# American Institute of Mathematical Sciences

doi: 10.3934/era.2021026
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## Instability and bifurcation of a cooperative system with periodic coefficients

 1 School of Mathematical Sciences University of Science and Technology of China, Hefei, Anhui 230026, China 2 School of Mathematical Sciences, Huaqiao University, Quanzhou, Fujian 362021, China

* Corresponding author: Tian Hou

Received  November 2020 Revised  January 2021 Early access March 2021

Fund Project: This work is supported by NSF of China No. 11825106, 11871231 and 11771414, CAS Wu Wen-Tsun Key Laboratory of Mathematics, University of Science and Technology of China

In this paper, we focus on a linear cooperative system with periodic coefficients proposed by Mierczyński [SIAM Review 59(2017), 649-670]. By introducing a switching strategy parameter $\lambda$ in the periodic coefficients, the bifurcation of instability and the optimization of the switching strategy are investigated. The critical value of unstable branches is determined by appealing to the theory of monotone dynamical system. A bifurcation diagram is presented and numerical examples are given to illustrate the effectiveness of our theoretical result.

Citation: Tian Hou, Yi Wang, Xizhuang Xie. Instability and bifurcation of a cooperative system with periodic coefficients. Electronic Research Archive, doi: 10.3934/era.2021026
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##### References:
The red region U (consists of U1 and U2) is the unstable region, the green region S is the asymptotically stable region, and the boundary between them are the Lyapunov stable region
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