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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

Jiu-Gang Dong 1,, , Seung-Yeal Ha 2, and  Doheon Kim 3,

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.

Keywords: Cucker-Smale model, flocking, communication delays, multi-agent systems, digraphs.
Mathematics Subject Classification: 34D05, 39A12, 70F99.
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. Ahn, H. Choi, S.-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. Google Scholar

[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. Google Scholar

[3]

M. Aouchiche, O. 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. Google Scholar

[4]

M. Ballerini, N. Cabibbo, R. Candelier, A. Cavagna, E. Cisbani, I. Giardina, V. Lecomte, A. Orlandi, G. Parisi, A. Procaccini, M. 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. Google Scholar

[5]

R. W. Beard, J. 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. Google Scholar

[6]

F. Bolley, J. 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. Google Scholar

[7]

J. A. Canizo, J. 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. Google Scholar

[8]

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. Google Scholar

[9]

J. Cho, S.-Y. Ha, F. Huang, C. 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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[12]

J. Cortés, S. Martinez, T. Karatas and F. Bullo, Coverage control for mobile sensing networks, IEEE Trans. Robot. Autom., 20 (2004), 243-255. Google Scholar

[13]

I. D. Couzin, J. Krause, N. 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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[16]

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

[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. Google Scholar

[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. Google Scholar

[19]

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

[20]

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

[21]

M. Fornasier, J. 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. Google Scholar

[22]

S.-Y. Ha, D. 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. Google Scholar

[23]

S.-Y. Ha, K. 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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[27]

A. Jadbabaie, J. 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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[32]

J. Park, H. 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. Google Scholar

[33]

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

[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.Google Scholar

[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. Google Scholar

[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. Google Scholar

[37]

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

[38]

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

[39]

H. G. Tanner, A. 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. Google Scholar

[40]

G. Vásárhelyi, C. Virágh, G. Somorjai, N. Tarcai, T. Szörényi, T. Nepusz and T. Vicsek, Outdoor flocking and formation flight with autonomous aerial robots, Proc. IEEE/RSJ Int. Conf. Intell. Robots Syst., (2014), 3866-3873. Google Scholar

[41]

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

[42]

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

[43]

C. Virágh, G. Vásárhelyi, N. Tarcai, T. Szörényi, G. Somorjai, T. Nepusz and T. Vicsek, Flocking algorithm for autonomous flying robots, Bioinspir. Biomim., 9 (2014), 025012. Google Scholar

show all references

References:
[1]

S. Ahn, H. Choi, S.-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. Google Scholar

[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. Google Scholar

[3]

M. Aouchiche, O. 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. Google Scholar

[4]

M. Ballerini, N. Cabibbo, R. Candelier, A. Cavagna, E. Cisbani, I. Giardina, V. Lecomte, A. Orlandi, G. Parisi, A. Procaccini, M. 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. Google Scholar

[5]

R. W. Beard, J. 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. Google Scholar

[6]

F. Bolley, J. 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. Google Scholar

[7]

J. A. Canizo, J. 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. Google Scholar

[8]

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. Google Scholar

[9]

J. Cho, S.-Y. Ha, F. Huang, C. 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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[12]

J. Cortés, S. Martinez, T. Karatas and F. Bullo, Coverage control for mobile sensing networks, IEEE Trans. Robot. Autom., 20 (2004), 243-255. Google Scholar

[13]

I. D. Couzin, J. Krause, N. 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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[16]

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

[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. Google Scholar

[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. Google Scholar

[19]

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

[20]

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

[21]

M. Fornasier, J. 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. Google Scholar

[22]

S.-Y. Ha, D. 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. Google Scholar

[23]

S.-Y. Ha, K. 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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[27]

A. Jadbabaie, J. 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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[32]

J. Park, H. 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. Google Scholar

[33]

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

[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.Google Scholar

[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. Google Scholar

[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. Google Scholar

[37]

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

[38]

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

[39]

H. G. Tanner, A. 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. Google Scholar

[40]

G. Vásárhelyi, C. Virágh, G. Somorjai, N. Tarcai, T. Szörényi, T. Nepusz and T. Vicsek, Outdoor flocking and formation flight with autonomous aerial robots, Proc. IEEE/RSJ Int. Conf. Intell. Robots Syst., (2014), 3866-3873. Google Scholar

[41]

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

[42]

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

[43]

C. Virágh, G. Vásárhelyi, N. Tarcai, T. Szörényi, G. Somorjai, T. Nepusz and T. Vicsek, Flocking algorithm for autonomous flying robots, Bioinspir. Biomim., 9 (2014), 025012. Google Scholar

Figure 1.  Digraph connection topology $ \mathcal C $
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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
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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
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Figure 4.  The convergence trajectories of the first component velocities satisfying the condition in Corollary 3.1
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Figure 5.  The trajectories of the first component velocities not satisfying the condition in Corollary 3.1
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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) $
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$ \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) $
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$ \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)$
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$\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) $
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$ \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) $
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