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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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. Google Scholar

[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). Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

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

[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.  Google Scholar

[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.  Google Scholar

[27]

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

[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.  Google Scholar

[29]

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

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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. Google Scholar

[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). Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

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

[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.  Google Scholar

[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.  Google Scholar

[27]

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

[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.  Google Scholar

[29]

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

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)
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