American Institute of Mathematical Sciences

Advanced
October  2016, 21(8): 2587-2599. doi: 10.3934/dcdsb.2016062

A new discrete Cucker-Smale flocking model under hierarchical leadership

Chun-Hsien Li 1, and  Suh-Yuh Yang 2,

1. 

Department of Mathematics, National Kaohsiung Normal University, Yanchao District, Kaohsiung City 82444, Taiwan
2. 

Department of Mathematics, National Central University, Jhongli District, Taoyuan City 32001, Taiwan

Received  July 2015 Revised  May 2016 Published  September 2016

This paper studies the flocking behavior in a new discrete-time Cucker-Smale model under hierarchical leadership. The features of this model are that each individual has its own intrinsic nonlinear dynamics and the interaction between individuals follows a hierarchical leadership structure. Based on a specific matrix norm, we prove that the conditional flocking indeed occurs. Numerical experiments are given to confirm the theoretical results.
Keywords: discrete-time system., Flocking, emergence behavior, Cucker-Smale model, hierarchical leadership.
Mathematics Subject Classification: Primary: 92D25, 92D50; Secondary: 92B25, 93C5.
Citation: Chun-Hsien Li, Suh-Yuh Yang. A new discrete Cucker-Smale flocking model under hierarchical leadership. Discrete & Continuous Dynamical Systems - B, 2016, 21 (8) : 2587-2599. doi: 10.3934/dcdsb.2016062
References:
[1]

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

[2]

F. Cucker and S. Smale, Best choices for regularization parameters in learning theory: On the bias-variance problem, Found. Comput. Math., 2 (2002), 413-428. doi: 10.1007/s102080010030.  Google Scholar

[3]

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

[4]

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

[5]

F. Dalmao and E. Moedecki, Cucker-Smale flocking under hierarchical leadership and random interactions, SIAM J. Appl. Math., 71 (2011), 1307-1316. doi: 10.1137/100785910.  Google Scholar

[6]

J. Dong, Flocking under hierarchical leadership with a free-will leader, Internat. J. Robust Nonlinear Control, 23 (2013), 1891-1898.  Google Scholar

[7]

S. Ha, T. Ha and J. Kim, Emergent behavior of a Cucker-Smale type particle model with nonlinear velocity coupling, IEEE Trans. Automat. Control, 55 (2010), 1679-1683. doi: 10.1109/TAC.2010.2046113.  Google Scholar

[8]

R. A. Horn and C. R. Johnson, Topics in Matrix Analysis, Cambridge University Press, Cambridge, 1991. doi: 10.1017/CBO9780511840371.  Google Scholar

[9]

R. A. Horn and C. R. Johnson, Matrix Analysis, second edition, Cambridge University Press, Cambridge, 2013.  Google Scholar

[10]

Z. Li, Effectual leadership in flocks with hierarchy and individual preference, Discrete Contin. Dyn. Syst., 34 (2014), 3683-3702. doi: 10.3934/dcds.2014.34.3683.  Google Scholar

[11]

Z. Li and S. Ha, Cucker-Smale flocking with alternating leaders, Quart. Appl. Math., 73 (2015), 693-709. doi: 10.1090/qam/1401.  Google Scholar

[12]

Z. Li, S. Ha and X. Xue, Emergent phenomena in an ensemble of Cucker-Smale particles under joint rooted leadership, Math. Models Methods Appl. Sci., 24 (2014), 1389-1419. doi: 10.1142/S0218202514500043.  Google Scholar

[13]

Z. Li and X. Xue, Cucker-Smale flocking under rooted leadership with free-will agents, Phys. A, 410 (2014), 205-217. doi: 10.1016/j.physa.2014.05.008.  Google Scholar

[14]

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

[15]

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

[16]

J. Park, H. Kim and S. Ha, Cucker-Smale flocking with inter-particle bonding forces, IEEE Trans. Automat. Control, 55 (2010), 2617-2623. doi: 10.1109/TAC.2010.2061070.  Google Scholar

[17]

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

[18]

Q. Song, F. Liu, J. Cao and J. Qiu, Cucker-Smale flocking with bounded cohesive and repulsive forces, Abstr. Appl. Anal., 2013, Art. ID 783279, 9 pp.  Google Scholar

[19]

J. Zhou, X. Wu, W. Yu, M. Small and J. Lu, Flocking of multi-agent dynamical systems based on pseudo-leader mechanism, Systems Control Lett., 61 (2012), 195-202. doi: 10.1016/j.sysconle.2011.10.006.  Google Scholar

show all references

References:
[1]

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

[2]

F. Cucker and S. Smale, Best choices for regularization parameters in learning theory: On the bias-variance problem, Found. Comput. Math., 2 (2002), 413-428. doi: 10.1007/s102080010030.  Google Scholar

[3]

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

[4]

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

[5]

F. Dalmao and E. Moedecki, Cucker-Smale flocking under hierarchical leadership and random interactions, SIAM J. Appl. Math., 71 (2011), 1307-1316. doi: 10.1137/100785910.  Google Scholar

[6]

J. Dong, Flocking under hierarchical leadership with a free-will leader, Internat. J. Robust Nonlinear Control, 23 (2013), 1891-1898.  Google Scholar

[7]

S. Ha, T. Ha and J. Kim, Emergent behavior of a Cucker-Smale type particle model with nonlinear velocity coupling, IEEE Trans. Automat. Control, 55 (2010), 1679-1683. doi: 10.1109/TAC.2010.2046113.  Google Scholar

[8]

R. A. Horn and C. R. Johnson, Topics in Matrix Analysis, Cambridge University Press, Cambridge, 1991. doi: 10.1017/CBO9780511840371.  Google Scholar

[9]

R. A. Horn and C. R. Johnson, Matrix Analysis, second edition, Cambridge University Press, Cambridge, 2013.  Google Scholar

[10]

Z. Li, Effectual leadership in flocks with hierarchy and individual preference, Discrete Contin. Dyn. Syst., 34 (2014), 3683-3702. doi: 10.3934/dcds.2014.34.3683.  Google Scholar

[11]

Z. Li and S. Ha, Cucker-Smale flocking with alternating leaders, Quart. Appl. Math., 73 (2015), 693-709. doi: 10.1090/qam/1401.  Google Scholar

[12]

Z. Li, S. Ha and X. Xue, Emergent phenomena in an ensemble of Cucker-Smale particles under joint rooted leadership, Math. Models Methods Appl. Sci., 24 (2014), 1389-1419. doi: 10.1142/S0218202514500043.  Google Scholar

[13]

Z. Li and X. Xue, Cucker-Smale flocking under rooted leadership with free-will agents, Phys. A, 410 (2014), 205-217. doi: 10.1016/j.physa.2014.05.008.  Google Scholar

[14]

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

[15]

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

[16]

J. Park, H. Kim and S. Ha, Cucker-Smale flocking with inter-particle bonding forces, IEEE Trans. Automat. Control, 55 (2010), 2617-2623. doi: 10.1109/TAC.2010.2061070.  Google Scholar

[17]

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

[18]

Q. Song, F. Liu, J. Cao and J. Qiu, Cucker-Smale flocking with bounded cohesive and repulsive forces, Abstr. Appl. Anal., 2013, Art. ID 783279, 9 pp.  Google Scholar

[19]

J. Zhou, X. Wu, W. Yu, M. Small and J. Lu, Flocking of multi-agent dynamical systems based on pseudo-leader mechanism, Systems Control Lett., 61 (2012), 195-202. doi: 10.1016/j.sysconle.2011.10.006.  Google Scholar

[1]

Seung-Yeal Ha, Jinwook Jung, Peter Kuchling. Emergence of anomalous flocking in the fractional Cucker-Smale model. Discrete & Continuous Dynamical Systems, 2019, 39 (9) : 5465-5489. doi: 10.3934/dcds.2019223
[2]

Agnieszka B. Malinowska, Tatiana Odzijewicz. Optimal control of the discrete-time fractional-order Cucker-Smale model. Discrete & Continuous Dynamical Systems - B, 2018, 23 (1) : 347-357. doi: 10.3934/dcdsb.2018023
[3]

Lining Ru, Xiaoping Xue. Flocking of Cucker-Smale model with intrinsic dynamics. Discrete & Continuous Dynamical Systems - B, 2019, 24 (12) : 6817-6835. doi: 10.3934/dcdsb.2019168
[4]

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
[5]

Mauro Rodriguez Cartabia. Cucker-Smale model with time delay. Discrete & Continuous Dynamical Systems, 2021  doi: 10.3934/dcds.2021195
[6]

Young-Pil Choi, Cristina Pignotti. Emergent behavior of Cucker-Smale model with normalized weights and distributed time delays. Networks & Heterogeneous Media, 2019, 14 (4) : 789-804. doi: 10.3934/nhm.2019032
[7]

Jan Haskovec, Ioannis Markou. Asymptotic flocking in the Cucker-Smale model with reaction-type delays in the non-oscillatory regime. Kinetic & Related Models, 2020, 13 (4) : 795-813. doi: 10.3934/krm.2020027
[8]

Chiun-Chuan Chen, Seung-Yeal Ha, Xiongtao Zhang. The global well-posedness of the kinetic Cucker-Smale flocking model with chemotactic movements. Communications on Pure & Applied Analysis, 2018, 17 (2) : 505-538. doi: 10.3934/cpaa.2018028
[9]

Young-Pil Choi, Jan Haskovec. Cucker-Smale model with normalized communication weights and time delay. Kinetic & Related Models, 2017, 10 (4) : 1011-1033. doi: 10.3934/krm.2017040
[10]

Martin Friesen, Oleksandr Kutoviy. Stochastic Cucker-Smale flocking dynamics of jump-type. Kinetic & Related Models, 2020, 13 (2) : 211-247. doi: 10.3934/krm.2020008
[11]

Hyeong-Ohk Bae, Young-Pil Choi, Seung-Yeal Ha, Moon-Jin Kang. Asymptotic flocking dynamics of Cucker-Smale particles immersed in compressible fluids. Discrete & Continuous Dynamical Systems, 2014, 34 (11) : 4419-4458. doi: 10.3934/dcds.2014.34.4419
[12]

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
[13]

Young-Pil Choi, Seung-Yeal Ha, Jeongho Kim. Propagation of regularity and finite-time collisions for the thermomechanical Cucker-Smale model with a singular communication. Networks & Heterogeneous Media, 2018, 13 (3) : 379-407. doi: 10.3934/nhm.2018017
[14]

Yu-Jhe Huang, Zhong-Fu Huang, Jonq Juang, Yu-Hao Liang. Flocking of non-identical Cucker-Smale models on general coupling network. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 1111-1127. doi: 10.3934/dcdsb.2020155
[15]

Young-Pil Choi, Samir Salem. Cucker-Smale flocking particles with multiplicative noises: Stochastic mean-field limit and phase transition. Kinetic & Related Models, 2019, 12 (3) : 573-592. doi: 10.3934/krm.2019023
[16]

Seung-Yeal Ha, Doheon Kim, Weiyuan Zou. Slow flocking dynamics of the Cucker-Smale ensemble with a chemotactic movement in a temperature field. Kinetic & Related Models, 2020, 13 (4) : 759-793. doi: 10.3934/krm.2020026
[17]

Zhisu Liu, Yicheng Liu, Xiang Li. Flocking and line-shaped spatial configuration to delayed Cucker-Smale models. Discrete & Continuous Dynamical Systems - B, 2021, 26 (7) : 3693-3716. doi: 10.3934/dcdsb.2020253
[18]

Seung-Yeal Ha, Shi Jin. Local sensitivity analysis for the Cucker-Smale model with random inputs. Kinetic & Related Models, 2018, 11 (4) : 859-889. doi: 10.3934/krm.2018034
[19]

Marco Caponigro, Massimo Fornasier, Benedetto Piccoli, Emmanuel Trélat. Sparse stabilization and optimal control of the Cucker-Smale model. Mathematical Control & Related Fields, 2013, 3 (4) : 447-466. doi: 10.3934/mcrf.2013.3.447
[20]

Seung-Yeal Ha, Dongnam Ko, Yinglong Zhang. Remarks on the critical coupling strength for the Cucker-Smale model with unit speed. Discrete & Continuous Dynamical Systems, 2018, 38 (6) : 2763-2793. doi: 10.3934/dcds.2018116

2020 Impact Factor: 1.327

Tools

Metrics

  • PDF downloads (299)
  • HTML views (0)
  • Cited by (3)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy