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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. 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. doi: 10.1007/s102080010030. Google Scholar

[3]

F. Cucker and S. Smale, Emergent behavior in flocks,, IEEE Trans. Automat. Control, 52 (2007), 852. 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. 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. 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. 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. doi: 10.1109/TAC.2010.2046113. Google Scholar

[8]

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

[9]

R. A. Horn and C. R. Johnson, Matrix Analysis,, second edition, (2013). Google Scholar

[10]

Z. Li, Effectual leadership in flocks with hierarchy and individual preference,, Discrete Contin. Dyn. Syst., 34 (2014), 3683. 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. 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. 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. 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. 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. 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. doi: 10.1109/TAC.2010.2061070. Google Scholar

[17]

J. Shen, Cucker-Smale flocking under hierarchical leadership,, SIAM J. Appl. Math., 68 (2007), 694. 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). 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. 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. 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. doi: 10.1007/s102080010030. Google Scholar

[3]

F. Cucker and S. Smale, Emergent behavior in flocks,, IEEE Trans. Automat. Control, 52 (2007), 852. 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. 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. 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. 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. doi: 10.1109/TAC.2010.2046113. Google Scholar

[8]

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

[9]

R. A. Horn and C. R. Johnson, Matrix Analysis,, second edition, (2013). Google Scholar

[10]

Z. Li, Effectual leadership in flocks with hierarchy and individual preference,, Discrete Contin. Dyn. Syst., 34 (2014), 3683. 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. 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. 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. 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. 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. 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. doi: 10.1109/TAC.2010.2061070. Google Scholar

[17]

J. Shen, Cucker-Smale flocking under hierarchical leadership,, SIAM J. Appl. Math., 68 (2007), 694. 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). 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. doi: 10.1016/j.sysconle.2011.10.006. Google Scholar

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