Hopf-pitchfork singularities in coupled systems

It is known from the literature that a family consisting of two brusselators linearly coupled by diffusion unfolds strange attractors due to the generic occurrence of a 4-dimensional nilpotent singularity of codimension 4. In this paper the attention is placed on the Hopf-pitchfork singularities which are unfolded by that coupled system. We will see that the associated map of bifurcations is very rich and includes configurations which could also play the role of organizing centers of chaotic dynamics. As it happens in the case of two brusselators, the occurrence of Hopf-pitchfork singularities is expected when Hopf bifurcations are coupled by a diffusion mechanism. On the other hand, one of the most interesting problems in the context of coupled systems is the understanding of processes of synchronization/desynchronization. We will also illustrate the role of Hopf-pitchfork singularities as organizing centers of these processes.