Hopf Bifurcation and Its Normal Form of Reaction Diffusion Systems Defined on Directed Networks

Compared with the real Laplacian eigenvalues of undirected networks, the ones of asymmetrical directed networks might be complex, which is able to trigger ad-ditional 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 frame-work, 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.

Automated summary

Compared with undirected networks, the eigenvalues of asymmetrical directed networks may be complex, leading to additional collective dynamics such as oscillatory behaviors. However, the high dimensionality of reaction-diffusion systems on directed networks poses challenges for in-depth dynamic analysis. In this paper, the Hopf normal form of general two-species reaction-diffusion systems on directed networks is derived, revealing noteworthy differences from the derivation on undirected networks. Using the obtained theoretical framework, a rigorous Hopf bifurcation analysis is conducted for an SI reaction-diffusion system on directed networks, and the numerical simulations are consistent with the theoretical analysis. Undoubtedly, our work offers an important approach to studying oscillations in directed networks.