Classification of the nonlinear dynamics and bifurcation structure of ultrasound contrast agents excited at higher multiples of their resonance frequency

Through numerical simulation of the Hoff model we show that when ultrasound contrast agents (UCAs) are excited at frequencies which are close to integer (m>2$m>2$) multiples of their natural resonance frequency, the bifurcation structure of the UCA oscillations as a function of pressure may be characterized by 3 general distinct regions. The UCA behavior starts with initial period one oscillations which undergoes a saddle node bifurcation to m   coexisting attractors for an acoustic pressure above a threshold, _Pt1$P_{t_1}$. Further increasing the pressure above a second threshold _Pt2$P_{t_2}$, is followed by a sudden transition to period 1 oscillations.