The numerical approximation of the normal form coefficients for a double Hopf bifurcation.

I discuss an analysis of the primary bifurcations that occur in a mathematical model of the differentially heated rotating fluid annulus. More specifically, I present the analysis of the double Hopf bifurcations that occur along the transition between axisymmetric steady solutions and non-axisymmetric rotating waves. Center manifold reduction and normal

forms are used to deduce the local behaviour of the full system of partial differential equations from a low-dimensional system of ordinary differential equations.

Analytically, the coefficients of the normal form equations can be found in terms of certain unknown functions, namely the steady axisymmetric solution, the eigenfunctions and some Taylor coefficients of the center manifold function. The unknown functions can be numerically approximated from steady partial differential equations in two spatial dimensions.

Thus, a combination of analytical and numerical methods are used to obtain numerical approximations of the coefficients of the normal form equations. The numerical results are validated by comparison with experimental observations.