Periodic orbit theory applied to a chaotically oscillating gas bubble in water

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Abstract
This study investigates the dynamics of an acoustically driven air bubble in water. Depending on the values of external parameters, the radial oscillations of the bubble can be either stable or chaotic. The necessary condition of chaotic behaviour is identified to be the non-zero amplitude of the bubble's afterbounces at the beginning of the next acoustic cycle, which brings memory into the system. We show that for some parameter values in the chaotic regime the dynamics can be reduced to a unimodal map. At these parameter values the periodic orbit theory is successfully applied to calculate averages of relevant physical quantities, such as the air concentration at which the bubble is in diffusive equilibrium with the surrounding liquid. Finally we investigate the convergence of the calculated quantities.
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