Formulation of the normal form of Turing-Hopf bifurcation in partial functional differential equations

Turing-Hopf bifurcation is considered as an important mechanism for generating complex spatio-temporal patterns in dynamical systems. In this work, the normal form up to the third order for the Hopf-steady state bifurcation, which includes the Turing-Hopf bifurcation, of a general system of partial functional differential equations (PFDEs) in a bounded spatial region of any dimension, is derived based on the center manifold and the normal form theories of PFDEs. The explicit formula for the coefficients in the normal form of Hopf-steady state bifurcation are presented concisely in a matrix form, which makes it more convenient in not only symbolic derivation but also numerical implementation. This provides an approach of showing the existence and stability of stationary and time-periodic solutions with spatial heterogeneity. The derived formulas are also applicable to partial differential equations and functional differential equations.