DYNAMICS OF COUPLED BUBBLE OSCILLATORS WITH DELAY

We investigate the stability of the in-phase mode in a system of two delay-coupled bubble oscillators. The bubble oscillator model is based on a 1956 paper by Keller and Kolodner. Delay coupling is due to the time it takes for a signal to travel from one bubble to another through the liquid medium that surrounds them. Using techniques from the theory of delay-differential equations as well as the method of averaging, we show that the equilibrium of the in-phase mode can be made unstable if the delay is long enough and if the coupling strength is large enough, resulting in a Hopf bifurcation. This work is motivated by medical applications involving noninvasive localized drug delivery via microbubbles.