Chaotic dynamics in Z(2)-equivariant unfoldings of codimension three singularities of vector fields in R-3

We study the most generic nilpotent singularity of a vector field in R-3 wh ich is equivariant under reflection with respect to a line, say the z-axis. We prove the existence of eight equivalence classes for C-0-equivalence, a ll determined by the 2-jet. We also show that in certain cases, the Z(2)-eq uivariant unfoldings generically contain codimension one heteroclinic cycle s which are comparable to the Skil'nikov-type homoclinic cycle in non-equiv ariant unfoldings. The heteroclinic cycles are accompanied by infinitely ma ny horseshoes and also have a reasonable possibility of generating suspensi ons of Henon-Like attractors, and even Lorenz-like attractors.