Effect of fast harmonic excitation on frequency-locking in a van der Pol–Mathieu–Duffing oscillator

We study the effect of high-frequency harmonic excitation on the entrainment area of the main resonance in a van der Pol–Mathieu–Duffing oscillator. An averaging technique is used to derive a self- and parametrically driven equation governing the slow dynamic of the oscillator. The multiple scales method is then performed on the slow dynamic near the main resonance to obtain a reduced autonomous slow flow equations governing the modulation of amplitude and phase of the slow dynamic. These equations are used to determine the steady state response, bifurcation and frequency–response curves. A second multiple scales expansion is used for each of the dependent variables of the slow flow to obtain slow slow flow modulation equations. Analysis of non-trivial equilibrium of this slow slow flow provides approximation of the slow flow limit cycle corresponding to quasi-periodic motion of the slow dynamic of the original system. Results show that fast harmonic excitation can change the nonlinear characteristic spring behavior and affect significantly the entrainment region. Numerical simulations are used to confirm the analytical results.