Bifurcation and chaos in shape and volume oscillations of a periodically driven bubble with two-to-one internal resonance

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Abstract
We study the dynamic sof gas or vapour bubbles when the volume mode of oscillation is coupled with one of the shape modes through quadratic resonance. In particular, the frequency ratio of the volume mode and the shape mode is assumed to be close to two-to-one. The analysis is based upon the use of a two-timescale asymptotic approximation, combined with domain perturbation theory. The viscous effect of the fluid is included by using a rigorous treatment of weak viscosity. Through solvability conditions, amplitude equations governing the slow-timescale dynamics of the resonant modes are obtained. Bifurcation analysis of these amplitude equations reveals interesting phenomena. When volume oscillations are forced by oscillations of the external pressure, we find that the volume mode may lose stability for sufficiently large amplitudes of oscillation, and this instability may lead to chaotic oscillations of both the volume and the shape modes. However, we find that for chaos to occur, a critical degree of detuning is required between the shape and volume modes, in the sense that their natural frequencies must differ by more than a critical value. When a shape mode is forced by oscillations of anisotropic components of the external pressure, we find that chaos can occur even for exact resonance of the two modes. The physical significance of this result is also given.