The dynamical behavior of a two degree-of-freedom system made up of a linearoscillator and a coupled nonlinear energy sink with nonlinear global and localpotentials is studied. The nonlinear global potential of the energy sink performsdirect interactions with the linear oscillator, while its local potential dependsonly on its own behavior during vibratory energy exchanges between two oscillators.A time multiple scale method around 1:1:1 resonance is used to detect slowinvariant manifold of the system, its equilibrium and singular points. Detectedequilibrium points permit us to predict periodic regime(s) while singular pointscan lead the system to strongly modulated responses characterized by persistentbifurcations. Several possible scenarios occurring during these strongly modulatedregimes are highlighted. All analytical predictions are compared withthose which are obtained by direct numerical integration of system equations.