# American Institute of Mathematical Sciences

• Previous Article
Global analysis of a model of competition in the chemostat with internal inhibitor
• DCDS-B Home
• This Issue
• Next Article
Numerical study of vanishing and spreading dynamics of chemotaxis systems with logistic source and a free boundary
February  2021, 26(2): 1111-1127. doi: 10.3934/dcdsb.2020155

## 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

Received  July 2019 Revised  December 2019 Published  May 2020

Fund Project: 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

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.

Citation: 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
##### References:

show all references

##### References:
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 $r = 2$
The graph of $G$
(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)
(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)
(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)
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)
 [1] Maoli Chen, Xiao Wang, Yicheng Liu. Collision-free flocking for a time-delay system. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 1223-1241. doi: 10.3934/dcdsb.2020251 [2] Rong Wang, Yihong Du. Long-time dynamics of a diffusive epidemic model with free boundaries. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020360 [3] Yoichi Enatsu, Emiko Ishiwata, Takeo Ushijima. Traveling wave solution for a diffusive simple epidemic model with a free boundary. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 835-850. doi: 10.3934/dcdss.2020387 [4] Alberto Bressan, Sondre Tesdal Galtung. A 2-dimensional shape optimization problem for tree branches. Networks & Heterogeneous Media, 2020  doi: 10.3934/nhm.2020031 [5] Shahede Omidi, Jafar Fathali. Inverse single facility location problem on a tree with balancing on the distance of server to clients. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2021017 [6] Min Xi, Wenyu Sun, Jun Chen. Survey of derivative-free optimization. Numerical Algebra, Control & Optimization, 2020, 10 (4) : 537-555. doi: 10.3934/naco.2020050 [7] Claude-Michel Brauner, Luca Lorenzi. Instability of free interfaces in premixed flame propagation. Discrete & Continuous Dynamical Systems - S, 2021, 14 (2) : 575-596. doi: 10.3934/dcdss.2020363 [8] Aurelia Dymek. Proximality of multidimensional $\mathscr{B}$-free systems. Discrete & Continuous Dynamical Systems - A, 2021  doi: 10.3934/dcds.2021013 [9] Emre Esentürk, Juan Velazquez. Large time behavior of exchange-driven growth. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 747-775. doi: 10.3934/dcds.2020299 [10] José Luiz Boldrini, Jonathan Bravo-Olivares, Eduardo Notte-Cuello, Marko A. Rojas-Medar. Asymptotic behavior of weak and strong solutions of the magnetohydrodynamic equations. Electronic Research Archive, 2021, 29 (1) : 1783-1801. doi: 10.3934/era.2020091 [11] Maho Endo, Yuki Kaneko, Yoshio Yamada. Free boundary problem for a reaction-diffusion equation with positive bistable nonlinearity. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3375-3394. doi: 10.3934/dcds.2020033 [12] Chueh-Hsin Chang, Chiun-Chuan Chen, Chih-Chiang Huang. Traveling wave solutions of a free boundary problem with latent heat effect. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021028 [13] Wei Feng, Michael Freeze, Xin Lu. On competition models under allee effect: Asymptotic behavior and traveling waves. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5609-5626. doi: 10.3934/cpaa.2020256 [14] Veena Goswami, Gopinath Panda. Optimal customer behavior in observable and unobservable discrete-time queues. Journal of Industrial & Management Optimization, 2021, 17 (1) : 299-316. doi: 10.3934/jimo.2019112 [15] Junyong Eom, Kazuhiro Ishige. Large time behavior of ODE type solutions to nonlinear diffusion equations. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3395-3409. doi: 10.3934/dcds.2019229 [16] Lin Shi, Dingshi Li, Kening Lu. Limiting behavior of unstable manifolds for spdes in varying phase spaces. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021020 [17] Huijuan Song, Bei Hu, Zejia Wang. Stationary solutions of a free boundary problem modeling the growth of vascular tumors with a necrotic core. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 667-691. doi: 10.3934/dcdsb.2020084 [18] Lei Yang, Lianzhang Bao. Numerical study of vanishing and spreading dynamics of chemotaxis systems with logistic source and a free boundary. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 1083-1109. doi: 10.3934/dcdsb.2020154 [19] Jingjing Wang, Zaiyun Peng, Zhi Lin, Daqiong Zhou. On the stability of solutions for the generalized vector quasi-equilibrium problems via free-disposal set. Journal of Industrial & Management Optimization, 2021, 17 (2) : 869-887. doi: 10.3934/jimo.2020002 [20] Yichen Zhang, Meiqiang Feng. A coupled $p$-Laplacian elliptic system: Existence, uniqueness and asymptotic behavior. Electronic Research Archive, 2020, 28 (4) : 1419-1438. doi: 10.3934/era.2020075

2019 Impact Factor: 1.27

## Tools

Article outline

Figures and Tables