Analytical study of the transition curves in the bi-linear Mathieu equation

The current work is primarily devoted to the asymptotic analysis of the instability zones existing in the bi-linear Mathieu equation. In this study, we invoke the common asymptotical techniques such as the method of averaging and the method of multiple time scales to derive relatively simple analytical expressions for the transition curves corresponding to the 1:n resonances. In contrast to the classical Mathieu equation, its bi-linear counterpart possesses additional instability zones (e.g. for n > 2). In this study, we demonstrate analytically the formation of these zones when passing from linear to bi-linear models as well as show the effect of the stiffness asymmetry parameter on their width in the limit of low amplitude parametric excitation. We show that using the analytical prediction devised in this study one can fully control the width of the resonance regions through the choice of asymmetry parameter resulting in either maximum possible width or it's complete annihilation. Results of the analysis show an extremely good correspondence with the numerical simulations of the model.