Quasi-periodic solutions can arise in assemblies of nonlinear oscillators as a consequence of Neimark-Sacker bifurcations. In this work, the appearance of Neimark-Sacker bifurcations is investigated analytically and numerically in the specific case of a system of two coupled oscillators featuring a 1:2 internal resonance. More specifically, the locus of Neimark-Sacker points is analytically derived and its evolution with respect to the system parameters is highlighted. The backbone curves, solution of the conservative system, are first investigated, showing in particular the existence of two families of periodic orbits, denoted as parabolic modes. The behaviour of these modes, when the detuning between the eigenfrequencies of the system is varied, is underlined. The non-vanishing limit value, at the origin of one solution family, allows explaining the appearance of isolated solutions for the damped-forced system. The results are then applied to a Micro-Electro-Mechanical System-like shallow arch structure, to show how the analytical expression of the Neimark-Sacker boundary curve can be used for rapid prediction of the appearance of quasiperiodic regime, and thus frequency combs, in Micro-Electro-Mechanical System dynamics.

作为 Neimark-Sacker 分岔的结果，在非线性振荡器组件中可能出现准周期解。在这项工作中，在具有 1:2 内部共振的两个耦合振荡器系统的特定情况下，分析和数值研究了 Neimark-Sacker 分岔的出现。更具体地说，Neimark-Sacker 点的轨迹是通过分析推导出来的，并且突出显示了其相对于系统参数的演变。骨架曲线，保守系统的解，首先被研究，特别表明存在两个周期轨道族，表示为抛物线模式。当系统的特征频率之间的失谐变化时，这些模式的行为被加下划线。非零极限值，在一个解族的原点，允许解释受阻系统的孤立解的出现。然后将结果应用于类似微机电系统的浅拱结构，以展示如何使用 Neimark-Sacker 边界曲线的解析表达式来快速预测准周期状态的出现，从而快速预测频率梳，在微机电系统动力学。