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Erratum and addendum to "A forward Ergodic Closing Lemma and the Entropy Conjecture for nonsingular endomorphisms away from tangencies" (Volume 40, Number 4, 2020, 2285-2313)
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Self polarization and traveling wave in a model for cell crawling migration
Cucker-Smale model with time delay
IMAS (UBA-CONICET) and Departamento de Matemática, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, Ciudad Universitaria, 1428 Buenos Aires, Argentina |
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.
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. |



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