October  2019, 24(10): 5569-5596. 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, 2019, 24 (10) : 5569-5596. 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.  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. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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

[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.  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. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  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. 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.  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. 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.  Google Scholar

[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.   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. 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.  Google Scholar

[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.   Google Scholar

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

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.  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. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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

[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.  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. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  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. 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.  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. 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.  Google Scholar

[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.   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. 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.  Google Scholar

[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.   Google Scholar

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

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]

A. M. Elaiw, N. H. AlShamrani, A. Abdel-Aty, H. Dutta. Stability analysis of a general HIV dynamics model with multi-stages of infected cells and two routes of infection. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020441

[2]

Stefan Ruschel, Serhiy Yanchuk. The spectrum of delay differential equations with multiple hierarchical large delays. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 151-175. doi: 10.3934/dcdss.2020321

[3]

Hirokazu Ninomiya. Entire solutions of the Allen–Cahn–Nagumo equation in a multi-dimensional space. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 395-412. doi: 10.3934/dcds.2020364

[4]

Shun Zhang, Jianlin Jiang, Su Zhang, Yibing Lv, Yuzhen Guo. ADMM-type methods for generalized multi-facility Weber problem. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020171

[5]

Ripeng Huang, Shaojian Qu, Xiaoguang Yang, Zhimin Liu. Multi-stage distributionally robust optimization with risk aversion. Journal of Industrial & Management Optimization, 2021, 17 (1) : 233-259. doi: 10.3934/jimo.2019109

[6]

Hongguang Ma, Xiang Li. Multi-period hazardous waste collection planning with consideration of risk stability. Journal of Industrial & Management Optimization, 2021, 17 (1) : 393-408. doi: 10.3934/jimo.2019117

[7]

Lingju Kong, Roger Nichols. On principal eigenvalues of biharmonic systems. Communications on Pure & Applied Analysis, 2021, 20 (1) : 1-15. doi: 10.3934/cpaa.2020254

[8]

Peizhao Yu, Guoshan Zhang, Yi Zhang. Decoupling of cubic polynomial matrix systems. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 13-26. doi: 10.3934/naco.2020012

[9]

Laurence Cherfils, Stefania Gatti, Alain Miranville, Rémy Guillevin. Analysis of a model for tumor growth and lactate exchanges in a glioma. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020457

[10]

Laurent Di Menza, Virginie Joanne-Fabre. An age group model for the study of a population of trees. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020464

[11]

Eduard Feireisl, Elisabetta Rocca, Giulio Schimperna, Arghir Zarnescu. Weak sequential stability for a nonlinear model of nematic electrolytes. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 219-241. doi: 10.3934/dcdss.2020366

[12]

Ilyasse Lamrani, Imad El Harraki, Ali Boutoulout, Fatima-Zahrae El Alaoui. Feedback stabilization of bilinear coupled hyperbolic systems. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020434

[13]

Felix Finster, Jürg Fröhlich, Marco Oppio, Claudio F. Paganini. Causal fermion systems and the ETH approach to quantum theory. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020451

[14]

Xiyou Cheng, Zhitao Zhang. Structure of positive solutions to a class of Schrödinger systems. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020461

[15]

Simon Hochgerner. Symmetry actuated closed-loop Hamiltonian systems. Journal of Geometric Mechanics, 2020, 12 (4) : 641-669. doi: 10.3934/jgm.2020030

[16]

Javier Fernández, Cora Tori, Marcela Zuccalli. Lagrangian reduction of nonholonomic discrete mechanical systems by stages. Journal of Geometric Mechanics, 2020, 12 (4) : 607-639. doi: 10.3934/jgm.2020029

[17]

Lingwei Ma, Zhenqiu Zhang. Monotonicity for fractional Laplacian systems in unbounded Lipschitz domains. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 537-552. doi: 10.3934/dcds.2020268

[18]

Peter H. van der Kamp, D. I. McLaren, G. R. W. Quispel. Homogeneous darboux polynomials and generalising integrable ODE systems. Journal of Computational Dynamics, 2021, 8 (1) : 1-8. doi: 10.3934/jcd.2021001

[19]

Yuri Fedorov, Božidar Jovanović. Continuous and discrete Neumann systems on Stiefel varieties as matrix generalizations of the Jacobi–Mumford systems. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020375

[20]

Weiwei Liu, Jinliang Wang, Yuming Chen. Threshold dynamics of a delayed nonlocal reaction-diffusion cholera model. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020316

2019 Impact Factor: 1.27

Metrics

  • PDF downloads (99)
  • HTML views (359)
  • Cited by (1)

Other articles
by authors

[Back to Top]