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Complete characterization of flocking versus nonflocking of Cucker–Smale model with nonlinear velocity couplings

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  • Kim, Jong-Ho
  • Park, Jea-Hyun

Abstract

We consider the Cucker–Smale model with a regular communication rate and nonlinear velocity couplings, which can be understood as the parabolic equations for the discrete p-Laplacian (p ≥ 1) with nonlinear weights involving a parameter β( > 0). For this model, we study the initial data and the ranges of p and β to characterize when flocking and nonflocking occur. Specifically, we analyze the nonflocking case, subdividing it into semi-nonflocking (only velocity alignment holds) and full nonflocking (group formation and velocity alignment do not hold). More precisely, we show that if β ∈ (0, 1], p ∈ [1, 3), then flocking occurs for any initial data. If β ∈ (0, 1], p ∈ [3, ∞), then semi-nonflocking occurs for any initial data. If β ∈ (1, ∞), p ∈ [1, 3), then flocking occurs for some initial data. In the case β ∈ (1, ∞) and p ∈ [3, ∞), we observe alternative states. Finally, we have numerically verified the conclusions obtained by analytical calculations.

Suggested Citation

  • 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).
  • Handle: RePEc:eee:chsofr:v:134:y:2020:i:c:s0960077920301168
    DOI: 10.1016/j.chaos.2020.109714
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    References listed on IDEAS

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    1. Chehabi, Hamza & Chakrone, Omar & Chehabi, Mohammed, 2019. "On the antimaximum principle for the discrete p-Laplacian with sign-changing weight," Applied Mathematics and Computation, Elsevier, vol. 342(C), pages 112-117.
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    Cited by:

    1. Lee, Dongsun & Lee, Chaeyoung, 2022. "Numerical solutions of the Allen–Cahn equation with the p-Laplacian," Applied Mathematics and Computation, Elsevier, vol. 434(C).
    2. Kim, Jong-Ho & Park, Jea-Hyun, 2023. "Analysis of mono- and multi-cluster flocking for a nonlinear Cucker–Smale model with external force," Chaos, Solitons & Fractals, Elsevier, vol. 168(C).

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