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There are numerous physical situations in which a hole or leak is introduced in an otherwise closed chaotic system. The leak can have a natural origin, it can mimic measurement devices, and it can also be used to reveal dynamical properties of the closed system. A unified treatment of leaking systems is provided and applications to different physical problems, in both the classical and quantum pictures, are reviewed. The treatment is based on the transient chaos theory of open systems, which is essential because real leaks have finite size and therefore estimations based on the closed system differ essentially from observations. The field of applications reviewed is very broad, ranging from planetary astronomy and hydrodynamical flows to plasma physics and quantum fidelity. The theory is expanded and adapted to the case of partial leaks (partial absorption and/or transmission) with applications to room acoustics and optical microcavities in mind. Simulations in the lima çon family of billiards illustrate the main text. Regarding billiard dynamics, it is emphasized that a correct discrete-time representation can be given only in terms of the so-called true-time maps, while traditional Poincaré maps lead to erroneous results. Perron-Frobenius-type operators are generalized so that they describe true-time maps with partial leaks

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The dynamics of a passive particle in a hydrodynamical flow behind a cylinder is investigated. The velocity field has been determined both by a numerical simulation of the Navier-Stokes flow and by an analytically defined model flow. To analyze the Lagrangian dynamics, we apply methods coming from chaotic scattering: periodic orbits, time delay function, decay statistics. The asymptotic delay time statistics are dominated by the influence of the boundary conditions on the wall and exhibit algebraic decay. The short time behavior is exponential and represents hyperbolic effects.

The weak-noise limit of Fokker-Planck models is studied for the case where the steady-state probability density in that limit cannot be represented by a continuously differentiable nonequilibrium potential. In a previous paper [J.Stat. Phys. 35, 729 (1984)], we have shown that this corresponds to the general case in systems outside thermodynamic equilibrium. By using an extremum principle, the nondifferentiable potential is constructed, which generalizes the differentiable case. The relation of approximate differentiable potentials to the exact nondifferentiable potential is considered and discussed for two examples with attracting limit cycles, a periodically forced nonlinear oscillator, end two phase-coupled nonlinear oscillators. The relevance of nondifferentiable potentials for nonequilibrium thermodynamics is pointed out.

The authors argue that the concept of snapshot attractors and of their natural probability distributions are the only available tools by means of which mathematically sound statements can be made about averages, variances, etc., for a given time instant in a changing climate. A basic advantage of the snapshot approach, which relies on the use of an ensemble, is that the natural distribution and thus any statistics based on it are independent of the particular ensemble used, provided it is initiated in the past earlier than a convergence time. To illustrate these concepts, a tutorial presentation is given within the framework of a low-order model in which the temperature contrast parameter over a hemisphere decreases linearly in time. Furthermore, the averages and variances obtained from the snapshot attractor approach are demonstrated to strongly differ from the traditional 30-yr temporal averages and variances taken along single realizations. The authors also claim that internal variability can be quantified by the natural distribution since it characterizes the chaotic motion represented by the snapshot attractor. This experience suggests that snapshot-attractor-based calculations might be appropriate to be evaluated in any large-scale climate model, and that the application of 30-yr temporal averages taken along single realizations should be complemented with this more appealing tool for the characterization of climate changes, which seems to be practically feasible with moderate ensemble sizes.

In order to simulate observational and experimental situations, we consider a leak in the phase space of a chaotic dynamical system. We obtain an expression for the escape rate of the survival probability applying the theory of transient chaos. This expression improves previous estimates based on the properties of the closed system and explains dependencies on the position and size of the leak and on the initial ensemble. With a subtle choice of the initial ensemble, we obtain an equivalence to the classical problem of Poincaré recurrences in closed systems, which is treated in the same framework. Finally, we show how our results apply to weakly chaotic systems and justify a split of the invariant saddle in hyperbolic and nonhyperbolic components, related, respectively, to the intermediate exponential and asymptotic power-law decays of the survival probability.