A model of a nonlinear beam with a piezoelectric patch linked to a nonlinear circuit is considered. The physical and mechanical parameters of the system are such that it presents a 1:3 internal resonance. The aim is to study, for different time scales, the effect of the nonlinearity of the electrical circuit on energy exchanges between the resonant modes. In fact, we would like to master energy channelling between two internally resonant modes of a composite beam via a nonlinear circuit. The investigations are carried out on the projected system equations on its internally resonant modes. The system behaviours at different scales of time are studied by coupled methods of complexification and multiple scale. Analytical developments permit to reveal different system dynamics characterized by periodic and/or non periodic regimes. Finally, the paper is accompanied by comparisons between systems equipped with a resonant circuit (linear) and a nonlinear one.