TY - JOUR
T1 - Hopf Bifurcation and Its Normal Form of Reaction Diffusion Systems Defined on Directed Networks
AU - Cui , Jianmeng
AU - Gou , Wei
AU - Jin , Zhen
JO - CSIAM Transactions on Applied Mathematics
VL - 3
SP - 619
EP - 652
PY - 2023
DA - 2023/04
SN - 4
DO - http://doi.org/10.4208/csiam-am.SO-2022-0047
UR - https://global-sci.org/intro/article_detail/csiam-am/21644.html
KW - Directed network, reaction-diffusion system, Hopf bifurcation, normal form, SI epidemic system.
AB - <p style="text-align: justify;">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.</p>