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doi: 10.3934/dcdsb.2019072

Interplay of time-delay and velocity alignment in the Cucker-Smale model on a general digraph

1. 

Department of Mathematics, Harbin Institute of Technology, Harbin 150001, China

2. 

Department of Mathematical Sciences and Research Institute of Mathematics, Seoul National University, Seoul 08826, Republic of Korea

3. 

School of Mathematics, Korea Institute for Advanced Study, Seoul 02455, Republic of Korea

* Corresponding author: Jiu-Gang Dong

Received  August 2018 Revised  November 2018 Published  April 2019

Fund Project: The work of S.-Y. Ha was supported by the Samsung Science and Technology Foundation under Project Number SSTF-BA1401-03. The work of J.-G. Dong was supported in part by NSFC grant 11671109

We study dynamic interplay between time-delay and velocity alignment in the ensemble of Cucker-Smale (C-S) particles(or agents) on time-varying networks which are modeled by digraphs containing spanning trees. Time-delayed dynamical systems often appear in mathematical models from biology and control theory, and they have been extensively investigated in literature. In this paper, we provide sufficient frameworks for the mono-cluster flocking to the continuous and discrete C-S models, which are formulated in terms of system parameters and initial data. In our proposed frameworks, we show that the continuous and discrete C-S models exhibit exponential flocking estimates. For the explicit C-S communication weights which decay algebraically, our results exhibit threshold phenomena depending on the decay rate and depth of digraph. We also provide several numerical examples and compare them with our analytical results.

Citation: Jiu-Gang Dong, Seung-Yeal Ha, Doheon Kim. Interplay of time-delay and velocity alignment in the Cucker-Smale model on a general digraph. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2019072
References:
[1]

S. AhnH. ChoiS.-Y. Ha and H. Lee, On the collision avoiding initial-configurations to the Cucker-Smale type flocking models, Comm. Math. Sci., 10 (2012), 625-643. doi: 10.4310/CMS.2012.v10.n2.a10.

[2]

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.

[3]

M. AouchicheO. Favaron and P. Hansen, Variable neighborhood search for extremal graphs. 22. Extending bounds for independence to upper irredundance, Discret Appl. Math., 157 (2009), 3497-3510. doi: 10.1016/j.dam.2009.04.004.

[4]

M. BalleriniN. CabibboR. CandelierA. CavagnaE. CisbaniI. GiardinaV. LecomteA. OrlandiG. ParisiA. ProcacciniM. Viale and V. Zdravkovic, Interaction ruling animal collective behavior depends on topological rather than metric distance: evidence from a field study, Proc. Natl. Acad. Sci. USA, 105 (2008), 1232-1237. doi: 10.1073/pnas.0711437105.

[5]

R. W. BeardJ. Lawton and and F. Y. Hadaegh, A coordination architecture for spacecraft formation control, IEEE Trans. Control Syst. Technol., 9 (2001), 777-790. doi: 10.1109/87.960341.

[6]

F. BolleyJ. A. Canizo and J. A. Carrillo, Stochastic mean-field limit: Non-Lipschitz forces and swarming, Math. Mod. Meth. Appl. Sci., 21 (2011), 2179-2210. doi: 10.1142/S0218202511005702.

[7]

J. A. CanizoJ. A. 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.

[8]

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.

[9]

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.

[10]

Y.-P. Choi and J. Haskovec, Cucker-Smale model with normalized communication weights and time delay, Kinetic Relat. Models, 10 (2017), 1011-1033. doi: 10.3934/krm.2017040.

[11]

Y.-P. Choi and Z. Li, Emergent behavior of Cucker-Smale flocking particles with time-lags, Appl. Math. Lett., 86 (2018), 49-56. doi: 10.1016/j.aml.2018.06.018.

[12]

J. CortésS. MartinezT. Karatas and F. Bullo, Coverage control for mobile sensing networks, IEEE Trans. Robot. Autom., 20 (2004), 243-255.

[13]

I. D. CouzinJ. KrauseN. R. Franks and S. Levin, Effective leadership and decision making in animal groups on the move, Nature, 433 (2005), 513-516. doi: 10.1038/nature03236.

[14]

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

[15]

F. Cucker and E. Mordecki, Flocking in noisy environments, J. Math. Pure Appl., 89 (2008), 278-296. doi: 10.1016/j.matpur.2007.12.002.

[16]

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

[17]

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

[18]

J.-G. Dong and L. Qiu, Flocking of the Cucker-Smale model on general digraphs, IEEE Trans. Automat. Control, 62 (2017), 5234-5239. doi: 10.1109/TAC.2016.2631608.

[19]

R. DuanM. Fornasier and G. Toscani, A kinetic flocking model with diffusion, Commun. Math. Phys., 300 (2010), 95-145. doi: 10.1007/s00220-010-1110-z.

[20]

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

[21]

M. FornasierJ. Haskovec and G. Toscani, Fluid dynamic description of flocking via Povzner-Boltzmann equation, Phys. D, 240 (2011), 21-31. doi: 10.1016/j.physd.2010.08.003.

[22]

S.-Y. HaD. Ko and Y. Zhang, Critical coupling strength of the Cucker-Smale model for flocking, Math. Models Methods Appl. Sci., 27 (2017), 1051-1087. doi: 10.1142/S0218202517400097.

[23]

S.-Y. HaK. Lee and D. Levy, Emergence of time-asymptotic flocking in a stochastic Cucker-Smale system, Commun. Math. Sci., 7 (2009), 453-469. doi: 10.4310/CMS.2009.v7.n2.a9.

[24]

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.

[25]

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

[26]

J. K. Hale and S. M. V. Lunel, Introduction to Functional Differential Equations, Applied Mathematical Sciences, 99, Springer, 1993. doi: 10.1007/978-1-4612-4342-7.

[27]

A. JadbabaieJ. Lin and A. Morse, Coordination of groups of mobile autonomous agents using nearest neighbor rules,, IEEE Trans. Automat. Control, 48 (2003), 988-1001. doi: 10.1109/TAC.2003.812781.

[28]

Z. Li and X. Xue, Cucker-Smale flocking under rooted leadership with fixed and switching topologies, SIAM J. Appl. Math., 70 (2010), 3156-3174. doi: 10.1137/100791774.

[29]

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.

[30]

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.

[31]

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.

[32]

J. ParkH. Kim and S.-Y. Ha, Cucker-Smale flocking with inter-particle bonding forces, IEEE Tran. Automat. Control, 55 (2010), 2617-2623. doi: 10.1109/TAC.2010.2061070.

[33]

L. PereaP. Elosegui and G. Gómez, Extension of the Cucker-Smale control law to space fight formations, J. Guidance Contr. Dyn., 32 (2009), 526-536.

[34]

C. Pignotti and E. Trelat, Convergence to consensus of the general finite-dimensional Cucker-Smale model with time-varying delays, preprint, arXiv: 1707.05020.

[35]

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.

[36]

D. Poyato and J. Soler, Euler-type equations and commutators in singular and hyperbolic limits of kinetic Cucker-Smale models, Math. Mod. Meth. Appl. Sci., 27 (2017), 1089-1152. doi: 10.1142/S0218202517400103.

[37]

C. W. Reynolds, Flocks, herds, and schools: A distributed behavioral model, Comput. Graph, 21 (1987), 25-34. doi: 10.1145/37401.37406.

[38]

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

[39]

H. G. TannerA. Jadbabaie and G. J. Pappas, Flocking in fixed and switching networks, IEEE Trans. Automat. Control, 52 (2007), 863-868. doi: 10.1109/TAC.2007.895948.

[40]

G. VásárhelyiC. VirághG. SomorjaiN. TarcaiT. SzörényiT. Nepusz and T. Vicsek, Outdoor flocking and formation flight with autonomous aerial robots, Proc. IEEE/RSJ Int. Conf. Intell. Robots Syst., (2014), 3866-3873.

[41]

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.

[42]

T. Vicsek and A. Zefeiris, Collective motion, Phys. Rep., 517 (2012), 71-140. doi: 10.1016/j.physrep.2012.03.004.

[43]

C. VirághG. VásárhelyiN. TarcaiT. SzörényiG. SomorjaiT. Nepusz and T. Vicsek, Flocking algorithm for autonomous flying robots, Bioinspir. Biomim., 9 (2014), 025012.

show all references

References:
[1]

S. AhnH. ChoiS.-Y. Ha and H. Lee, On the collision avoiding initial-configurations to the Cucker-Smale type flocking models, Comm. Math. Sci., 10 (2012), 625-643. doi: 10.4310/CMS.2012.v10.n2.a10.

[2]

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.

[3]

M. AouchicheO. Favaron and P. Hansen, Variable neighborhood search for extremal graphs. 22. Extending bounds for independence to upper irredundance, Discret Appl. Math., 157 (2009), 3497-3510. doi: 10.1016/j.dam.2009.04.004.

[4]

M. BalleriniN. CabibboR. CandelierA. CavagnaE. CisbaniI. GiardinaV. LecomteA. OrlandiG. ParisiA. ProcacciniM. Viale and V. Zdravkovic, Interaction ruling animal collective behavior depends on topological rather than metric distance: evidence from a field study, Proc. Natl. Acad. Sci. USA, 105 (2008), 1232-1237. doi: 10.1073/pnas.0711437105.

[5]

R. W. BeardJ. Lawton and and F. Y. Hadaegh, A coordination architecture for spacecraft formation control, IEEE Trans. Control Syst. Technol., 9 (2001), 777-790. doi: 10.1109/87.960341.

[6]

F. BolleyJ. A. Canizo and J. A. Carrillo, Stochastic mean-field limit: Non-Lipschitz forces and swarming, Math. Mod. Meth. Appl. Sci., 21 (2011), 2179-2210. doi: 10.1142/S0218202511005702.

[7]

J. A. CanizoJ. A. 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.

[8]

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.

[9]

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.

[10]

Y.-P. Choi and J. Haskovec, Cucker-Smale model with normalized communication weights and time delay, Kinetic Relat. Models, 10 (2017), 1011-1033. doi: 10.3934/krm.2017040.

[11]

Y.-P. Choi and Z. Li, Emergent behavior of Cucker-Smale flocking particles with time-lags, Appl. Math. Lett., 86 (2018), 49-56. doi: 10.1016/j.aml.2018.06.018.

[12]

J. CortésS. MartinezT. Karatas and F. Bullo, Coverage control for mobile sensing networks, IEEE Trans. Robot. Autom., 20 (2004), 243-255.

[13]

I. D. CouzinJ. KrauseN. R. Franks and S. Levin, Effective leadership and decision making in animal groups on the move, Nature, 433 (2005), 513-516. doi: 10.1038/nature03236.

[14]

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

[15]

F. Cucker and E. Mordecki, Flocking in noisy environments, J. Math. Pure Appl., 89 (2008), 278-296. doi: 10.1016/j.matpur.2007.12.002.

[16]

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

[17]

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

[18]

J.-G. Dong and L. Qiu, Flocking of the Cucker-Smale model on general digraphs, IEEE Trans. Automat. Control, 62 (2017), 5234-5239. doi: 10.1109/TAC.2016.2631608.

[19]

R. DuanM. Fornasier and G. Toscani, A kinetic flocking model with diffusion, Commun. Math. Phys., 300 (2010), 95-145. doi: 10.1007/s00220-010-1110-z.

[20]

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

[21]

M. FornasierJ. Haskovec and G. Toscani, Fluid dynamic description of flocking via Povzner-Boltzmann equation, Phys. D, 240 (2011), 21-31. doi: 10.1016/j.physd.2010.08.003.

[22]

S.-Y. HaD. Ko and Y. Zhang, Critical coupling strength of the Cucker-Smale model for flocking, Math. Models Methods Appl. Sci., 27 (2017), 1051-1087. doi: 10.1142/S0218202517400097.

[23]

S.-Y. HaK. Lee and D. Levy, Emergence of time-asymptotic flocking in a stochastic Cucker-Smale system, Commun. Math. Sci., 7 (2009), 453-469. doi: 10.4310/CMS.2009.v7.n2.a9.

[24]

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.

[25]

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

[26]

J. K. Hale and S. M. V. Lunel, Introduction to Functional Differential Equations, Applied Mathematical Sciences, 99, Springer, 1993. doi: 10.1007/978-1-4612-4342-7.

[27]

A. JadbabaieJ. Lin and A. Morse, Coordination of groups of mobile autonomous agents using nearest neighbor rules,, IEEE Trans. Automat. Control, 48 (2003), 988-1001. doi: 10.1109/TAC.2003.812781.

[28]

Z. Li and X. Xue, Cucker-Smale flocking under rooted leadership with fixed and switching topologies, SIAM J. Appl. Math., 70 (2010), 3156-3174. doi: 10.1137/100791774.

[29]

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.

[30]

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.

[31]

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.

[32]

J. ParkH. Kim and S.-Y. Ha, Cucker-Smale flocking with inter-particle bonding forces, IEEE Tran. Automat. Control, 55 (2010), 2617-2623. doi: 10.1109/TAC.2010.2061070.

[33]

L. PereaP. Elosegui and G. Gómez, Extension of the Cucker-Smale control law to space fight formations, J. Guidance Contr. Dyn., 32 (2009), 526-536.

[34]

C. Pignotti and E. Trelat, Convergence to consensus of the general finite-dimensional Cucker-Smale model with time-varying delays, preprint, arXiv: 1707.05020.

[35]

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.

[36]

D. Poyato and J. Soler, Euler-type equations and commutators in singular and hyperbolic limits of kinetic Cucker-Smale models, Math. Mod. Meth. Appl. Sci., 27 (2017), 1089-1152. doi: 10.1142/S0218202517400103.

[37]

C. W. Reynolds, Flocks, herds, and schools: A distributed behavioral model, Comput. Graph, 21 (1987), 25-34. doi: 10.1145/37401.37406.

[38]

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

[39]

H. G. TannerA. Jadbabaie and G. J. Pappas, Flocking in fixed and switching networks, IEEE Trans. Automat. Control, 52 (2007), 863-868. doi: 10.1109/TAC.2007.895948.

[40]

G. VásárhelyiC. VirághG. SomorjaiN. TarcaiT. SzörényiT. Nepusz and T. Vicsek, Outdoor flocking and formation flight with autonomous aerial robots, Proc. IEEE/RSJ Int. Conf. Intell. Robots Syst., (2014), 3866-3873.

[41]

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.

[42]

T. Vicsek and A. Zefeiris, Collective motion, Phys. Rep., 517 (2012), 71-140. doi: 10.1016/j.physrep.2012.03.004.

[43]

C. VirághG. VásárhelyiN. TarcaiT. SzörényiG. SomorjaiT. Nepusz and T. Vicsek, Flocking algorithm for autonomous flying robots, Bioinspir. Biomim., 9 (2014), 025012.

Figure 1.  Digraph connection topology $ \mathcal C $
Figure 2.  The convergence trajectories of the first component velocities satisfying the condition (6.1). Left: Digraph $ \mathcal C $ and right: all-to-all graph
Figure 3.  The convergence trajectories of the first component velocities not satisfying the condition (6.1). Left: Digraph $ {\mathcal C} $ and right: all-to-all graph
Figure 4.  The convergence trajectories of the first component velocities satisfying the condition in Corollary 3.1
Figure 5.  The trajectories of the first component velocities not satisfying the condition in Corollary 3.1
Table1 
$ \boldsymbol x_1(t) $ $ (1, 0) $ $ \boldsymbol v_1(t) $ $ \frac{e^{-10}}{672 \sqrt{2}}(1, -2) $ $ \boldsymbol x_2(t) $ $ (0, 1) $ $ \boldsymbol v_2(t) $ $ \frac{e^{-10}}{672 \sqrt{2}}(3, -4) $
$ \boldsymbol x_3(t) $ $ (-1, 0) $ $ \boldsymbol v_3(t) $ $ \frac{e^{-10}}{672 \sqrt{2}}(5, 6) $ $ \boldsymbol x_4(t) $ $ (0, -1) $ $ \boldsymbol v_4(t) $ $ \frac{e^{-10}}{672 \sqrt{2}}(-7, -8) $
$ \boldsymbol x_1(t) $ $ (1, 0) $ $ \boldsymbol v_1(t) $ $ \frac{e^{-10}}{672 \sqrt{2}}(1, -2) $ $ \boldsymbol x_2(t) $ $ (0, 1) $ $ \boldsymbol v_2(t) $ $ \frac{e^{-10}}{672 \sqrt{2}}(3, -4) $
$ \boldsymbol x_3(t) $ $ (-1, 0) $ $ \boldsymbol v_3(t) $ $ \frac{e^{-10}}{672 \sqrt{2}}(5, 6) $ $ \boldsymbol x_4(t) $ $ (0, -1) $ $ \boldsymbol v_4(t) $ $ \frac{e^{-10}}{672 \sqrt{2}}(-7, -8) $
Table2 
$ \boldsymbol x_1(t) $ $ (1, 0) $ $ \boldsymbol v_1(t) $ $ (1, -2) $ $ \boldsymbol x_2(t) $ $ (0, 1) $ $ \boldsymbol v_2(t) $ $ (3, -4) $
$ \boldsymbol x_3(t) $ $ (-1, 0) $ $ \boldsymbol v_3(t) $ $ (5, 6) $ $ \boldsymbol x_4(t) $ $ (0, -1) $ $ \boldsymbol v_4(t) $ $ (-7, -8) $
$ \boldsymbol x_1(t) $ $ (1, 0) $ $ \boldsymbol v_1(t) $ $ (1, -2) $ $ \boldsymbol x_2(t) $ $ (0, 1) $ $ \boldsymbol v_2(t) $ $ (3, -4) $
$ \boldsymbol x_3(t) $ $ (-1, 0) $ $ \boldsymbol v_3(t) $ $ (5, 6) $ $ \boldsymbol x_4(t) $ $ (0, -1) $ $ \boldsymbol v_4(t) $ $ (-7, -8) $
Table3 
$\boldsymbol x_1(t)$$(1, 0)$$\boldsymbol v_1(t)$$\frac{e^{-10}}{7056 \sqrt{2}}(1, -2)$$\boldsymbol x_2(t)$$(0, 1)$$\boldsymbol v_2(t)$$\frac{e^{-10}}{7056 \sqrt{2}}(3, -4)$
$\boldsymbol x_3(t)$$(-1, 0)$$\boldsymbol v_3(t)$$\frac{e^{-10}}{7056 \sqrt{2}}(5, 6)$$\boldsymbol x_4(t)$$(0, -1)$$\boldsymbol v_4(t)$$\frac{e^{-10}}{7056 \sqrt{2}}(-7, -8)$
$\boldsymbol x_1(t)$$(1, 0)$$\boldsymbol v_1(t)$$\frac{e^{-10}}{7056 \sqrt{2}}(1, -2)$$\boldsymbol x_2(t)$$(0, 1)$$\boldsymbol v_2(t)$$\frac{e^{-10}}{7056 \sqrt{2}}(3, -4)$
$\boldsymbol x_3(t)$$(-1, 0)$$\boldsymbol v_3(t)$$\frac{e^{-10}}{7056 \sqrt{2}}(5, 6)$$\boldsymbol x_4(t)$$(0, -1)$$\boldsymbol v_4(t)$$\frac{e^{-10}}{7056 \sqrt{2}}(-7, -8)$
Table4 
$ \boldsymbol x_1(t) $ $ (1, 0) $ $ \boldsymbol v_1(t) $ $ (1, -2) $ $ \boldsymbol x_2(t) $ $ (0, 1) $ $ \boldsymbol v_2(t) $ $ (3, -4) $
$ \boldsymbol x_3(t) $ $ (-1, 0) $ $ \boldsymbol v_3(t) $ $ (5, 6) $ $ \boldsymbol x_4(t) $ $ (0, -1) $ $ \boldsymbol v_4(t) $ $ (-7, -8) $
$ \boldsymbol x_1(t) $ $ (1, 0) $ $ \boldsymbol v_1(t) $ $ (1, -2) $ $ \boldsymbol x_2(t) $ $ (0, 1) $ $ \boldsymbol v_2(t) $ $ (3, -4) $
$ \boldsymbol x_3(t) $ $ (-1, 0) $ $ \boldsymbol v_3(t) $ $ (5, 6) $ $ \boldsymbol x_4(t) $ $ (0, -1) $ $ \boldsymbol v_4(t) $ $ (-7, -8) $
[1]

Martial Agueh, Reinhard Illner, Ashlin Richardson. Analysis and simulations of a refined flocking and swarming model of Cucker-Smale type. Kinetic & Related Models, 2011, 4 (1) : 1-16. doi: 10.3934/krm.2011.4.1

[2]

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