Stability and bifurcation of Mathieu–Duffing equation

Abstract

Various phenomena in science, physics, and engineering result in the Mathieu equation with cubic nonlinear term, known as the Mathieu–Duffing equation. In previous works, different perturbation methods have been used to investigate the stability and bifurcation of this equation in the vicinity of the first unstable tongue and for relatively small values of natural frequency. The primary goal of this paper is to adapt the Strained Parameters Method to investigate the stability and bifurcation associated with stability change around the second unstable tongue. In addition, this work shows that the Strained Parameters Method is able to obtain the same results previously obtained by other perturbation techniques with minimum computational effort. An inductive approach is used to express the multipliers of the transition curves and the location of the newborn equilibria as a function of the parametric frequency. Lastly, the Floquet theory and Poincaré map are used to validate the analytical results.