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Turing instability and Hopf bifurcation of a spatially discretized diffusive Brusselator model with zero-flux boundary conditions

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Abstract

In the present paper, a spatially discretized diffusive Brusselator model with zero-flux boundary conditions is considered. Firstly, the global existence and uniqueness of the positive solution are proved. Then the local stability of the unique spatially homogeneous steady state is considered by analyzing the relevant eigenvalue problem with the aid of decoupling method. Hence, the occurrence conditions of Turing bifurcation and Hopf bifurcation for the model at this steady state are obtained. Meanwhile, the comparative simulations on the stability regions of the steady state between the spatially discretized diffusive Brusselator model and its counterpart in continuous space are given. Furthermore, the approximate expressions of the bifurcating periodic solutions are derived according to Hopf bifurcation theorem. The bifurcating spatially nonhomogeneous periodic solutions show the formation of a special kind of periodic structures for this model. Finally, numerical simulations are given to demonstrate the theoretical results.

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Acknowledgements

The authors are grateful to the reviewer’s valuable comments, which led to the improvement of this article.

Funding

This work is partially supported by National Natural Science Foundation of China (Nos.11971143,11601384,12071074) and Tianjin Municipal Education Commission Research Project (No.2018KJ147).

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Authors and Affiliations

  1. Department of Mathematics, Tianjin University of Technology, Tianjin, 300384, China

    Zunxian Li

  2. School of Mathematics, Hangzhou Normal University, Hangzhou, 311121, Zhejiang, China

    Yongli Song

  3. Department of Mathematics, Foshan University, Foshan, 528000, Guangdong, China

    Chufen Wu

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  1. Zunxian Li
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Li, Z., Song, Y. & Wu, C. Turing instability and Hopf bifurcation of a spatially discretized diffusive Brusselator model with zero-flux boundary conditions. Nonlinear Dyn 111, 713–731 (2023). https://doi.org/10.1007/s11071-022-07863-z

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