Chaos is present in non-linear systems that describe temporal phenomena in the physical and life sciences. Well-known instances of chaos include the logistic map, the Lorenz equations, and forced non-linear oscillators. The regular and chaotic dynamics of free gas bubbles and gas-encapsulated microbubbles (contrast agents) are of immense importance in the efficient implementation of ultrasonic contrast imaging. A modification of the Keller-Herring model, which takes into account the elastic properties of the encapsulating shell, is investigated with respect to the bifurcation structure of the time-dependent microbubble radius. Numerical simulations show that the radial oscillations can be periodic as well as chaotic in appropriate parameter domains. Several investigations on chaotic aspects of dynamics presented here are new and highlighted appropriately. The influence of the acoustic field and shell parameters are investigated to identify values where the oscillations undergo bifurcations and the bubble response becomes chaotic. An analysis of chaotic behaviour in multiple-bubble systems completes the investigation and reveals the significant influence the free gas bubbles have on the acoustic response of their encapsulated counterparts.

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Numerical integration of an excitable reaction-diffusion ͑RD͒ system on a sphere is presented. The evolution of counterrotating double spiral waves on this manifold is studied and it is shown that tips of the spiral can either perform a meandering motion or rigidly rotate around a fixed center, depending on the system control parameter. This transition in dynamics is also illustrated by considering the phase plane of the solutions of the RD system. ͓S1063-651X͑97͒03410-7͔

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