Avoided crossings in three coupled oscillators as a model system of   acoustic bubbles
  The resonance frequencies and oscillation phases of three acoustically coupled bubbles are examined to show that avoided crossings can appear in a multibubble system. Via a simple coupled oscillator model, we show that if at least three bubbles exist, it is possible for their resonance frequencies as functions of the separation distances between the bubbles to experience an avoided crossing. Furthermore, by focusing our attention on the oscillation phases and based on analysis of the transition frequencies [M. Ida, Phys. Lett. A 297, 210 (2002); M. Ida, J. Phys. Soc. Jpn. 71, 1214 (2002)] of the coupled bubbles, we show that a distinct state exchange takes place between the bubbles at a point in the avoided crossing region, where a resonance frequency of the triple-bubble system crosses with a transition frequency not corresponding to the resonance frequencies. 
Masato Ida