Analysis of transition regions of bi-linear, parametrically driven, two-oscillator model: Resonant excitation in the neighborhood of non-similar NNMs

Present study is concerned with the analysis of special family of parametric instability tongues emerging in the dissipative bi-linear model of two oscillators subjected to parametric excitation. The study focuses on analytical approximation and characterization of instability regions corresponding to the resonant, non-similar NNMs. In spite of the complexity of the problem under investigation, we succeeded in devising a novel and efficient analytical approach, which provides an explicit analytical approximation of transition regions corresponding to the special family of most significant sub-harmonic (m:1) resonance tongues, corresponding to IP and OP NNMs. Analytical study of the model reveals some interesting and unusual properties of instability regions of damped-driven, bi-linear system. Presently devised analytical procedure shows a fairly good correspondence with the direct numerical simulations of the full model.