Bifurcation structure of bubble oscillators

Abstract

Methods from chaos physics are applied to a model of a driven spherical gas bubble in water to determine its dynamic properties, especially its resonance behavior and bifurcation structure. The dynamic properties are described in a growing level of abstraction by radius-time curves, trajectories in state space, strange attractors in the Poincaré plane, basins of attraction, bifurcation diagrams, winding number diagrams, and phase diagrams. A sequence of bifurcation diagrams is given, exemplifying the recurrent pattern in the bifurcation set and its relation to the resonances of the system. Period-doubling cascades to chaos and back (‘‘period bubbling’’) are a prominent recurring feature connected with each resonance (demonstrated for period-1, period-2, and period-3 resonances, and observed for some higher-order resonances). The recurrent nature of the bifurcation set is most easily seen in the phase diagrams given. A similar structure of the bifurcation set has also been found for other nonlinear oscillators (Duffing, Toda, laser, and Morse).