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.

当前的工作主要致力于双线性Mathieu方程中存在的不稳定性区域的渐近分析。在这项研究中，我们调用常见的渐近技术，例如求平均值的方法和多个时间尺度的方法，以得出对应于1：n共振的跃迁曲线的相对简单的解析表达式。与经典Mathieu方程相反，其双线性对等物具有附加的不稳定性区域（例如，对于n > 2）。在这项研究中，我们从线性模型转换为双线性模型时，分析地证明了这些区域的形成，并显示了在低振幅参数激励的极限下，刚度不对称参数对其宽度的影响。我们表明，使用本研究中设计的分析预测，可以通过选择不对称参数来完全控制谐振区域的宽度，从而获得最大可能的宽度或完全an没。分析结果显示出与模型的数值模拟极为一致。