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

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.

Automated summary

This study investigates the vibratory energy transfer between a linear and a nonlinear oscillator at different time scales. The nonlinear system is characterized by a time-dependent periodic restoring forcing function. By detecting the dynamics of the system at fast and slow time scales, different dynamical characteristics are revealed, including slow invariant manifold, equilibrium points, and singularities. The study demonstrates that the time-dependent nonlinearity affects the phase-dependent slow invariant manifold, frequency responses, and the stability borders of the slow invariant manifold and singularities zones. The presence of backbone curves and isolas in the system is also identified, which should be carefully considered for system control purposes.