Stability Analysis of a Parametric Duffing Oscillator

Abstract

The Duffing oscillator represents an important example to describe, mathematically, the nonlinear behavior of several phenomena in physics and engineering. In view of its diverse applications in science and engineering, the current work investigates the stability analysis of the parametric Duffing oscillator. This equation represents a second-order ordinary differential equation with cubic nonlinearity and periodic coefficient. A coupling of the homotopy perturbation method (HPM) and Laplace transform (LT), an approximate solution is derived. The HPM is adapted to find another accurate approximate solution. The latter analysis reveals the exact solution of the cubic Duffing equation. In addition, an expanded frequency parameter is achieved to find another approximate periodic solution. Therefore, this method is exercised to govern the stability criteria of the problem. Finally, the multiple time scales with the HPM is used to judge the stability criteria. The analyses include the resonance as well as nonresonance cases. Numerical estimations are performed to confirm, graphically, the perturbed solutions together with the stability examination. It is shown that the damped parameter and the cubic stiffness parameter have a destabilizing influence. In contrast, the natural and parametric frequencies are of stabilizing influences.