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Emergence of anomalous flocking in the fractional Cucker-Smale model

  • * Corresponding author: Jinwook Jung

    * Corresponding author: Jinwook Jung 
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  • In this paper, we study the emergent behaviors of the Cucker-Smale (C-S) ensemble under the interplay of memory effect and flocking dynamics. As a mathematical model incorporating aforementioned interplay, we introduce the fractional C-S model which can be obtained by replacing the usual time derivative by the Caputo fractional time derivative. For the proposed fractional C-S model, we provide a sufficient framework which admits the emergence of anomalous flocking with the algebraic decay and an $\ell^2$-stability estimate with respect to initial data. We also provide several numerical examples and compare them with our theoretical results.

    Mathematics Subject Classification: Primary: 70F99, 92D25.


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  • Figure 1.  Initial configurations for $ \psi_m >0 $.

    Figure 2.  Slow velocity alignment for $ \psi_m >0 $

    Figure 3.  Relaxation rate toward velocity alignment for $ \psi_m>0 $

    Figure 4.  Initial configurations for each case, when $ \psi $ is just nonnegative.

    Figure 5.  Slow velocity alignment when $ \psi $ is just nonnegative

    Figure 6.  Relaxation rate toward velocity alignment when $ \psi $ is just nonnegative

    Figure 7.  Non-flocking result when $ \psi $ is just nonnegative

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