Blue-Sky Catastrophic Bifurcations Behind Emergence and Disappearance of Quasiperiodic Rubbing Oscillations in a Piecewise Smooth Rotor–Stator System

Abstract

In this paper, an approach combining the methods of multigrid Point Mapping under Cell Reference and Stagger-and-step is proposed, which can efficiently determine both attractors and unstable saddle-type invariant sets in higher-dimensional dynamical systems. Then, the bifurcations around quasiperiodic partial rub oscillations of a piecewise smooth rotor–stator system are investigated, which are categorized into two classes depending on whether Nonlinear Normal Modes (NNMs) exist or not. It is found that the quasiperiodic rubbing oscillations, within the parameter region with an NNM, emerge through a Hopf bifurcation from a periodic rubbing oscillation and disappear through a collision with a saddle torus with an accompanying variation of system parameters. Within the parameter region without NNM, the bifurcation scenarios of two sequences of quasiperiodic rubbing oscillations, which emerge subsequently with the increasing of the rotating speed, are similar but somewhat complicated. That is, the quasiperiodic rubbing oscillation emerges through an inverse doubling of torus from a chaotic attractor, which is produced from a chaotic saddle after chaotic crisis. The quasiperiodic rubbing oscillation disappears through a collision between the stable and the unstable quasiperiodic tori. The insight into blue-sky catastrophic bifurcations in the quasiperiodic rubbing oscillations of the piecewise smooth rotor–stator rubbing system is thus obtained through this study.