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Acoustically excited bubbles are involved in a wide range of phenomena andapplications ranging from oceanography to sonoluminescence; they haveapplications in chemistry, medical imaging, and therapeutic ultrasound. Thecomplexity of bubble dynamics and the limited understanding of their behaviorrestricts the exploration of their full potential. The bubble oscillator is ahighly nonlinear system, which makes it difficult to generate a comprehensiveunderstanding of its oscillatory behavior. One method used to investigate suchcomplex dynamical systems is the bifurcation analysis. Numerous investigationshave employed the method of bifurcation diagrams to study the effect ofdifferent control parameters on the bubble behavior. These studies, however,focused mainly on investigating the subharmonic (SH) and chaotic oscillationsof the bubbles. Super-harmonic (SuH) and ultra-harmonic (UH) bubbleoscillations remain under-investigated. One reason is that the conventionalmethod used for generating bifurcation diagrams cannot reliably identifyfeatures that are responsible for the identification of SuH and UHoscillations. Additionally, the conventional method cannot distinguish betweenthe UHs and SHs. We introduce a simple procedure for the generation ofbifurcation diagrams to address this shortcoming. This method selects themaxima of the bubble oscillatory response and plots them alongside thetraditional bifurcation points for the corresponding control parameter. Throughapplying this method, the oscillatory behavior of the bubble oscillator isanalyzed, and stable SuH and UH bubble oscillations are investigated. Based onthis new analysis, the conditions for the generation and amplification of UHand SuH regimes are discussed.

A comprehensive bifurcation method to analyze the super-harmonic and ultra-harmonic behavior of the acoustically excited bubble oscillator