Detection of different dynamics of two coupled oscillators including a time-dependent cubic nonlinearity

Vibratory energy channelling between a linear and nonlinear oscillator is studied at different time scales. The system possesses time-dependent periodic restoring forcing function. Detection of fast slow dynamics leads to revealing dynamical characteristics, namely invariant manifold, equilibrium singular points. We show that the nonlinearity produces phase-dependent frequency responses, modifications concerning stability borders its manifold singularities zones. backbone curves also isola are detected; latter should be taken into account carefully if aim control.