TITLE Global Dynamics and Fast Indicators AUTHOR Carles Sim\'o Departament de Matem\`atica Aplicada i An\`alisi Universitat de Barcelona Gran Via de les Corts Catalanes, 585 08007 Barcelona e-mail:carles@maia.ub.es To appear in {\em Global Analysis of Dynamical Systems}, edited by Henk W. Broer, Bernd Krauskopf and Gert Vegter, IOP ABSTRACT: Dynamical systems play an important role in understanding many problems in science. The variety of difficulties that they present to the dynamicist is huge. Local problems around some well known object (a point, a periodic or quasi-periodic orbit, an invariant manifold, etc) can be studied by different methods. A combination of analytic, geometric and topological tools provides a detailed account of this local dynamics and the bifurcations which occur when changing parameters. On the other hand, more global problems, facing to a big part of the phase space or to a large set in the parameter space, can be studied by using probabilistic methods and by computing several numeric indicators. But it can happen that we would like to combine both things: a relatively detailed knowledge of the dynamics in a large set. To this end it is useful \begin{itemize} \item to extend the local analysis to larger domains, say by using normal forms up to some relatively large order, so that they can give good quantitative information. Unfolding of the bifurcations found. This analysis provides a guidance to the numeric experiments to be done, \item to do systematic numeric experiments, like computation of invariant objects: fixed points, periodic orbits, tori, etc, and, if it applies, the related stable, unstable and centre manifolds. Intersections of the manifolds (homoclinic and heteroclinic phenomena) and quantitative measures associated to them. Continuation of these objects with respect to parameters and detection and analysis of the bifurcations. These experiments, in turn, give hints on new phenomena to be investigated. \end{itemize} This programme has been carried out previously in several cases, like \cite{BRS,SBR}, \cite{BST}, \cite{CSnam}, \cite{CSaa}, \cite{SS}, \cite{Schi}. However, both approaches can require an important effort. It is suitable to have {\it fast indicators} aiming at a significant knowledge of the dynamics in a quick way (with the help of arrays of processors). Both the design of the indicators and the interpretation of the results must be guided by \begin{itemize} \vspace*{-2mm} \item the known dynamical phenomena on the considered class of systems, \item the role of the numerical errors, \item computational efficiency. \end{itemize} In this paper we sketch some tools. They are presented by showing how they apply to some examples, restricting to conservative systems, and summarizing part of the results. But are described with generality enough so that they can be used in many other problems in ``experimental'' mathematics.