Formulation of the normal forms of Turing-Hopf bifurcation in reaction-diffusion systems with time delay

TL;DR

This paper derives explicit third-order normal forms for Turing-Hopf bifurcations in reaction-diffusion systems with delay, facilitating analysis of complex spatiotemporal patterns near bifurcation points.

Contribution

It provides a comprehensive, explicit formulation of normal forms for a specific codimension-two bifurcation in delayed reaction-diffusion systems, aiding stability and pattern analysis.

Findings

Normal forms explicitly expressed as functions of derivatives and characteristic functions.

Reduction of PFDEs to a three-dimensional ODE system near bifurcation.

Application to analyze existence and stability of patterned solutions.

Abstract

The normal forms up to the third order for a Hopf-steady state bifurcation of a general system of partial functional differential equations (PFDEs) is derived based on the center manifold and normal form theory of PFDEs. This is a codimension-two degenerate bifurcation with the characteristic equation having a pair of simple purely imaginary roots and a simple zero root, and the corresponding eigenfunctions may be spatially inhomogeneous. The PFDEs are reduced to a three-dimensional system of ordinary differential equations and precise dynamics near bifurcation point can be revealed by two unfolding parameters. The normal forms are explicitly written as functions of the Fr\'echet derivatives up to the third orders and characteristic functions of the original PFDEs, and they are presented in a concise matrix notation, which greatly eases the applications to the original PFDEs and is convenient for computer implementation. This provides a user-friendly approach of showing the existence and stability of patterned stationary and time-periodic solutions with spatial heterogeneity when the parameters are near a Turing-Hopf bifurcation point, and it can also be applied to reaction-diffusion systems without delay and the retarded functional differential equations without diffusion.