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Computational and topological aspects of dynamics |
Dynamical
systems appear as mathematical models in every area of the natural sciences. In
recent years our knowledge of dynamical systems and its links with topology has
made tremendous progress. Advances in geometry, knot theory and computational
topology are finding applications in the study of dynamical systems. The
topological information turns out to have essential implications for observed
phenomena such as stationary, periodic and chaotic solutions. In order to understand the observed long term behaviour in a dynamical
system, it is crucial to analyse the invariant dynamics of the system and its
dependence on parameters. The reason is that the invariant dynamics, the
bounded motions which exist for all (forward and backward) time, often form the
global attractor for all motions. Usually these orbits have better properties
than arbitrary orbits, exhibiting topological structures which carry essential
information and/or converting the study of an infinite dimensional system to a
finite dimensional setting. An important aspect is the computability of topological information. Numerical simulations are a relatively easy way to explore the phenomena exhibited by a particular dynamical systems, but such computations do not reveal any of the reasons for the observed behaviour. On the other hand, topological techniques give a deeper understanding of the underlying structure and capture the behaviour of broad families of systems. To combine these two approaches one must be able to find topological information in a computationally robust manner. Arguments from topology can then be used to justify specific features seen in the numerical simulations. Advances in applications of topology to dynamics and the rapid increase
of computer power make this an ideal opportunity to bring together experts from
the fields of topology, dynamics and computational homology to foster
collaborations and to develop new links. [Back] |