In this work the Hoff model for microbubble oscillations was used together with bifurcation diagrams to investigate microbubble behavior for a range of the system control parameters (driving frequency and pressure, microbubble size and shell parameters). We have shown that if the microbubble sizes are optimized so that their resonance frequency is a fraction of the sonication frequency (1/3, 1/4...), above a certain pressure threshold, the microbubbles can emit higher order subharmonics at (1/3, 2/3...) of the sonication frequency. Experimental results (using Artenga and in-house made lipid microbubbles with a 25 MHz, 30 cycle pulse using 0.1-2.5 MPa pressures) showed subpopulations of the microbubbles that were able to show this behaviour by scattering ultrasound according to the theoretical predictions (e.g. having two frequency maxima for subharmonics at f/3 and 2f/3). The ring-down analysis of the natural free oscillations of these microbubbles confirmed the theoretical resonance predictions.