science and Technology, 23955 Thuwal, KSA

* Corresponding author: Jan Haskovec

Received  August 2016 Revised  January 2017 Published  March 2017

Fund Project: YPC was supported by Engineering and Physical Sciences Research Council (EP/K00804/1) and ERC-Starting grant HDSPCONTR "High-Dimensional Sparse Optimal Control". He was also supported by the Alexander Humboldt Foundation through the Humboldt Research Fellowship for Postdoctoral Researchers. JH was supported by KAUST baseline funds and KAUST grant no. 1000000193

We study a Cucker-Smale-type system with time delay in which agents interact with each other through normalized communication weights. We construct a Lyapunov functional for the system and provide sufficient conditions for asymptotic flocking, i.e., convergence to a common velocity vector. We also carry out a rigorous limit passage to the mean-field limit of the particle system as the number of particles tends to infinity. For the resulting Vlasov-type equation we prove the existence, stability and large-time behavior of measure-valued solutions. This is, to our best knowledge, the first such result for a Vlasov-type equation with time delay. We also present numerical simulations of the discrete system with few particles that provide further insights into the flocking and oscillatory behaviors of the particle velocities depending on the size of the time delay.

Citation: Young-Pil Choi, Jan Haskovec. Cucker-Smale model with normalized communication weights and time delay. Kinetic & Related Models, 2017, 10 (4) : 1011-1033. doi: 10.3934/krm.2017040
References:
[1]

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

[2]

J. CarrilloY.-P. Choi and M. Hauray, Local well-posedness of the generalized Cucker-Smale model with singular kernels, ESAIM Proc., 47 (2014), 17-35. doi: 10.1051/proc/201447002.

[3]

J. A. Carrillo, Y.-P. Choi and M. Hauray, The derivation of swarming models: Mean-field limit and Wasserstein distances, Collective Dynamics from Bacteria to Crowds: An Excursion Through Modeling, Analysis and Simulation (eds. A. Muntean and F. Toschi), Springer Series: CISM International Centre for Mechanical Sciences, 533 (2014), 1–46. doi: 10.1007/978-3-7091-1785-9_1.

[4]

J. Carrillo, Y. -P. Choi, M. Hauray and S. Salem, Mean-field limit for collective behavior models with sharp sensitivity regions, preprint, arXiv: 1510.02315.

[5]

J. Carrillo, Y. -P. Choi, P. B. Mucha and J. Peszek, Sharp conditions to avoid collisions in singular Cucker-Smale interactions, preprint, arXiv: 1609.03447.

[6]

J. Carrillo, Y. -P. Choi and S. Pérez, A review an attractive-repulsive hydrodynamics for consensus in collective behavior, in Active Particles Vol. Ⅰ -Advances in Theory, Models, Applications (eds. N. Bellomo, P. Degond and E. Tadmor), Birkhäuser-Springer Series: Modelling and Simulation in Science and Technology, (2017).

[7]

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

[8]

J. 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.

[9]

J. Carrillo, M. Fornasier, G. Toscani and F. Vecil, Particle, kinetic, and hydrodynamic models of swarming, in Mathematical Modeling of Collective Behaviour in Socio-Economic and Life Sciences (eds. G. Naldi, L. Pareschi and G. Toscani), Birkhäuser Series: Modelling and Simulation in Science and Technology, (2010), 297–336. doi: 10.1007/978-0-8176-4946-3_12.

[10]

Y.-P. Choi, Global classical solutions of the Vlasov-Fokker-Planck equation with local alignment forces, Nonlinearity, 29 (2016), 1887-1916. doi: 10.1088/0951-7715/29/7/1887.

[11]

Y. -P. Choi, S. -Y. Ha and Z. Li, Emergent dynamics of the Cucker-Smale flocking model and its variants, in Active Particles Vol. Ⅰ -Advances in Theory, Models, Applications (eds. N. Bellomo, P. Degond and E. Tadmor), Birkhäuser-Springer Series: Modelling and Simulation in Science and Technology, 2017.

[12]

F. Cucker and S. Smale, Emergent behaviour in flocks, IEEE T. on Automat. Contr., 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]

R. DuanM. Fornasier and G. Toscani, A kinetic flocking model with diffusion, Comm. Math. Phys., 300 (2010), 95-145. doi: 10.1007/s00220-010-1110-z.

[15]

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

[16]

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

[17]

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

[18]

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

[19]

A. Halanay, Differential Equations: Stability, Oscillations, Time Lags Academic Press, New York London, 1966.

[20]

J. Haskovec, Flocking dynamics and mean-field limit in the Cucker-Smale-type model with topological interactions, Physica D, 261 (2013), 42-51. doi: 10.1016/j.physd.2013.06.006.

[21]

P.-E. Jabin, A review of the mean field limits for Vlasov equations, Kinetic and Related models, 7 (2014), 661-711. doi: 10.3934/krm.2014.7.661.

[22]

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.

[23]

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

[24]

L. Pareschi and G. Toscani, Interacting Multiagent Systems: Kinetic equations and Monte Carlo methods Oxford University Press, 2014.

[25]

J. Peszek, Existence of piecewise weak solutions of a discrete Cucker-Smale's flocking model with a singular communication weight, J. Differ. Equat., 257 (2014), 2900-2925. doi: 10.1016/j.jde.2014.06.003.

[26]

H. Smith, An Introduction to Delay Differential Equations with Applications to the Life Sciences Springer, New York Dordrecht Heidelberg London, 2011. doi: 10.1007/978-1-4419-7646-8.

[27]

D. Sumpter, Collective Animal Behavior Princeton University Press, 2010. doi: 10.1515/9781400837106.

[28]

T. TonN. Linh and A. Yagi, Flocking and non-flocking behaviour in a stochastic Cucker-Smale system, Anal. Appl., 12 (2014), 63-73. doi: 10.1142/S0219530513500255.

[29]

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

show all references

References:
[1]

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

[2]

J. CarrilloY.-P. Choi and M. Hauray, Local well-posedness of the generalized Cucker-Smale model with singular kernels, ESAIM Proc., 47 (2014), 17-35. doi: 10.1051/proc/201447002.

[3]

J. A. Carrillo, Y.-P. Choi and M. Hauray, The derivation of swarming models: Mean-field limit and Wasserstein distances, Collective Dynamics from Bacteria to Crowds: An Excursion Through Modeling, Analysis and Simulation (eds. A. Muntean and F. Toschi), Springer Series: CISM International Centre for Mechanical Sciences, 533 (2014), 1–46. doi: 10.1007/978-3-7091-1785-9_1.

[4]

J. Carrillo, Y. -P. Choi, M. Hauray and S. Salem, Mean-field limit for collective behavior models with sharp sensitivity regions, preprint, arXiv: 1510.02315.

[5]

J. Carrillo, Y. -P. Choi, P. B. Mucha and J. Peszek, Sharp conditions to avoid collisions in singular Cucker-Smale interactions, preprint, arXiv: 1609.03447.

[6]

J. Carrillo, Y. -P. Choi and S. Pérez, A review an attractive-repulsive hydrodynamics for consensus in collective behavior, in Active Particles Vol. Ⅰ -Advances in Theory, Models, Applications (eds. N. Bellomo, P. Degond and E. Tadmor), Birkhäuser-Springer Series: Modelling and Simulation in Science and Technology, (2017).

[7]

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

[8]

J. 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.

[9]

J. Carrillo, M. Fornasier, G. Toscani and F. Vecil, Particle, kinetic, and hydrodynamic models of swarming, in Mathematical Modeling of Collective Behaviour in Socio-Economic and Life Sciences (eds. G. Naldi, L. Pareschi and G. Toscani), Birkhäuser Series: Modelling and Simulation in Science and Technology, (2010), 297–336. doi: 10.1007/978-0-8176-4946-3_12.

[10]

Y.-P. Choi, Global classical solutions of the Vlasov-Fokker-Planck equation with local alignment forces, Nonlinearity, 29 (2016), 1887-1916. doi: 10.1088/0951-7715/29/7/1887.

[11]

Y. -P. Choi, S. -Y. Ha and Z. Li, Emergent dynamics of the Cucker-Smale flocking model and its variants, in Active Particles Vol. Ⅰ -Advances in Theory, Models, Applications (eds. N. Bellomo, P. Degond and E. Tadmor), Birkhäuser-Springer Series: Modelling and Simulation in Science and Technology, 2017.

[12]

F. Cucker and S. Smale, Emergent behaviour in flocks, IEEE T. on Automat. Contr., 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]

R. DuanM. Fornasier and G. Toscani, A kinetic flocking model with diffusion, Comm. Math. Phys., 300 (2010), 95-145. doi: 10.1007/s00220-010-1110-z.

[15]

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

[16]

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

[17]

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

[18]

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

[19]

A. Halanay, Differential Equations: Stability, Oscillations, Time Lags Academic Press, New York London, 1966.

[20]

J. Haskovec, Flocking dynamics and mean-field limit in the Cucker-Smale-type model with topological interactions, Physica D, 261 (2013), 42-51. doi: 10.1016/j.physd.2013.06.006.

[21]

P.-E. Jabin, A review of the mean field limits for Vlasov equations, Kinetic and Related models, 7 (2014), 661-711. doi: 10.3934/krm.2014.7.661.

[22]

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.

[23]

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

[24]

L. Pareschi and G. Toscani, Interacting Multiagent Systems: Kinetic equations and Monte Carlo methods Oxford University Press, 2014.

[25]

J. Peszek, Existence of piecewise weak solutions of a discrete Cucker-Smale's flocking model with a singular communication weight, J. Differ. Equat., 257 (2014), 2900-2925. doi: 10.1016/j.jde.2014.06.003.

[26]

H. Smith, An Introduction to Delay Differential Equations with Applications to the Life Sciences Springer, New York Dordrecht Heidelberg London, 2011. doi: 10.1007/978-1-4419-7646-8.

[27]

D. Sumpter, Collective Animal Behavior Princeton University Press, 2010. doi: 10.1515/9781400837106.

[28]

T. TonN. Linh and A. Yagi, Flocking and non-flocking behaviour in a stochastic Cucker-Smale system, Anal. Appl., 12 (2014), 63-73. doi: 10.1142/S0219530513500255.

[29]

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

Figure 1.  The system with two particles: Particle velocities $v_1(t)$, $v_2(t)$ as solututions of (30) (left panels) and velocity diameters $d_V(t)$ (right panels, logarithmic scale) for $\tau=0.25$ (first row) and $\tau=1$ (second row). The initial condition is constant, $v_1(t)\equiv 1$, $v_2(t)\equiv -1$ for $t\in[-\tau,0]$
Figure 2.  The system with three particles: particle velocities $v_1(t)$, $v_2(t), v_3(t)$ as solutions of (5)–(7) (left panels) and velocity diameters $d_V(t)$ (right panels, logarithmic scale) for $\tau=0.25$ (first row) and $\tau=1$ (second row), with exponentially decaying influence function $\psi(s) = e^{-s}$. The initial condition is in both cases given by (33)–(34)
Figure 3.  The system with four particles: particle velocities as solutions of (5)–(7) (left panels) and velocity diameters $d_V(t)$ (right panels, logarithmic scale) for $\tau=0.25$ (first row) and $\tau=1$ (second row), with the influence function $\psi(s) = {(1+s^2)^{-4}}$. The initial condition is in both cases given by (35)–(36)
[1]

Hyeong-Ohk Bae, Young-Pil Choi, Seung-Yeal Ha, Moon-Jin Kang. Asymptotic flocking dynamics of Cucker-Smale particles immersed in compressible fluids. Discrete & Continuous Dynamical Systems - A, 2014, 34 (11) : 4419-4458. doi: 10.3934/dcds.2014.34.4419

[2]

Young-Pil Choi, Seung-Yeal Ha, Jeongho Kim. Propagation of regularity and finite-time collisions for the thermomechanical Cucker-Smale model with a singular communication. Networks & Heterogeneous Media, 2018, 13 (3) : 379-407. doi: 10.3934/nhm.2018017

[3]

Martial Agueh, Reinhard Illner, Ashlin Richardson. Analysis and simulations of a refined flocking and swarming model of Cucker-Smale type. Kinetic & Related Models, 2011, 4 (1) : 1-16. doi: 10.3934/krm.2011.4.1

[4]

Chun-Hsien Li, Suh-Yuh Yang. A new discrete Cucker-Smale flocking model under hierarchical leadership. Discrete & Continuous Dynamical Systems - B, 2016, 21 (8) : 2587-2599. doi: 10.3934/dcdsb.2016062

[5]

Seung-Yeal Ha, Jinwook Jung, Peter Kuchling. Emergence of anomalous flocking in the fractional Cucker-Smale model. Discrete & Continuous Dynamical Systems - A, 2019, 39 (9) : 5465-5489. doi: 10.3934/dcds.2019223

[6]

Jiu-Gang Dong, Seung-Yeal Ha, Doheon Kim. Interplay of time-delay and velocity alignment in the Cucker-Smale model on a general digraph. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 1-28. doi: 10.3934/dcdsb.2019072

[7]

Chiun-Chuan Chen, Seung-Yeal Ha, Xiongtao Zhang. The global well-posedness of the kinetic Cucker-Smale flocking model with chemotactic movements. Communications on Pure & Applied Analysis, 2018, 17 (2) : 505-538. doi: 10.3934/cpaa.2018028

[8]

Agnieszka B. Malinowska, Tatiana Odzijewicz. Optimal control of the discrete-time fractional-order Cucker-Smale model. Discrete & Continuous Dynamical Systems - B, 2018, 23 (1) : 347-357. doi: 10.3934/dcdsb.2018023

[9]

Laure Pédèches. Asymptotic properties of various stochastic Cucker-Smale dynamics. Discrete & Continuous Dynamical Systems - A, 2018, 38 (6) : 2731-2762. doi: 10.3934/dcds.2018115

[10]

Young-Pil Choi, Samir Salem. Cucker-Smale flocking particles with multiplicative noises: Stochastic mean-field limit and phase transition. Kinetic & Related Models, 2019, 12 (3) : 573-592. doi: 10.3934/krm.2019023

[11]

Seung-Yeal Ha, Shi Jin. Local sensitivity analysis for the Cucker-Smale model with random inputs. Kinetic & Related Models, 2018, 11 (4) : 859-889. doi: 10.3934/krm.2018034

[12]

Marco Caponigro, Massimo Fornasier, Benedetto Piccoli, Emmanuel Trélat. Sparse stabilization and optimal control of the Cucker-Smale model. Mathematical Control & Related Fields, 2013, 3 (4) : 447-466. doi: 10.3934/mcrf.2013.3.447

[13]

Seung-Yeal Ha, Dongnam Ko, Yinglong Zhang. Remarks on the critical coupling strength for the Cucker-Smale model with unit speed. Discrete & Continuous Dynamical Systems - A, 2018, 38 (6) : 2763-2793. doi: 10.3934/dcds.2018116

[14]

Seung-Yeal Ha, Dongnam Ko, Yinglong Zhang, Xiongtao Zhang. Emergent dynamics in the interactions of Cucker-Smale ensembles. Kinetic & Related Models, 2017, 10 (3) : 689-723. doi: 10.3934/krm.2017028

[15]

Ioannis Markou. Collision-avoiding in the singular Cucker-Smale model with nonlinear velocity couplings. Discrete & Continuous Dynamical Systems - A, 2018, 38 (10) : 5245-5260. doi: 10.3934/dcds.2018232

[16]

Seung-Yeal Ha, Jeongho Kim, Xiongtao Zhang. Uniform stability of the Cucker-Smale model and its application to the Mean-Field limit. Kinetic & Related Models, 2018, 11 (5) : 1157-1181. doi: 10.3934/krm.2018045

[17]

Ewa Girejko, Luís Machado, Agnieszka B. Malinowska, Natália Martins. On consensus in the Cucker–Smale type model on isolated time scales. Discrete & Continuous Dynamical Systems - S, 2018, 11 (1) : 77-89. doi: 10.3934/dcdss.2018005

[18]

Keng Deng, Yixiang Wu. Asymptotic behavior for a reaction-diffusion population model with delay. Discrete & Continuous Dynamical Systems - B, 2015, 20 (2) : 385-395. doi: 10.3934/dcdsb.2015.20.385

[19]

Rong Yang, Li Chen. Mean-field limit for a collision-avoiding flocking system and the time-asymptotic flocking dynamics for the kinetic equation. Kinetic & Related Models, 2014, 7 (2) : 381-400. doi: 10.3934/krm.2014.7.381

[20]

Zhipeng Qiu, Jun Yu, Yun Zou. The asymptotic behavior of a chemostat model. Discrete & Continuous Dynamical Systems - B, 2004, 4 (3) : 721-727. doi: 10.3934/dcdsb.2004.4.721

2018 Impact Factor: 1.38

Metrics

  • PDF downloads (14)
  • HTML views (84)
  • Cited by (4)

Other articles
by authors

[Back to Top]