October  2022, 17(5): 753-782. doi: 10.3934/nhm.2022025

Uniform stability of the Cucker–Smale and thermodynamic Cucker–Smale ensembles with singular kernels

Department of Mathematical Sciences, and Research Institute of Mathematics, Seoul National University, Seoul, Republic of Korea

* Corresponding author: Hyunjin Ahn

Received  March 2022 Revised  May 2022 Published  October 2022 Early access  June 2022

Fund Project: This work was supported by the National Research Foundation of Korea(NRF) grant funded by the Korea government(MSIT) (2022R1C12007321). The author would like to thank Dr. Woojoo Shim for the suggestion of the main idea in the proof of Theorem 5.3

This paper presents several sufficient frameworks for a collision avoidance and flocking dynamics of the Cucker–Smale (CS) model and thermodynamic CS (TCS) model with arbitrary dimensions and singular interaction kernels. In general, unlike regular kernels, singular kernels usually interfere with the global well-posedness of the targeted models from the perspective of the standard Cauchy–Lipschitz theory due to the possibility of a finite-in-time blow-up. Therefore, according to the intensity of the singularity of a kernel (strong or weak), we provide a detailed framework for the global well-posedness and emergent dynamics for each case. Finally, we provide an admissible set in terms of system parameters and initial data for the uniform stability of the $ d $-dimensional TCS with a singular kernel, which can be reduced to a sufficient framework for the uniform stability of the $ d $-dimensional CS with singular kernel if all agents have the same initial temperature.

Citation: Hyunjin Ahn. Uniform stability of the Cucker–Smale and thermodynamic Cucker–Smale ensembles with singular kernels. Networks and Heterogeneous Media, 2022, 17 (5) : 753-782. doi: 10.3934/nhm.2022025
References:
[1]

J. A. AcebronL. L. BonillaC. J. P. Pérez VicenteF. Ritort and R. Spigler, The Kuramoto model: A simple paradigm for synchronization phenomena, Rev. Mod. Phys., 77 (2005), 137-185. 

[2]

H. Ahn, Emergent behaviors of thermodynamic Cucker–Smale ensemble with unit-speed constraint, Submitted.

[3]

H. AhnS.-Y. HaM. Kang and W. Shim, Emergent behaviors of relativistic flocks on Riemannian manifolds, Phys. D., 427 (2021), 133011.  doi: 10.1016/j.physd.2021.133011.

[4]

H. AhnS.-Y. HaD. KimF. Schlöder and W. Shim, The mean-field limit of the Cucker–Smale model on Riemannian manifolds, Q. Appl. Math., 80 (2022), 403-450. 

[5]

H. AhnS.-Y. Ha and J. Kim, Uniform stability of the relativistic Cucker-Smale model and its application to a mean-field limit, Commun. Pure Appl. Anal., 20 (2021), 4209-4237.  doi: 10.3934/cpaa.2021156.

[6]

G. AlbiN. BellomoL. FermoS.-Y. HaL. PareschiD. Poyato and J. Soler, Vehicular traffic, crowds, and swarms. On the kinetic theory approach towards research perspective, Math. Models Methods Appl. Sci., 29 (2019), 1901-2005.  doi: 10.1142/S0218202519500374.

[7]

J. Buck and E. Buck, Biology of synchronous flashing of fireflies, Nature, 211 (1966), 562-564. 

[8]

J. ByeonS.-Y. Ha and J. Kim, Asymptotic flocking dynamics of a relativistic Cucker–Smale flock under singular communications, J. Math. Phys., 63 (2022), 012702.  doi: 10.1063/5.0062745.

[9]

J. A. CarrilloY.-P. ChoiP. B. Muncha and J. Peszek, Sharp conditions to avoid collisions in singular Cucker–Smale interactions, Nonlinear Anal. Real World Appl., 37 (2017), 317-328.  doi: 10.1016/j.nonrwa.2017.02.017.

[10]

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.

[11]

P. CattiauxF. Delebecque and L. Pédéches, Stochastic Cucker–Smale models: Old and new, Ann. Appl. Probab., 28 (2018), 3239-3286.  doi: 10.1214/18-AAP1400.

[12]

H. ChoJ.-G. Dong and S.-Y. Ha, Emergent behaviors of a thermodynamic Cucker–Smale flock with a time-delay on a general digraph, Math. Methods Appl. Sci., 45 (2022), 164-196.  doi: 10.1002/mma.7771.

[13]

S.-H. Choi and S.-Y. Ha, Interplay of the unit-speed constraint and time-delay in Cucker–Smale flocking, J. Math. Phys., 59 (2018), 082701.  doi: 10.1063/1.4996788.

[14]

S.-H. Choi and S.-Y. Ha, Emergence of flocking for a multi-agent system moving with constant speed, Commun. Math. Sci., 14 (2016), 953-972.  doi: 10.4310/CMS.2016.v14.n4.a4.

[15]

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.

[16]

Y.-P. ChoiS.-Y. Ha and J. Kim, Propagation of regularity and finite-time collisions for the thermomechanical Cucker–Smale model with a singular communication, Netw. Heterog. Media, 13 (2018), 379-407.  doi: 10.3934/nhm.2018017.

[17]

Y.-P. ChoiS.-Y. Ha and Z. Li, Emergent dynamics of the Cucker–Smale flocking model and its variants, Active Particles. Advances in Theory, Models, and Applications, Model. Simul. Sci. Eng. Technol, 1 (2017), 299-331. 

[18]

Y.-P. ChoiD. KalsieJ. Peszek and A. Peters, A collisionless singular Cucker–Smale model with decentralized formation control, SIAM J. Appl. Dyn. Syst., 18 (2019), 1954-1981.  doi: 10.1137/19M1241799.

[19]

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.

[20]

Y.-P. Choi and X. Zhang, One dimensional singular Cucker-Smale model: Uniform-in-time mean-field limit and contractivity, J. Differ. Equ., 287 (2021), 428-459.  doi: 10.1016/j.jde.2021.04.002.

[21]

J. ChoS.-Y. HaF. HuangC. Jin and D. Ko, Emergence of bi-cluster flocking for agent-based models with unit speed constraint, Anal. Appl. (Singap.), 14 (2016), 39-73.  doi: 10.1142/S0219530515400023.

[22]

J. ChoS.-Y. HaF. HuangC. Jin and D. Ko, Emergence of bi-cluster flocking for the Cucker–Smale model, Math. Models Methods Appl. Sci., 26 (2016), 1191-1218.  doi: 10.1142/S0218202516500287.

[23]

F. Cucker and J.-G. Dong, A conditional, collision-avoiding, model for swarming, Discrete Contin. Dyn. Syst., 34 (2014), 1009-1020.  doi: 10.3934/dcds.2014.34.1009.

[24]

F. Cucker and J.-G. Dong, A general collision-avoiding flocking framework, IEEE Trans. Automat. Control, 56 (2011), 1124-1129.  doi: 10.1109/TAC.2011.2107113.

[25]

F. Cucker and J.-G. Dong, Avoiding collisions in flocks, IEEE Trans. Automat. Control, 55 (2010), 1238-1243.  doi: 10.1109/TAC.2010.2042355.

[26]

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

[27]

P. Degond and S. Motsch, Large-scale dynamics of the persistent turning walker model of fish behavior, J. Stat. Phys., 131 (2008), 989-1021.  doi: 10.1007/s10955-008-9529-8.

[28]

G. B. Ermentrout, An adaptive model for synchrony in the firefly Pteroptyx malaccae, J. Math. Biol., 29 (1991), 571-585.  doi: 10.1007/BF00164052.

[29]

E. FerranteA. E. TurgutA. StranieriC. Pinciroli and M. Dorigo, Self-organized flocking with a mobile robot swarm: A novel motion control method, Adapt. Behav., 20 (2012), 460-477. 

[30]

A. Figalli and M. Kang, A rigorous derivation from the kinetic Cucker–Smale model to the pressureless Euler system with nonlocal alignment, Anal. PDE., 12 (2019), 843-866.  doi: 10.2140/apde.2019.12.843.

[31]

S.-Y. HaM.-J. Kang and J. Kim, Hydrodynamic limit of the kinetic thermomechanical Cucker–Smale model in a strong local alignment regime, Commun. Pure Appl. Anal., 19 (2019), 1233-1256.  doi: 10.3934/cpaa.2020057.

[32]

S.-Y. HaM.-J. Kang and B. Kwon, A hydrodynamic model for the interaction of Cucker–Smale particles and incompressible fluid, Math. Models. Methods Appl. Sci., 24 (2014), 2311-2359.  doi: 10.1142/S0218202514500225.

[33]

S.-Y. HaJ. KimC. MinT. Ruggeri and X. Zhang, Uniform stability and mean-field limit of a thermodynamic Cucker–Smale model, Quart. Appl. Math., 77 (2019), 131-176.  doi: 10.1090/qam/1517.

[34]

S.-Y. HaJ. Kim and T. Ruggeri, Emergent behaviors of thermodynamic Cucker-Smale particles, SIAM J. Math. Anal., 50 (2018), 3092-3121.  doi: 10.1137/17M111064X.

[35]

S.-Y. HaJ. Kim and T. Ruggeri, From the relativistic mixture of gases to the relativistic Cucker-Smale flocking, Arch. Rational Mech. Anal., 235 (2020), 1661-1706.  doi: 10.1007/s00205-019-01452-y.

[36]

S.-Y. HaJ. Kim and X. Zhang, Uniform stability of the Cucker–Smale model and its application to the mean-field limit, Kinet. Relat. Models, 11 (2018), 1157-1181.  doi: 10.3934/krm.2018045.

[37]

S.-Y. HaD. Ko and Y. Zhang, Remarks on the coupling strength for the Cucker–Smale with unit speed, Discrete Contin. Dyn. Syst., 38 (2018), 2763-2793.  doi: 10.3934/dcds.2018116.

[38]

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. 

[39]

S.-Y. Ha and T. Ruggeri, Emergent dynamics of a thermodynamically consistent particle model, Arch. Ration. Mech. Anal., 223 (2017), 1397-1425.  doi: 10.1007/s00205-016-1062-3.

[40]

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

[41]

T. K. KarperA. Mellet and K. Trivisa, Hydrodynamic limit of the kinetic Cucker–Smale flocking model, Math. Models Methods Appl. Sci., 25 (2015), 131-163.  doi: 10.1142/S0218202515500050.

[42]

P. B. Mucha and J. Peszek, The Cucker–Smale equation: Singular communication weight, measure-valued solutions and weak-atomic uniqueness,, Arch. Rational Mech. Anal., 227 (2018), 273-308.  doi: 10.1007/s00205-017-1160-x.

[43]

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

[44]

J. ParkH. 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.

[45]

J. Peszek, Discrete Cucker–Smale flocking model with a weakly singular kernel, SIAM J. Math. Anal., 47 (2015), 3671-3686.  doi: 10.1137/15M1009299.

[46]

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

[47] A. PikovskyM. Rosenblum and J. Kurths, Synchronization: A Universal Concept in Nonlinear Sciences, Cambridge University Press, Cambridge, 2001.  doi: 10.1017/CBO9780511755743.
[48]

L. RuX. LiY. Liu and X. Wang, Flocking of Cucker–Smale model with unit speed on general digraphs, Proc. Am. Math. Soc., 149 (2021), 4397-4409.  doi: 10.1090/proc/15594.

[49]

S. H. Strogatz, From Kuramoto to Crawford: Exploring the onset of synchronization in populations of coupled oscillators, Phys. D, 143 (2000), 1-20.  doi: 10.1016/S0167-2789(00)00094-4.

[50]

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

[51]

C. M. Topaz and A. L. Bertozzi, Swarming patterns in a two-dimensional kinematic model for biological groups, SIAM J. Appl. Math., 65 (2004), 152-174.  doi: 10.1137/S0036139903437424.

[52]

T. VicsekA. CzirókE. Ben-JacobI. Cohen and O. Schochet, 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.

[53]

T. Vicsek and A. Zefeiris, Collective motion, Phys. Rep., 517 (2012), 71-140. 

[54]

A. T. Winfree, The Geometry of Biological Time, Springer, New York, 1980.

[55]

A. T. Winfree, Biological rhythms and the behavior of populations of coupled oscillators, J. Theor. Biol., 16 (1967), 15-42. 

show all references

References:
[1]

J. A. AcebronL. L. BonillaC. J. P. Pérez VicenteF. Ritort and R. Spigler, The Kuramoto model: A simple paradigm for synchronization phenomena, Rev. Mod. Phys., 77 (2005), 137-185. 

[2]

H. Ahn, Emergent behaviors of thermodynamic Cucker–Smale ensemble with unit-speed constraint, Submitted.

[3]

H. AhnS.-Y. HaM. Kang and W. Shim, Emergent behaviors of relativistic flocks on Riemannian manifolds, Phys. D., 427 (2021), 133011.  doi: 10.1016/j.physd.2021.133011.

[4]

H. AhnS.-Y. HaD. KimF. Schlöder and W. Shim, The mean-field limit of the Cucker–Smale model on Riemannian manifolds, Q. Appl. Math., 80 (2022), 403-450. 

[5]

H. AhnS.-Y. Ha and J. Kim, Uniform stability of the relativistic Cucker-Smale model and its application to a mean-field limit, Commun. Pure Appl. Anal., 20 (2021), 4209-4237.  doi: 10.3934/cpaa.2021156.

[6]

G. AlbiN. BellomoL. FermoS.-Y. HaL. PareschiD. Poyato and J. Soler, Vehicular traffic, crowds, and swarms. On the kinetic theory approach towards research perspective, Math. Models Methods Appl. Sci., 29 (2019), 1901-2005.  doi: 10.1142/S0218202519500374.

[7]

J. Buck and E. Buck, Biology of synchronous flashing of fireflies, Nature, 211 (1966), 562-564. 

[8]

J. ByeonS.-Y. Ha and J. Kim, Asymptotic flocking dynamics of a relativistic Cucker–Smale flock under singular communications, J. Math. Phys., 63 (2022), 012702.  doi: 10.1063/5.0062745.

[9]

J. A. CarrilloY.-P. ChoiP. B. Muncha and J. Peszek, Sharp conditions to avoid collisions in singular Cucker–Smale interactions, Nonlinear Anal. Real World Appl., 37 (2017), 317-328.  doi: 10.1016/j.nonrwa.2017.02.017.

[10]

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.

[11]

P. CattiauxF. Delebecque and L. Pédéches, Stochastic Cucker–Smale models: Old and new, Ann. Appl. Probab., 28 (2018), 3239-3286.  doi: 10.1214/18-AAP1400.

[12]

H. ChoJ.-G. Dong and S.-Y. Ha, Emergent behaviors of a thermodynamic Cucker–Smale flock with a time-delay on a general digraph, Math. Methods Appl. Sci., 45 (2022), 164-196.  doi: 10.1002/mma.7771.

[13]

S.-H. Choi and S.-Y. Ha, Interplay of the unit-speed constraint and time-delay in Cucker–Smale flocking, J. Math. Phys., 59 (2018), 082701.  doi: 10.1063/1.4996788.

[14]

S.-H. Choi and S.-Y. Ha, Emergence of flocking for a multi-agent system moving with constant speed, Commun. Math. Sci., 14 (2016), 953-972.  doi: 10.4310/CMS.2016.v14.n4.a4.

[15]

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.

[16]

Y.-P. ChoiS.-Y. Ha and J. Kim, Propagation of regularity and finite-time collisions for the thermomechanical Cucker–Smale model with a singular communication, Netw. Heterog. Media, 13 (2018), 379-407.  doi: 10.3934/nhm.2018017.

[17]

Y.-P. ChoiS.-Y. Ha and Z. Li, Emergent dynamics of the Cucker–Smale flocking model and its variants, Active Particles. Advances in Theory, Models, and Applications, Model. Simul. Sci. Eng. Technol, 1 (2017), 299-331. 

[18]

Y.-P. ChoiD. KalsieJ. Peszek and A. Peters, A collisionless singular Cucker–Smale model with decentralized formation control, SIAM J. Appl. Dyn. Syst., 18 (2019), 1954-1981.  doi: 10.1137/19M1241799.

[19]

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.

[20]

Y.-P. Choi and X. Zhang, One dimensional singular Cucker-Smale model: Uniform-in-time mean-field limit and contractivity, J. Differ. Equ., 287 (2021), 428-459.  doi: 10.1016/j.jde.2021.04.002.

[21]

J. ChoS.-Y. HaF. HuangC. Jin and D. Ko, Emergence of bi-cluster flocking for agent-based models with unit speed constraint, Anal. Appl. (Singap.), 14 (2016), 39-73.  doi: 10.1142/S0219530515400023.

[22]

J. ChoS.-Y. HaF. HuangC. Jin and D. Ko, Emergence of bi-cluster flocking for the Cucker–Smale model, Math. Models Methods Appl. Sci., 26 (2016), 1191-1218.  doi: 10.1142/S0218202516500287.

[23]

F. Cucker and J.-G. Dong, A conditional, collision-avoiding, model for swarming, Discrete Contin. Dyn. Syst., 34 (2014), 1009-1020.  doi: 10.3934/dcds.2014.34.1009.

[24]

F. Cucker and J.-G. Dong, A general collision-avoiding flocking framework, IEEE Trans. Automat. Control, 56 (2011), 1124-1129.  doi: 10.1109/TAC.2011.2107113.

[25]

F. Cucker and J.-G. Dong, Avoiding collisions in flocks, IEEE Trans. Automat. Control, 55 (2010), 1238-1243.  doi: 10.1109/TAC.2010.2042355.

[26]

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

[27]

P. Degond and S. Motsch, Large-scale dynamics of the persistent turning walker model of fish behavior, J. Stat. Phys., 131 (2008), 989-1021.  doi: 10.1007/s10955-008-9529-8.

[28]

G. B. Ermentrout, An adaptive model for synchrony in the firefly Pteroptyx malaccae, J. Math. Biol., 29 (1991), 571-585.  doi: 10.1007/BF00164052.

[29]

E. FerranteA. E. TurgutA. StranieriC. Pinciroli and M. Dorigo, Self-organized flocking with a mobile robot swarm: A novel motion control method, Adapt. Behav., 20 (2012), 460-477. 

[30]

A. Figalli and M. Kang, A rigorous derivation from the kinetic Cucker–Smale model to the pressureless Euler system with nonlocal alignment, Anal. PDE., 12 (2019), 843-866.  doi: 10.2140/apde.2019.12.843.

[31]

S.-Y. HaM.-J. Kang and J. Kim, Hydrodynamic limit of the kinetic thermomechanical Cucker–Smale model in a strong local alignment regime, Commun. Pure Appl. Anal., 19 (2019), 1233-1256.  doi: 10.3934/cpaa.2020057.

[32]

S.-Y. HaM.-J. Kang and B. Kwon, A hydrodynamic model for the interaction of Cucker–Smale particles and incompressible fluid, Math. Models. Methods Appl. Sci., 24 (2014), 2311-2359.  doi: 10.1142/S0218202514500225.

[33]

S.-Y. HaJ. KimC. MinT. Ruggeri and X. Zhang, Uniform stability and mean-field limit of a thermodynamic Cucker–Smale model, Quart. Appl. Math., 77 (2019), 131-176.  doi: 10.1090/qam/1517.

[34]

S.-Y. HaJ. Kim and T. Ruggeri, Emergent behaviors of thermodynamic Cucker-Smale particles, SIAM J. Math. Anal., 50 (2018), 3092-3121.  doi: 10.1137/17M111064X.

[35]

S.-Y. HaJ. Kim and T. Ruggeri, From the relativistic mixture of gases to the relativistic Cucker-Smale flocking, Arch. Rational Mech. Anal., 235 (2020), 1661-1706.  doi: 10.1007/s00205-019-01452-y.

[36]

S.-Y. HaJ. Kim and X. Zhang, Uniform stability of the Cucker–Smale model and its application to the mean-field limit, Kinet. Relat. Models, 11 (2018), 1157-1181.  doi: 10.3934/krm.2018045.

[37]

S.-Y. HaD. Ko and Y. Zhang, Remarks on the coupling strength for the Cucker–Smale with unit speed, Discrete Contin. Dyn. Syst., 38 (2018), 2763-2793.  doi: 10.3934/dcds.2018116.

[38]

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. 

[39]

S.-Y. Ha and T. Ruggeri, Emergent dynamics of a thermodynamically consistent particle model, Arch. Ration. Mech. Anal., 223 (2017), 1397-1425.  doi: 10.1007/s00205-016-1062-3.

[40]

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

[41]

T. K. KarperA. Mellet and K. Trivisa, Hydrodynamic limit of the kinetic Cucker–Smale flocking model, Math. Models Methods Appl. Sci., 25 (2015), 131-163.  doi: 10.1142/S0218202515500050.

[42]

P. B. Mucha and J. Peszek, The Cucker–Smale equation: Singular communication weight, measure-valued solutions and weak-atomic uniqueness,, Arch. Rational Mech. Anal., 227 (2018), 273-308.  doi: 10.1007/s00205-017-1160-x.

[43]

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

[44]

J. ParkH. 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.

[45]

J. Peszek, Discrete Cucker–Smale flocking model with a weakly singular kernel, SIAM J. Math. Anal., 47 (2015), 3671-3686.  doi: 10.1137/15M1009299.

[46]

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

[47] A. PikovskyM. Rosenblum and J. Kurths, Synchronization: A Universal Concept in Nonlinear Sciences, Cambridge University Press, Cambridge, 2001.  doi: 10.1017/CBO9780511755743.
[48]

L. RuX. LiY. Liu and X. Wang, Flocking of Cucker–Smale model with unit speed on general digraphs, Proc. Am. Math. Soc., 149 (2021), 4397-4409.  doi: 10.1090/proc/15594.

[49]

S. H. Strogatz, From Kuramoto to Crawford: Exploring the onset of synchronization in populations of coupled oscillators, Phys. D, 143 (2000), 1-20.  doi: 10.1016/S0167-2789(00)00094-4.

[50]

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

[51]

C. M. Topaz and A. L. Bertozzi, Swarming patterns in a two-dimensional kinematic model for biological groups, SIAM J. Appl. Math., 65 (2004), 152-174.  doi: 10.1137/S0036139903437424.

[52]

T. VicsekA. CzirókE. Ben-JacobI. Cohen and O. Schochet, 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.

[53]

T. Vicsek and A. Zefeiris, Collective motion, Phys. Rep., 517 (2012), 71-140. 

[54]

A. T. Winfree, The Geometry of Biological Time, Springer, New York, 1980.

[55]

A. T. Winfree, Biological rhythms and the behavior of populations of coupled oscillators, J. Theor. Biol., 16 (1967), 15-42. 

[1]

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