Parametrically Excited Vibrations in a Nonlinear Damped Triple-Well Oscillator with Resonant Frequency

Purpose  This paper deals with the parametrically excited vibrations and mode transitions of a nonlinear damped triple-well oscillator in detail. The multiple timescale structure of the triple-well oscillator with resonant frequency is revealed. We present the correct predictions of the region of parameter space where the periodic orbits alternate between epochs of fast and slow motions in such excited vibrations. 
Methods  Typically, one can treats the cosinoidal part of the perturbation as a singular variable of the whole system, which only remains autonomous form. We study the critical manifold and bifurcation structure of this system. Since the attracting manifold terminated at a fold leading to the change of local stability, the corresponding singular orbit would jump to another stable branch. Characteristic features of this method are folded singularities and structure of the critical manifold. 
Result  Parametrically excited vibrations with resonant frequency could reflect multiple timescale structure. A singular periodic orbit can be constructed, which consists of distinct segments. Starting along the critical manifold with small damped oscillations, then there is a fast transition that flows to another attracting branch for the folded singularity. As the fifth stiffness value varies, mixed-mode oscillations with more jumps are produced. Moreover, the jump-doubling behaviors can be presented under the flip of excitation frequency ratio. 
Conclusion  As we show in the paper, damping and external excitations with resonant frequency ratio in nonlinear vibration system could construct singular periodic orbits. It is possible for the trajectory to jump away from the folded singularity with finite speed. Then, it performs small damped oscillations about one attracting branch of stable foci. By taking advantage of time scale separation, we demonstrate the origin of such bursting behaviors.