Unimodal order in the image of the simplest equivariant chaotic system

The simplest equivariant chaotic dynamics is investigated in terms of its image, i.e., under the 2-->1 mapping allowing one to obtain a projection of the dynamics without any residual symmetry. The inversion symmetry is therefore deleted. The bifurcation diagram can thus be predicted from the unimodal order although the first-return map computed in the original phase space exhibits three critical points. This feature is the same as the one observed on the Burke and Shaw system although this latter system has a rotation symmetry.