Predictability of Chaotic Systems
Chaotic dynamical systems are characterised by the fact that initially close trajectories eventually diverge. As a result, long-term predictions of systems like the atmosphere, which determines our everyday weather conditions, are difficult if not impossible. This would remain true even if it were possible to run a "perfect" computer model which incorporates all relevant physical and chemical processes. The reason for this is that the initial state of our weather system is not known precisely enough, for measurements of air pressure and wind speed, for example, are only made at certain locations.
Nevertheless, it is known that there are regimes of good predictability, relatively stable weather conditions (e. g. the so-called omega situation in Europe). From the point of view of dynamical systems, this means that not all initial conditions are equally well predictable. Therefore we investigate local (in state space) measures of predictability, in the hope of identifying either regions of states that can be predicted well or others that are less predictable than the average.
We study the difference between various measures of local predictability, assuming a perfect model scenario. These include measures based on a linearisation of the system (e. g. local Lyapunov exponents) as well as others that include the full nonlinear system dynamics (ensemble-related measures). This makes it possible to study the effects of the linearisation and the dependence on prediction time and initial separation of trajectories. Furthermore, we want to provide a way of determining a simple rule-of-thumb for determining the necessary size of an ensemble used to estimate local predictability.