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.