Hypernormal form for the Hopf-zero bifurcation

In this paper we present hypernormal forms up to an arbitrary order for equilibria of tridimensional systems having a linear degeneracy corresponding to a pair of pure imaginary eigenvalues and a third one zero. These simplest normal forms are obtained assuming some generic conditions on the quadratic terms, and using C-infinity-conjugacy as well as C-infinity-equivalence. Also, the case of Z(2)-symmetric systems is considered. In this situation, the hypernormal forms are characterized under generic conditions on the cubic terms. In all the cases, Ne provide recursive algorithms that compute explicitly the hypernormal form coefficients, in terms of the normal form coefficients.