# American Institute of Mathematical Sciences

## Emergent behaviors of the generalized Lohe matrix model

 1 Department of Mathematical Sciences and Research Institute of Mathematics, Seoul National University, Seoul 08826, Republic of Korea, and, Korea Institue for Advanced Study, Hoegiro 85, Seoul 02455, Republic of Korea 2 Department of Mathematical Sciences, Seoul National University, Seoul 08826, Republic of Korea

* Corresponding author: Hansol Park

Received  March 2020 Revised  July 2020 Published  October 2020

Fund Project: The work of S.-Y. Ha is supported by NRF-2020R1A2C3A01003881. The work of H. Park is supported by Basic Science Research Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Education (2019R1I1A1A01059585)

We present a first-order aggregation model on the space of complex matrices which can be derived from the Lohe tensor model on the space of tensors with the same rank and size. We call such matrix-valued aggregation model as "the generalized Lohe matrix model". For the proposed matrix model with two cubic coupling terms, we study several structural properties such as the conservation laws, solution splitting property. In particular, for the case of only one coupling, we reformulate the reduced Lohe matrix model into the Lohe matrix model with a diagonal frustration, and provide several sufficient frameworks leading to the complete and practical aggregations. For the estimates of collective dynamics, we use a nonlinear functional approach using an ensemble diameter which measures the degree of aggregation.

Citation: Seung-Yeal Ha, Hansol Park. Emergent behaviors of the generalized Lohe matrix model. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2020286
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