The pressure threshold for 1/2 order subharmonic (SH) emissions and period doubling during the oscillations of ultrasonically excited bubbles is thought to be minimum when the bubble is sonicated with twice its resonance frequency ($f_r$). This estimate is based on studies that simplified or neglected the effects of thermal damping. In this work, the nonlinear dynamics of ultrasonically excited bubbles is investigated accounting for the thermal dissipation. Results are visualized using bifurcation diagrams as a function of pressure. Here we show that, and depending on the gas, the pressure threshold for 1/2 order SHs can be minimum at a frequency between $0.5f_r\le f\le 0.6f_r$. In this frequency range, the generation of 1/2 order SHs are due to the occurrence of 5/2 order ultra-harmonic resonance. The stability of such oscillations is size dependent. For an air bubble immersed in water, only bubbles bigger than 1 $\mathrm{\mu }m$ in diameter are able to emit non-destructive SHs in these frequency ranges.

当气泡以其共振频率的两倍进行超声处理时，认为在超声激发的气泡振荡期间1/2级次谐波（SH）发射和周期加倍的压力阈值最小。$F_{\mathrm{[R}}$）。该估算基于简化或忽略了热阻尼效应的研究。在这项工作中，考虑了热耗散，研究了超声激发气泡的非线性动力学。使用分叉图将结果可视化为压力的函数。在这里，我们表明，根据气体，1/2阶SHs的压力阈值可以在$0。5F_{\mathrm{[R}}\le F\le 0。6F_{\mathrm{[R}}$。在此频率范围内，1/2阶SHs的产生是由于5/2阶超谐波共振的发生。这种振荡的稳定性取决于尺寸。对于浸入水中的气泡，只有大于1的气泡$\mathrm{\mu }米$ 在这些频率范围内，能够发射出无损SH的直径。