  The acoustic bubble is an example of a highly nonlinear system which is the
building block of several applications and phenomena ranging from underwater
acoustics to sonochemistry and medicine. Nonlinear behavior of bubbles, and
most importantly 1/2 order subharmonics (SH), are used to increase the contrast
to tissue ratio (CTR) in diagnostic ultrasound (US) and to monitor bubble
mediated therapeutic US. It is shown experimentally and numerically that when
bubbles are sonicated with their SH resonance frequency (fsh=2fr where fr is
the linear resonance frequency), SHs are generated at the lowest excitation
pressure. SHs then increase rapidly with pressure increase and reach an upper
limit of the achievable SH signal strength. Numerous studies have investigated
the pressure threshold of SH oscillations; however, conditions to enhance the
saturation level of SHs has not been investigated. In this paper nonlinear
dynamics of bubbles excited by frequencies in the range of fr<f<2fr is studied
for different sizes of bubbles (400nm-8 micron). We show that the SH resonance
frequency is pressure dependent and decreases as pressure increases. When a
bubble is sonicated with its pressure dependent SH resonance frequency,
oscillations undergo a saddle node bifurcation from a P1 or P2 regime to a P2
oscillation regime with higher amplitude. The saddle node bifurcation is
concomitant with over saturation of the SH and UH amplitude and eventual
enhancement of the upper limit of SH and UH strength (e.g. $\approx$ 7 dB in UH
amplitude). This can increase the CTR and signal to noise ratio in
applications. Here, we show that the highest non-destructive SH amplitude
occurs when f~=1.5-1.8fr.
