Flocking of non-identical Cucker-Smale models on general coupling network

Yu-Jhe Huang; Zhong-Fu Huang; Jonq Juang; Yu-Hao Liang

doi:10.3934/dcdsb.2020155

Article Contents

2021, Volume 26, Issue 2: 1111-1127. Doi: 10.3934/dcdsb.2020155

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Flocking of non-identical Cucker-Smale models on general coupling network

1.
Department of Applied Mathematics, National Chiao Tung University, Hsinchu, Taiwan, ROC
2.
Department of Applied Mathematics, National University of Kaohsiung, Kaohsiung 700, Taiwan, ROC

^* Corresponding author: Jonq Juang
^* Corresponding author: Jonq Juang

Received: July 2019

Revised: December 2019

Early access: May 2020

Published: February 2021

This work is partially supported by the Ministry of Science and Technology of the Republic of China under grant No.\ MOST 107-2115-M-009-011-MY2 and No. MOST 107-2115-M-390-006-MY2

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Abstract

The purpose of the paper is to investigate the flocking behavior of the discrete-time Cucker-Smale(C-S) model under general interaction network topologies with agents having their free-will accelerations. We prove theoretically that if the free-will accelerations of agents are summable, then, for any given initial conditions, the solution achieves flocking with a finite moving speed by suitably choosing the time step as well as the communication rate of the system or the strength of the interaction between agents. In particular, if the communication rate $ \beta $ of the system is subcritical, i.e., $ \beta $ is less than a critical value $ \beta_c $, then flocking holds for any initial conditions regardless of the strength of the interaction between agents. While, if the communication rate is critical ($ \beta = \beta_c $) or supercritical ($ \beta > \beta_c $), then flocking can only be achieved by making the strength of the interaction large enough. We also present some numerical simulations to support our obtained theoretical results.

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Mathematics Subject Classification: Primary: 91C20, 92D50; Secondary: 15B48.

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Figure 1. The network consisting of $ 9 $ vertices has a spanning tree. For this network, $ \mathcal{R} = \{1,2,3\} $, $ n = 9 $, $ \ell = 3 $, $ h = 3 $ and $ r = 2 $

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Figure 2. The graph of $ G $

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Figure 3. (Subcritical) Numerical simulation for model Eq. (4) with the network given in Fig. 1. This simulation result shows the solution achieves flocking. Here parameters in Eq. (4) are chosen as $ \beta = 1/10 $, $ \kappa = 10 $ and $ \varepsilon = 0.001 $. The initial conditions are randomly chosen and they satisfy $ a\approx 0.8932 $, $ b\approx 0.8104 $. Here $ a,b $ are defined in (20a)

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Figure 4. (Critical) Numerical simulation for model Eq. (4) with the network given in Fig. 1. This simulation result shows the solution achieves flocking. Here parameters in Eq. (4) are chosen as $ \beta = 1/6 $, $ \kappa = 120 $ and $ \varepsilon = 0.001 $. The initial conditions are randomly chosen and they satisfy $ a\approx 0.8932 $, $ b\approx 0.8104 $. Here $ a,b $ are defined in (20a)

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Figure 5. (Supercritical) Numerical simulation for model Eq. (4) with the network given in Fig. 1. This simulation result shows the solution achieves flocking. Here parameters in Eq. (4) are chosen as $ \beta = 1/3 $, $ \kappa = 200 $ and $ \varepsilon = 0.001 $. The initial conditions are randomly chosen and they satisfy $ a\approx 0.8932 $, $ b\approx 0.8104 $. Here $ a,b $ are defined in (20a)

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Figure 6. Numerical simulation for model Eq. (4) under the network provided in Fig. 1. This simulation result shows the solution does not achieve flocking. Here parameters in Eq. (4) are chosen as $ \beta = 1 $ (supercritical), $ \kappa = 0.1 $ and $ \varepsilon = 0.001 $. The initial conditions are randomly chosen and they satisfy $ a\approx 0.2872 $, $ b\approx 0.2850 $. Here $ a,b $ are defined in (20a). Such set of parameters and initial conditions do not satisfy the sufficient condition (23)

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