(untitled) (CSIAM Transactions on Applied Mathematics, 2023)
Compared with the real Laplacian eigenvalues of undirected networks, the ones of asymmetrical directed networks might be complex, which is able to trigger additional collective dynamics, including the oscillatory behaviors. However, the high dimensionality of the reaction-diffusion systems defined on directed networks makes it difficult to do in-depth dynamic analysis. In this paper, we strictly derive the Hopf normal form of the general two-species reaction-diffusion systems defined on directed networks, with revealing some noteworthy differences in the derivation process from the corresponding on undirected networks. Applying the obtained theoretical framework, we conduct a rigorous Hopf bifurcation analysis for an SI reaction-diffusion system defined on directed networks, where numerical simulations are well consistent with theoretical analysis. Undoubtedly, our work will provide an important way to study the oscillations in directed networks.