Hopf Bifurcation and Its Normal Form of Reaction Diffusion Systems Defined on Directed Networks (CSIAM Transactions on Applied Mathematics, 2023)
Compared with the real Laplacian eigenvalues of undirected networks, theones of asymmetrical directed networks might be complex, which is able to trigger additional collective dynamics, including the oscillatory behaviors. However, the highdimensionality of the reaction-diffusion systems defined on directed networks makesit difficult to do in-depth dynamic analysis. In this paper, we strictly derive the Hopfnormal form of the general two-species reaction-diffusion systems defined on directednetworks, with revealing some noteworthy differences in the derivation process fromthe corresponding on undirected networks. Applying the obtained theoretical framework, we conduct a rigorous Hopf bifurcation analysis for an SI reaction-diffusionsystem defined on directed networks, where numerical simulations are well consistentwith theoretical analysis. Undoubtedly, our work will provide an important way tostudy the oscillations in directed networks.
Keywords: Directed network; reaction-diffusion system; Hopf bifurcation; normal form; SI epidemic system.