Waiting time paradox applied to transient times

When working with chaotic systems, the average length of a transient phase before a particular event occurs is often required. Examples of this include the transient phase before a control mechanism can be activated @1,2# or the length of a chaotic transient after a crisis has occurred @3#. In both cases, the transient occurs before iterates fall into a small region. For controlling chaos, this region is a target region, typically around a fixed or periodic point, in which control can then be applied. In the case of a crisis, the region is a small escape region. There are different ways in which a transient phase can be computed numerically, with very different answers being obtained for the average length in some cases. We consider two methods for computing the average length of the transient which occurs before an iterate enters a particular small target or escape region. ~1! A large number of initial conditions is taken whose distribution is the same as the invariant probability distribution of the chaotic attractor. The lengths of the transient phases before entering the target region are then averaged. ~2! Let p be the measure of the chaotic attractor contained in the target region. Then p is also the probability of an iterate entering the target region. A standard calculation then gives that the average length of the transient phase is 1/p . The second method is often used when deriving exponents in scaling laws for transient times @2,3#. To illustrate the different methods, we consider the Hénon map