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

Abstract

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