Emergence of anomalous flocking in the fractional Cucker-Smale model

Seung-Yeal Ha; Jinwook Jung; Peter Kuchling

doi:10.3934/dcds.2019223

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2019, Volume 39, Issue 9: 5465-5489. Doi: 10.3934/dcds.2019223

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Emergence of anomalous flocking in the fractional Cucker-Smale model

1.
Department of Mathematical Sciences and Research Institute of Mathematics, Seoul National University, Seoul 08826, Republic of Korea
2.
Korea Institute for Advanced Study, Hoegiro 85, Seoul 02455, Republic of Korea
3.
Department of Mathematical Sciences, Seoul National University, Seoul, 08826, Republic of Korea
4.
Faculty of Mathematics, Bielefeld University, Bielefeld 33501, Germany

^* Corresponding author: Jinwook Jung
^* Corresponding author: Jinwook Jung

Received: October 31, 2018

Revised: February 28, 2019

Published: May 2019

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Abstract

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.

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

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

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

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

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

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

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

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

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