# American Institute of Mathematical Sciences

June  2018, 38(6): 2763-2793. doi: 10.3934/dcds.2018116

## Remarks on the critical coupling strength for the Cucker-Smale model with unit speed

 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 Department of Mathematical Sciences, Industrial and Mathematical Data Analytics Research Center, Seoul National University, Seoul 08826, Republic of Korea

* Corresponding author: Yinglong Zhang

Received  April 2017 Revised  January 2018 Published  April 2018

Fund Project: The work of S.-Y. Ha is supported by the Samsung Science and Technology Foundation under Project Number SSTF-BA1401-03. The work of D. Ko is supported by the fellowship of POSCO TJ Park Foundation. The work of Y. Zhang is partially supported by a National Research Foundation of Korea grant (2014R1A2A2A05002096) funded by the Korean government.

We present a non-trivial lower bound for the critical coupling strength to the Cucker-Smale model with unit speed constraint and short-range communication weight from the viewpoint of a mono-cluster(global) flocking. For a long-range communication weight, the critical coupling strength is zero in the sense that the mono-cluster flocking emerges from any initial configurations for any positive coupling strengths, whereas for a short-range communication weight, a mono-cluster flocking can emerge from an initial configuration only for a sufficiently large coupling strength. Our main interest lies on the condition of non-flocking. We provide a positive lower bound for the critical coupling strength. We also present numerical simulations for the upper and lower bounds for the critical coupling strength depending on initial configurations and compare them with analytical results.

Citation: Seung-Yeal Ha, Dongnam Ko, Yinglong Zhang. Remarks on the critical coupling strength for the Cucker-Smale model with unit speed. Discrete & Continuous Dynamical Systems, 2018, 38 (6) : 2763-2793. doi: 10.3934/dcds.2018116
##### References:

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##### References:
Position-velocity configurations
Diameter of positions $D({\bf{x}})$ along time axis
Temporal evolution of $\theta({\bf{x}}(t), {\bf{v}}(t))$
Randomly chosen Position-velocity distribution
Initial configuration and its evidence of non-flocking
Emergence of local flocking
Emergence of local flocking for smaller $\kappa$
Emergence of larger cluster flocking for much smaller $\kappa$
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