Effect of time delay on flocking dynamics

Hyeong-Ohk Bae; Seung Yeon Cho; Jane Yoo; Seok-Bae Yun; Hyeong-Ohk Bae; Seung Yeon Cho; Jane Yoo; Seok-Bae Yun

doi:10.3934/nhm.2022027

Networks and Heterogeneous Media

2022, Volume 17, Issue 5: 803-825. doi: 10.3934/nhm.2022027

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Effect of time delay on flocking dynamics

Hyeong-Ohk Bae ^{1
,},
Seung Yeon Cho ^{2
,},
Jane Yoo ^{3
,
,},
Seok-Bae Yun ^{4
,}

1.
Department of Financial Engineering, Ajou University, Republic of Korea
2.
Department of Mathematics and Research Institute of Natural Science, Gyeongsang National University, Republic of Korea
3.
Department of Financial Engineering, Ajou University, Republic of Korea
4.
Department of Mathematics, Sungkyunkwan University, Republic of Korea

Received: 01 February 2022 Published: 17 June 2022
Primary: 82C22, 91B70, 91G80

We propose a time-delayed Cucker-Smale type model(CS model), which can be applied to modeling (1) collective dynamics of self-propelling agents and (2) the dynamical system of stock return volatility in a financial market. For both models, we assume that it takes a certain amount of time to collect/process information about the current position/return configuration until velocity/volatility adjustment is made. We provide a sufficient condition under which flocking phenomena occur. We also identify the initial configuration for a two-agent case, in which collective behaviors are accelerated by changes in the delay parameter. Numerical illustrations and financial simulations are carried out to verify the validity of the model.
- Cucker–Smale model,
- flocking,
- volatility,
- time delay
Citation: Hyeong-Ohk Bae, Seung Yeon Cho, Jane Yoo, Seok-Bae Yun. Effect of time delay on flocking dynamics[J]. Networks and Heterogeneous Media, 2022, 17(5): 803-825. doi: 10.3934/nhm.2022027

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Abstract

We propose a time-delayed Cucker-Smale type model(CS model), which can be applied to modeling (1) collective dynamics of self-propelling agents and (2) the dynamical system of stock return volatility in a financial market. For both models, we assume that it takes a certain amount of time to collect/process information about the current position/return configuration until velocity/volatility adjustment is made. We provide a sufficient condition under which flocking phenomena occur. We also identify the initial configuration for a two-agent case, in which collective behaviors are accelerated by changes in the delay parameter. Numerical illustrations and financial simulations are carried out to verify the validity of the model.

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