Bifurcation structure of nonlinear modal coupling in quartz crystal resonator exhibiting internal resonance

This paper introduces a new use of a quartz crystal resonator to achieve internal resonance via nonlinear coupling of multiple eigenmodes to the externally excited mode. Specifically, we investigated the bifurcation structure of the system in terms of the amplitude and frequency of the harmonic driving force. We found that near the frequency of internal resonance a bifurcation occurs in which the stable limit cycle destabilizes and generates an invariant two-dimensional torus in the phase space of the system. Motion on the torus is either quasiperiodic or phase locked, depending on the choice of parameters. In addition, more complex hysteretic behavior was observed than what occurs in the uncoupled Duffing system, due to the presence of the internally resonant modes.