# American Institute of Mathematical Sciences

July  2021, 26(7): 3693-3716. doi: 10.3934/dcdsb.2020253

## Flocking and line-shaped spatial configuration to delayed Cucker-Smale models

 Department of Mathematics, College of Liberal Arts and Sciences, National University of Defense Technology, Changsha, Hunan 410073, China

* Corresponding author: Yicheng Liu

Received  July 2019 Revised  February 2020 Published  July 2021 Early access  August 2020

Fund Project: This work was supported by the NSFC (11701267;11671011;11801562) and Hunan Natural Science Excellent Youth Fund (2020JJ3029)

As we known, it is popular for a designed system to achieve a prescribed performance, which have remarkable capability to regulate the flow of information from distinct and independent components. Also, it is not well understand, in both theories and applications, how self propelled agents use only limited environmental information and simple rules to organize into an ordered motion. In this paper, we focus on analysis the flocking behaviour and the line-shaped pattern for collective motion involving time delay effects. Firstly, we work on a delayed Cucker-Smale-type system involving a general communication weight and a constant delay $\tau>0$ for modelling collective motion. In a result, by constructing a new Lyapunov functional approach, combining with two delayed differential inequalities established by $L^2$-analysis, we show that the flocking occurs for the general communication weight when $\tau$ is sufficiently small. Furthermore, to achieve the prescribed performance, we introduce the line-shaped inner force term into the delayed collective system, and analytically show that there is a flocking pattern with an asymptotic flocking velocity and line-shaped pattern. All results are novel and can be illustrated by numerical simulations using some concrete influence functions. Also, our results significantly extend some known theorems in the literature.

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

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##### References:
Time-domain behaviors of the state variables $x_{1}(t)$, $x_{2}(t)$, $v_{1}(t)$, $v_{2}(t)$ in Example 4.1
Time-domain behaviors of the state variables $x_{1}(t)$, $x_{2}(t)$, $v_{1}(t)$, $v_{2}(t)$ for the case $\beta = \frac{1}{2}$ in Example 4.2
Time-domain behaviors of the state variables $x_{1}(t)$, $x_{2}(t)$, $v_{1}(t)$, $v_{2}(t)$ for the case $\beta = 1$ in Example 4.2
The first is the initial position of each particle in the system; the second is the population distribution after iteration $200$ ($2s$). the third is the population distribution after iteration $2000$ ($20s$). The value of each parameter is given as: $N = 100, \lambda = 2, \tau = 0.2, \psi = \frac{1}{(1+r^2)^{1/3}}$, $\gamma = 1$
The first is the initial position of each particle in the system; the second is the population distribution after iteration $2000$ ($20s$); the third is the population distribution after iteration $5000$ ($50s$). The value of each parameter is given as: $N = 100, \lambda = 20, \tau = 0.2, \psi = \frac{1}{(1+r^2)^{1/2}}$, $\gamma = 1$
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