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 Poincar6 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 (Dutling, Toda, laser, and Morse).
The shock wave-induced collapse and jet formation of pre-existing air bubbles at the focus of an extracorporeal shock wave lithotripter is investigated using high-speed photography. The experimentally obtained collapse time, ranging from 1 to 9 μs for bubbles with an initial radius R0 of 0.15 to 1.2 mm, agrees well with numerical results obtained using the Gilmore model. The collapse time is not linearly dependent on the initial bubble diameter since the temporal profile of the lithotripter wave contains a stress wave. The bubbles, positioned below a thin plastic foil, show strong jet formation in the direction of wave propagation with peak velocities of up to 770 m/s at the moment of collapse. Bubbles of initial radii between 0.3 and 0.7 mm always induce perforation of the foil by the jet (hole diameter 80–300 μm). Averaging the jet flow speed over 5 μs immediately after the collapse results in velocities from nearly zero up to 210 m/s, depending on the initial bubble size, with a maximum at R0=550 μm. This maximum is related to the temporal profile of the shock wave and to the effective cross section of the bubble for shock wave energy transfer. As cavitation bubbles are generated in the focal region of the lithotripter, the results are discussed with respect to the processes in a cavitation bubble field, which are of importance in cavitation erosion as well as in extracorporeal shock wave lithotripsy.
In the cognitive sciences the study of complex rhythmic movements is a challenging problem which is a subject of extensive research. Experiments on bimanual movements are paradigms for studying the ability of humans on timing and coordination. Such experiments give insights into the control mechanisms of the central nervous system and also reflect the functional state and level of training of the person tested. In a recent study on bimanual polyrhythm production the existence of phase transitions in dependence on the speed of performance has been shown. In this paper we present an iterated map model to explain main findings of these experiments. The model consists of two iterated maps describing the dynamics of the finger movements. The essential properties of the model are a nonlinear correction function and a coupling mechanism between the two maps. Numerical simulations show that the model is in qualitative agreement with the experimentally observed phenomena.
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