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.