Notes on the two main routes of period doubling and 1/2 order subharmonic oscillations in a bubble oscillator.

Period doubling route to chaos is one of the well-known characteristics of nonlinear oscillators. A bubble is a highly nonlinear oscillator that exists in various phenomena and applications ranging from material science and underwater acoustics to medical ultrasound. The occurrence of period doubling (PD) in the oscillations of the bubbles is concomitant with the generation of 1/2 order subharmonics (SHs). SH oscillations are used to monitor the bubble activity and are employed as a measure of stable cavitation. As bubbles are intravascular contrast agents, SH oscillations are used in diagnostic ultrasound to increase the contrast of the blood vessels to the tissue. Despite the importance of PD in a bubble oscillator, the dynamical behavior of the bubble during stable SH and ultra-harmonic (UH) oscillations have not thoroughly studied yet. In this work, through applying a comprehensive bifurcation method, we study the nonlinear radial oscillations of the bubble oscillator as a function of ultrasound driving pressure. The frequency of the driving force is chosen as the linear resonance frequency (fr) and linear SH resonance frequency (fsh=2fr) of the bubble. Results show that, when the bubble is sonicated with 2fr, PD doubling occurrence is more likely to result in non-destructive oscillations. The evolution of the bubble stable P2 dynamics forms the shape of a bow-tie for bubbles with an initial diameter of 740nm and above. When f=2fr, the phase portrait of the P2 attractor is distinctly different from a P2 attractor when f=fr, and SH component of the backscattered pressure is maximum. When sonicated by 2fr, due to lower oscillation amplitude and gentler bubble collapse, the bubble can sustain stable P2 oscillations for a longer duration and over a broader range of applied acoustic pressure.