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2020, Volume 40, Issue 4: 2037-2060. Doi: 10.3934/dcds.2020105

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Emergent dynamics of an orientation flocking model for multi-agent system

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.
National Institute for Mathematical Sciences, 70, Yuseong-daero 1689 beon-gil, Yuseong-gu, Daejeon 34047, Republic of Korea
4.
The Research Institute of Basic Sciences, Seoul National University, Seoul 08826, Republic of Korea
5.
Department of Mathematics, Myongji University, 116 Myongjiro, Yong-In 17058, Republic of Korea

^* Corresponding author: Se Eun Noh
^* Corresponding author: Se Eun Noh

Received: October 2018

Revised: September 2019

Published: January 2020

The work of S.-Y. Ha was supported by the National Research Foundation of Korea (NRF-2017R1A5A1015626), the work of D. Kim was supported from the National Institute for Mathematical Sciences (NIMS) grant funded by the Korea government (MIST) (No.B19610000), and the work of S. E. Noh was supported by the National Research Foundation of Korea (NRF-2017R1C1B5018312)

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Abstract

We study the orientation flocking for the deterministic counterpart of a stochastic agent-based model introduced by Degond, Frouvelle and Merino-Aceituno in 2017, where the orientation is defined as a $ {\rm SO}(3) $ matrix. Their proposed model can be reduced to the other collective dynamics models such as the Lohe matrix model and the Viscek-type model as special cases. In this work, we study the emergent dynamics of the orientation flocking model in two frameworks. First, we present sufficient conditions leading to the orientation flocking when the natural frequency matrices are identical. To be precise, we prove that all orientation matrices asymptotically converge to the common one, and the spatial position diameter remains uniformly bounded. Second, we show the emergence of orientation-locked states for non-identical natural frequency matrices, that is, the difference of any two orientation matrices tends to the definite constant matrix. On the other hand, we establish the finite-in-time stability with respect to initial data of the proposed orientation flocking model. We also present the numerical results consistent with our rigorous analysis. Our work remains valid even for dimensions greater than three.

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Mathematics Subject Classification: Primary: 82C10, 82C22, 35B37, 34C15.

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Figure 1. Identical case: $ H_i \equiv H $

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Figure 2. Non-identical case

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