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

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 Fréchet 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.

On Turing-Turing bifurcation of partial functional differential equations and its induced superposition patterns

“…Applying normal form method, spatiotemporal dynamics resulting from Hopf bifurcation have been extensively investigated, see [29][30][31][32][33][34][35]. Furthermore, based on center manifold theory and normal form method, Jiang et al [36] recently derived several concise formulas of computing coefficients of normal forms for partial functional differential equations at Turing-Hopf singularity. And, these formulas make it easy for us to compute normal forms.…”

“…Inspired by Jiang et al [36], we discuss the calculation of normal forms of Turing-Turing bifurcation for some parameterized PFDEs, then investigate superposition patterns of a diffusive predator-prey system near Turing-Turing singularity by analyzing the obtained normal forms. Firstly, we derive the third-order normal form which is locally topologically equivalent to the original parameterized PFDEs at Turing-Turing singularity, based on Faria's work [26,27] and center manifold theory [24,25].…”