Properties of border states of transient chaos

Escape through a time-dependent hole in the doubling map

We investigate the escape dynamics of the doubling map with a time-periodic hole. Ulam's method was used to calculate the escape rate as a function of the control parameters. We consider two cases, oscillating or breathing holes, where the sides of the hole are moving in or out of phase respectively. We find out that the escape rate is well described by the overlap of the hole with its images, for holes centered at periodic orbits.

Strange nonchaotic repellers

We show the existence of strange nonchaotic repellers--that is, systems with transient dynamics whose nonattracting invariant set is fractal, but whose maximum Lyapunov coefficient is zero. We introduce the concept using a simple one-dimensional map and argue that strange nonchaotic repellers are a general phenomenon, occurring in bifurcation points of transient chaotic systems. All strange nonchaotic systems studied to date have been attractors; here, it is revealed that strange nonchaotic sets are also present in transient systems.

Comparison of averages of flows and maps

It is shown that in transient chaos there is no direct relation between averages in a continuous time dynamical system (flow) and averages using the analogous discrete system defined by the corresponding Poincaré map. In contrast to permanent chaos, results obtained from the Poincaré map can even be qualitatively incorrect. The reason is that the return time between intersections on the Poincaré surface becomes relevant. However, after introducing a true-time Poincaré map, quantities known from the usual Poincaré map, such as conditionally invariant measure and natural measure, can be generalized to this case. Escape rates and averages, e.g., Liapunov exponents and drifts, can be determined correctly using these measures. Significant differences become evident when we compare with results obtained from the usual Poincaré map.

Relationships among coefficients in deterministic and stochastic transient diffusion

Systems are studied in which transport is possible due to large extensions with open boundaries in certain directions, but the particles responsible for transport can disappear from it by leaving it in other directions, by chemical reaction or by adsorption. The connection of the total escape rate, the rate of the disappearance, and the diffusion coefficient is investigated. It leads to the observation that the diffusion coefficient defined by <x(2)> is in general different from the one present in the effective Fokker-Planck equation. The result makes it possible to generalize the Gaspard-Nicolis formula [Phys. Rev. Lett. 65, 1693 (1990)] to this transient case in deterministic systems.