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May  2022, 42(5): 2409-2432. doi: 10.3934/dcds.2021195

Cucker-Smale model with time delay

Mauro Rodriguez Cartabia

IMAS (UBA-CONICET) and Departamento de Matemática, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, Ciudad Universitaria, 1428 Buenos Aires, Argentina

Received  May 2021 Revised  October 2021 Published  May 2022 Early access  December 2021

We study the flocking model for continuous time introduced by Cucker and Smale adding a positive time delay $ \tau $. The goal of this article is to prove that the same unconditional flocking result for the non-delayed case is valid in the delayed case. A novelty is that we do not need to impose any restriction on the size of $ \tau $. Furthermore, when the unconditional flocking occurs, velocities converge exponentially fast to a common one.

Keywords: Flocking, delay differential equations, collective behavior, consensus reaching problem, Cucker–Smale model.
Mathematics Subject Classification: 93D20, 34K25, 92D50.
Citation: Mauro Rodriguez Cartabia. Cucker-Smale model with time delay. Discrete and Continuous Dynamical Systems, 2022, 42 (5) : 2409-2432. doi: 10.3934/dcds.2021195
References:
[1]

S. Ahn, H.-O. Bae, S.-Y. Ha, Y. Kim and H. Lim, Application of flocking mechanism to the modeling of stochastic volatility, Math. Models Methods Appl. Sci., 23 (2013), 1603-1628.  doi: 10.1142/S0218202513500176.

[2]

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.

[3]

Y.-P. Choi, S.-Y. Ha and Z. Li, Emergent dynamics of the cucker–smale flocking model and its variants, Active Particles, 1 (2017), 299-331. 

[4]

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.

[5]

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.

[6]

Y.-L. Chuang, Y. R. Huang, M. R. D'Orsogna and A. L. Bertozzi, Multi-vehicle flocking: Scalability of cooperative control algorithms using pairwise potentials, Proceedings 2007 IEEE International Conference on Robotics and Automation, (2007), 2292–2299. doi: 10.1109/ROBOT.2007.363661.

[7]

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

[8]

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

[9]

F. Cucker, S. Smale and D.-X. Zhou, Modeling language evolution, Found. Comput. Math., 4 (2004), 315-343.  doi: 10.1007/s10208-003-0101-2.

[10]

J.-G. Dong, S.-Y. Ha and D. 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.

[11]

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

[12]

S.-Y. Ha and D. Levy, Particle, kinetic and fluid models for phototaxis, Discrete Contin. Dyn. Syst. Ser. B, 12 (2009), 77-108.  doi: 10.3934/dcdsb.2009.12.77.

[13]

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.

[14]

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

[15]

J. Haskovec, Direct proof of unconditional asymptotic consensus in the Hegselmann–Krause model with transmission-type delay, Bull. Lond. Math. Soc., 53 (2021), 1312-1323.  doi: 10.1112/blms.12497.

[16]

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.

[17]

J. Haskovec, Well posedness and asymptotic consensus in the Hegselmann-Krause model with finite speed of information propagation, Proc. Amer. Math. Soc., 149 (2021), 3425-3437.  doi: 10.1090/proc/15522.

[18]

R. Hegselmann, U. Krause and et al., Opinion dynamics and bounded confidence models, analysis, and simulation, Journal of Artificial Societies and Social Simulation, 5 (2002).

[19]

V. Kolmanovskii and A. Myshkis, Introduction to the Theory and Applications of Functional Differential Equations, Mathematics and its Applications, 463. Kluwer Academic Publishers, Dordrecht, 1999. doi: 10.1007/978-94-017-1965-0.

[20]

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.

[21]

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.

[22]

S.-I. Niculescu, Delay Effects on Stability: A Robust Control Approach, Lecture Notes in Control and Information Sciences, 269. Springer-Verlag London, Ltd., London, 2001.

[23]

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

[24]

J. Park, H. J. 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.

[25]

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.

[26]

C. Pignotti and I. R. 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.

[27]

J. P. Pinasco, M. Rodriguez Cartabia and N. Saintier, Interacting particles systems with delay and random delay differential equations, Nonlinear Anal., 214 (2022), 112524.  doi: 10.1016/j.na.2021.112524.

[28]

C. W. Reynolds, Flocks, herds and schools: A distributed behavioral model, Seminal Graphics: Pioneering Efforts That Shaped the Field, (1998), 273–282. doi: 10.1145/280811.281008.

[29]

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

[30] D. J. Sumpter, Collective Animal Behavior, Princeton University Press, 2010. 
[31]

T. Vicsek, A. Czirók, E. Ben-Jacob, I. 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.

[32]

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, H.-O. Bae, S.-Y. Ha, Y. Kim and H. Lim, Application of flocking mechanism to the modeling of stochastic volatility, Math. Models Methods Appl. Sci., 23 (2013), 1603-1628.  doi: 10.1142/S0218202513500176.

[2]

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.

[3]

Y.-P. Choi, S.-Y. Ha and Z. Li, Emergent dynamics of the cucker–smale flocking model and its variants, Active Particles, 1 (2017), 299-331. 

[4]

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.

[5]

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.

[6]

Y.-L. Chuang, Y. R. Huang, M. R. D'Orsogna and A. L. Bertozzi, Multi-vehicle flocking: Scalability of cooperative control algorithms using pairwise potentials, Proceedings 2007 IEEE International Conference on Robotics and Automation, (2007), 2292–2299. doi: 10.1109/ROBOT.2007.363661.

[7]

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

[8]

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

[9]

F. Cucker, S. Smale and D.-X. Zhou, Modeling language evolution, Found. Comput. Math., 4 (2004), 315-343.  doi: 10.1007/s10208-003-0101-2.

[10]

J.-G. Dong, S.-Y. Ha and D. 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.

[11]

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

[12]

S.-Y. Ha and D. Levy, Particle, kinetic and fluid models for phototaxis, Discrete Contin. Dyn. Syst. Ser. B, 12 (2009), 77-108.  doi: 10.3934/dcdsb.2009.12.77.

[13]

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.

[14]

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

[15]

J. Haskovec, Direct proof of unconditional asymptotic consensus in the Hegselmann–Krause model with transmission-type delay, Bull. Lond. Math. Soc., 53 (2021), 1312-1323.  doi: 10.1112/blms.12497.

[16]

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.

[17]

J. Haskovec, Well posedness and asymptotic consensus in the Hegselmann-Krause model with finite speed of information propagation, Proc. Amer. Math. Soc., 149 (2021), 3425-3437.  doi: 10.1090/proc/15522.

[18]

R. Hegselmann, U. Krause and et al., Opinion dynamics and bounded confidence models, analysis, and simulation, Journal of Artificial Societies and Social Simulation, 5 (2002).

[19]

V. Kolmanovskii and A. Myshkis, Introduction to the Theory and Applications of Functional Differential Equations, Mathematics and its Applications, 463. Kluwer Academic Publishers, Dordrecht, 1999. doi: 10.1007/978-94-017-1965-0.

[20]

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.

[21]

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.

[22]

S.-I. Niculescu, Delay Effects on Stability: A Robust Control Approach, Lecture Notes in Control and Information Sciences, 269. Springer-Verlag London, Ltd., London, 2001.

[23]

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

[24]

J. Park, H. J. 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.

[25]

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.

[26]

C. Pignotti and I. R. 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.

[27]

J. P. Pinasco, M. Rodriguez Cartabia and N. Saintier, Interacting particles systems with delay and random delay differential equations, Nonlinear Anal., 214 (2022), 112524.  doi: 10.1016/j.na.2021.112524.

[28]

C. W. Reynolds, Flocks, herds and schools: A distributed behavioral model, Seminal Graphics: Pioneering Efforts That Shaped the Field, (1998), 273–282. doi: 10.1145/280811.281008.

[29]

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

[30] D. J. Sumpter, Collective Animal Behavior, Princeton University Press, 2010. 
[31]

T. Vicsek, A. Czirók, E. Ben-Jacob, I. 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.

[32]

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

Figure 3.  We plot two simulations using system 33. The imagines at the top represent the case of $ 4 $ particles with $ K = 5 $. The imagines at the bottom represent the case of $ 20 $ particles with $ K = 25 $. The black solid lines show the center of positions $ x_{\bf{c}} $ (left imagines) and the center of velocities $ v_{\bf{c}} $ (right imagines). In both cases we set $ \Delta t = 1/1000 $, $ \tau = 1 $ and the initial conditions are constant in the velocities, satisfying $ (d/dt) x^{(a)}_0 = v^{(a)}_0 $ for each $ a\in A $
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Figure 1.  Numerical simulation of Example 4.1. We use system 33 and set $ \tau = 1 $, $ \Delta t = 1/1000 $, $ N = 2 $ and $ K = 1 $
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Figure 2.  Numerical simulation of Example 4.2 with $ \tau = 1 $ and $ \varepsilon = 0.2 $. We use system 33 setting $ \Delta t = 1/1000 $, $ N = 2 $ and $ K = 1 $
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