CIM Thematic Term on Mathematics and the Environment
School and Workshop on Dynamical Systems
and Applications
Minicourse
Dynamical systems, numerical
experiments and super-computing
Carles Simó
(Univ. Barcelona, Spain)
Dynamical systems study the
evolution models of natural phenomena and the simplified models which help to
understand them. They can be given in deterministic form, either by means of
ordinary differential equations, partial differential equations or discrete
maps. They are useful in all domains of science and technology.
In their study tools from
all areas of mathematics are used. But for systems with some degree of
complexity it is impossible to produce a fairly complete description of the
evolution in the space of states, and its dependence with respect to
parameters, without using numerical techniques. They are essential for concrete
applications and very useful even for theoretical studies. They can be seen as
an experimental part of mathematics.
In the last years it has
become possible to achieve a generalisation in the systematic use of numerical
experiments, due to the availability of large arrays of processors working in
parallel with a reduced cost. But the impact of new algorithms has been even
larger. This makes feasible to face problems of larger and larger complexity.
In the lectures some
aspects of the general methodology and several examples will be presented.