doi: 10.3934/krm.2022022
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The stochastic delayed Cucker-Smale system in a harmonic potential field

Department of Applied Mathematics, Donghua University, Shanghai, China

*Corresponding author: Linglong Du

Received  March 2022 Revised  May 2022 Early access June 2022

Fund Project: This work is supported by Natural Science Foundation of China (No.12001097 and 12171082)

We propose a delayed Cucker-Smale system with multiplicative noise in a harmonic potential field and investigate its emergent dynamics. It exhibits a collective behavior "flocking and concentration" if the corresponding non-delay stochastic system admits the almost surely collective behavior and the delay is sufficiently small. We provide theoretical framework and numerical simulations.

Citation: Linglong Du, Xinyun Zhou. The stochastic delayed Cucker-Smale system in a harmonic potential field. Kinetic and Related Models, doi: 10.3934/krm.2022022
References:
[1]

G. AlbiN. BellomoL. FermoS.-Y. HaJ. KimL. PareschiD. Poyato and J. Soler, Vehicular traffic, crowds and swarms: From kinetic theory and multiscale methods to applications and research perspectives, Math. Models Methods Appl. Sci., 29 (2019), 1901-2005.  doi: 10.1142/S0218202519500374.

[2]

H.-O. Bae, S.-Y. Ha, M. Kang, H. Lim, Y. Kim and J. Yoo, Time-delayed stochastic volatility model, Phys. D, 430 (2022), 14 pp. doi: 10.1016/j.physd.2021.133088.

[3]

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.

[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]

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

[7]

L. Du and X. Zhou, Flocking and concentration behavior for the stochastic Cucker-Smale system in a harmonic field, preprint, 2022, arXiv: 2205.13232.

[8]

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

[9]

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

[10]

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.

[11]

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.

[12]

J. Haskovec and I. Markou, Asymptotic flocking in the Cucker-Smale model with reaction-type delays in the non-oscillatory regime, Kinet. Relat. Models, 13 (2020), 795-813.  doi: 10.3934/krm.2020027.

[13]

D. Y. KhusainovĬ. DiblikM. Ruzhichkova and Y. Lukacheva, A representation of the solution of the Cauchy problem for an oscillatory system with pure delay, Nonlinear Oscil. (N.Y.), 11 (2008), 276-285.  doi: 10.1007/s11072-008-0030-8.

[14]

X. Mao and A. Shah, Exponential stability of stochastic differential delay equations, Stochastics Stochastics Rep., 60 (1997), 135-153.  doi: 10.1080/17442509708834102.

[15]

L. Pareschi and G. Toscani, Interacting Multiagent Systems: Kinetic Equations and Monte Carlo Methods, Oxford University Press, London, 2014.

[16]

R. Shu and E. Tadmor, Flocking hydrodynamics with external potentials, Arch. Ration. Mech. Anal., 238 (2020), 347-381.  doi: 10.1007/s00205-020-01544-0.

[17]

J. Toner and Y. Tu, Flocks, herds, and schools: A quantitative theory of flocking, Phys. Rev. E (3), 58 (1998), 4828-4858.  doi: 10.1103/PhysRevE.58.4828.

show all references

References:
[1]

G. AlbiN. BellomoL. FermoS.-Y. HaJ. KimL. PareschiD. Poyato and J. Soler, Vehicular traffic, crowds and swarms: From kinetic theory and multiscale methods to applications and research perspectives, Math. Models Methods Appl. Sci., 29 (2019), 1901-2005.  doi: 10.1142/S0218202519500374.

[2]

H.-O. Bae, S.-Y. Ha, M. Kang, H. Lim, Y. Kim and J. Yoo, Time-delayed stochastic volatility model, Phys. D, 430 (2022), 14 pp. doi: 10.1016/j.physd.2021.133088.

[3]

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.

[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]

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

[7]

L. Du and X. Zhou, Flocking and concentration behavior for the stochastic Cucker-Smale system in a harmonic field, preprint, 2022, arXiv: 2205.13232.

[8]

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

[9]

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

[10]

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.

[11]

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.

[12]

J. Haskovec and I. Markou, Asymptotic flocking in the Cucker-Smale model with reaction-type delays in the non-oscillatory regime, Kinet. Relat. Models, 13 (2020), 795-813.  doi: 10.3934/krm.2020027.

[13]

D. Y. KhusainovĬ. DiblikM. Ruzhichkova and Y. Lukacheva, A representation of the solution of the Cauchy problem for an oscillatory system with pure delay, Nonlinear Oscil. (N.Y.), 11 (2008), 276-285.  doi: 10.1007/s11072-008-0030-8.

[14]

X. Mao and A. Shah, Exponential stability of stochastic differential delay equations, Stochastics Stochastics Rep., 60 (1997), 135-153.  doi: 10.1080/17442509708834102.

[15]

L. Pareschi and G. Toscani, Interacting Multiagent Systems: Kinetic Equations and Monte Carlo Methods, Oxford University Press, London, 2014.

[16]

R. Shu and E. Tadmor, Flocking hydrodynamics with external potentials, Arch. Ration. Mech. Anal., 238 (2020), 347-381.  doi: 10.1007/s00205-020-01544-0.

[17]

J. Toner and Y. Tu, Flocks, herds, and schools: A quantitative theory of flocking, Phys. Rev. E (3), 58 (1998), 4828-4858.  doi: 10.1103/PhysRevE.58.4828.

Figure 1.  The macroscopic system dynamics
Figure 2.  The microscopic system dynamics of constant symmetric communication function case
Figure 3.  The microscopic system dynamics of radially symmetric communication function case
Figure 4.  100 realizations for the microscopic system dynamics of constant symmetric communication function case
Figure 5.  100 realizations for the microscopic system dynamics of radially symmetric communication function case
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