In this paper, we explore the dynamical behaviors of the 1D two-grid coupled cellular neural networks. Assuming the boundary conditions of zero-flux type, the stability of the zero equilibrium is discussed by analyzing the relevant eigenvalue problem with the aid of the decoupling method, and the conditions for the occurrence of Turing instability and Hopf bifurcation at the zero equilibrium are derived. Furthermore, the approximate expressions of the bifurcating periodic solutions are also obtained by using the Hopf bifurcation theorem. Finally, numerical simulations are provided to demonstrate the theoretical results.

中文翻译：

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细胞神经网络中的图灵不稳定性和 Hopf 分岔

在本文中，我们探索了一维两网格耦合细胞神经网络的动力学行为。假设零通量型边界条件，通过解耦法分析相关特征值问题，讨论了零平衡的稳定性，推导了零平衡发生图灵不稳定性和Hopf分岔的条件. 此外，利用Hopf分岔定理，还得到了分岔周期解的近似表达式。最后，提供数值模拟来证明理论结果。