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Pattern formation of Cucker–Smale system with nonlinear velocity couplings

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  • Ren, Jianlong
  • Liu, Qiming
  • Li, Ping

Abstract

In this paper, we propose a novel Cucker–Smale system with nonlinear velocity couplings and a targeted driving force. Firstly, by imposing assumptions on the initial state, the system can achieve flocking behavior. Secondly, the collision avoidance results under various velocity coupling degrees are deduced by applying the triangle inequality. Thirdly, by applying the Barbălat’s lemma, all agents eventually reach the prescribed line-shaped formation with a targeted driving force. In particular, for 1/2<α<1 and 1<β<3/2, the finite-time and fixed-time line-shaped formation can be successfully achieved without the symbolic function, and an upper bound on the settling time is obtained. Finally, the results of the theoretical analysis are verified by numerical simulations, and the final formation’s trajectory equation is derived by using the least squares method.

Suggested Citation

  • Ren, Jianlong & Liu, Qiming & Li, Ping, 2025. "Pattern formation of Cucker–Smale system with nonlinear velocity couplings," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 233(C), pages 413-432.
    • Handle: RePEc:eee:matcom:v:233:y:2025:i:c:p:413-432
    DOI: 10.1016/j.matcom.2025.02.002
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    References listed on IDEAS

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    1. Li, Le & Yan, Lifen & Huang, Chuangxia & Cao, Jinde & Ding, Xiaodan, 2024. "Linear formation of Cucker–Smale model with distributed time delays," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 222(C), pages 296-310.
    2. Kim, Jong-Ho & Park, Jea-Hyun, 2020. "Complete characterization of flocking versus nonflocking of Cucker–Smale model with nonlinear velocity couplings," Chaos, Solitons & Fractals, Elsevier, vol. 134(C).
    3. Qizhen Xiao & Hongliang Liu & Zhenghua Xu & Zigen Ouyang, 2020. "On Collision Avoiding Fixed-Time Flocking with Measurable Diameter to a Cucker–Smale-Type Self-Propelled Particle Model," Complexity, Hindawi, vol. 2020, pages 1-12, May.
    4. Wu, Jun & Liu, Yicheng, 2021. "Flocking behaviours of a delayed collective model with local rule and critical neighbourhood situation," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 179(C), pages 238-252.
    5. Hongliang Liu & Xiao Wang & Xiang Li & Yicheng Liu, 2020. "Finite-time flocking and collision avoidance for second-order multi-agent systems," International Journal of Systems Science, Taylor & Francis Journals, vol. 51(1), pages 102-115, January.
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