Necessary and sufficient conditions for the occurrence of generalized synchronization of unidirectionally coupled dynamical systems are given in terms of asymptotic stability. The relation between generalized synchronization, predictability, and equivalence of dynamical systems is discussed. All theoretical results are illustrated by analytical and numerical examples. In particular, the existence of generalized synchronization in the case of parameter mismatch between coupled systems leads to a new interpretation of recent experimental results. Furthermore, the possible application of generalized synchronization for attractor reconstruction in nonlinear time series analysis is discussed.

Shock wave emission and cavitation bubble expansion after optical breakdown in water with Nd:YAG laser pulses of 30-ps and 6-ns duration is investigated for energies between 50 μJ and 10 mJ which are often used for intraocular laser surgery. Time-resolved photography is applied to measure the position of the shock front and the bubble wall as a function of time. The photographs are used to determine the shock front and bubble wall velocity as well as the shock wave pressure as a function of time or position. Calculations of the bubble formation and shock wave emission are performed using the Gilmore model of cavitation bubble dynamics and the Kirkwood–Bethe hypothesis. The calculations are based on the laser pulse duration, the size of the plasma, and the maximally expanded cavitation bubble, i.e., on easily measurable parameters. They yield the dynamics of the bubble wall, the pressure evolution inside the bubble, and pressure profiles in the surrounding liquid at fixed times after the start of the laser pulse. The results of the calculations agree well with the experimental data. A large percentage of the laser pulse energy (up to 72%) is transformed into the mechanical energy ES and EB of the shock wave and cavitation bubble, whereby the partitioning between ES and EB is approximately equal. 65%–85% of ES is dissipated during the first 10 mm of shock wave propagation. The pressure at the plasma rim ranges from 1300 MPa (50 μJ, 30 ps) to 7150 MPa (10 mJ, 6 ns). The calculated initial shock wave duration has values between 20 and 58 ns, the duration measured 10 mm away from the plasma is between 43 and 148 ns. A formation phase of the shock front occurs after the ns pulses, but not after the ps pulses where the shock front exists already 100 ps after the start of the laser pulse. After shock front formation, the pressure decays approximately proportional to r−2, and at pressure values below 100 MPa proportional to r−1.06. The maximum bubble wall velocity ranges from 390 to 2450 m/s. The calculations of bubble and shock wave dynamics can cover a large parameter range and may thus serve as a tool for the optimization of laser parameters in medical laser applications.

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The mutual interaction between small oscillating cavitation bubbles (R 0 Ͻ10 m) in a strong acoustic field ͑P a Ͼ1 bar, f ϭ20 kHz͒ is investigated numerically. We assume spherical symmetry and a coupling of the bubble oscillations. Our results show that the strength and even the directions of the resulting secondary Bjerknes forces differ considerably from predictions of the well-known linear theory. This is of immediate consequence for understanding and modeling structure formation processes in acoustic cavitation and multibubble sonoluminescence. ͓S1063-651X͑97͒01909-0͔

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