Program Outline

The workshop will promote an active and lively discussion of recent developments concerning the different types of Lyapunov-like vectors, their characterization and potential use in forecasting of spatially extended chaotic systems, covering both theoretical aspects and practical implementations in realistic applications.

Being able to predict systems' behaviour often becomes difficult when nonlinear interactions are present. There are a number of approaches to tackle the problem in low-dimensional systems. However, as the number of degrees of freedom increases the task of making a valid forecast becomes exceedingly difficult, not only from a computational point of view, but also due to fundamental limitations in what we can actually predict. Predicting often comes together with characterization, and so, developing good tools to explore and characterize complex dynamics does provide new forecasting techniques as a plus. Far from being an academic question, predicting the future state of high-dimensional systems from the present state has a paramount importance in practical applications including weather forecasting, hydrodynamics, turbulence, and oceanic flows. The economic impact of a prediction system is huge and can only be possibly tackled within the context of multidisciplinary approaches.

The Lyapunov analysis is a well established tool to explore chaotic or regular dynamics. However, up to very recently, a useful link between the computational tools and the mathematical theory of stable and unstable manifolds of the dynamics was absent. Just think of the concept of 'Lyapunov vector' as used in the very many practical applications. Everyday different types of so-called, in a generic fashion, Lyapunov vectors are computed in the most diverse applications ranging from engineering to atmospheric applications. Depending on the context, researchers define backward Lyapunov vectors, singular Lyapunov vectors, bred Lyapunov vectors, finite-time Lyapunov vectors, and so on. All these vectors often have different physical interpretation and ony provide access to pieces of partial information about the complexity puzzle. The reason being that they are just able to partially access different aspects of the dynamical information on the system future and/or remote past.

In particular, the most important areas that the workshop will cover are:

- Characterization of space-time or high-dimensional chaotic states.
- Predictability of chaotic space-time dynamics.
- Weather forecasting: ensembles of perturbations and assimilation.
- Oceanic flows, advection and mixing.
- Identification of global modes in collective dynamics.
- Hydrodynamic modes in molecular dynamics simulations.
- Characterization of hyperbolic/nonhyperbolic chaotic sets.
- Control of chaotic dynamics.