Backbone curves, Neimark-Sacker boundaries and appearance of quasi-periodicity in nonlinear oscillators: application to 1:2 internal resonance and frequency combs in MEMS

Abstract

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.