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Abstract In this paper, we prove the existence and uniqueness of the equilibrium solution of the system by using the M-matrix and the topological degree technique and study the boundedness and robustness of dynamics of reaction–diffusion high-order Markovian jump Cohen–Grossberg neural networks (CGNNs) with p-Laplacian diffusion, including the common reaction–diffusion CGNNs. The obtained criteria are applicable to computer Matlab LMI-toolbox, which is suitable for large-scale calculations of actual complex engineering. Finally, a numerical example demonstrates the effectiveness of the proposed method.