Theory and computation

There are many different variations on invariant manifolds such as periodic orbits, invariant tori (quasi-periodic solutions), normally hyperbolic invariant manifolds and the stable and unstable manifolds of those, as well as time-dependent versions of all of the above. All of these variants appear naturally in many applications, and each requires its own mathematics and its own algorithms. Nevertheless, there are many common ideas and tools.

The theory for these invariant objects was developed in the 1960’s-70’s (KAM theory and normally hyperbolic manifolds). The systematic computation of invariant manifolds is a more recent field. There are, by now many different methods, depending, of course, on the type of manifolds one is considering. In many cases the development of algorithms and of mathematical theory go hand in hand: The way to prove theorems is to describe an iterative procedure that converges to a true solution. This iterative procedure can be implemented as an algorithm, and the better proofs also lead to more efficient algorithms.

TraX is designed to make progress in this situation, getting closer to the (very long term) goal of a systematic theory and "black box'' reliable algorithms backed up by rigorous methods. The TraX participants have extensive expertise in two different types of algorithms: the normal form method or similar perturbative schemes, and the parameterization method. For both the TraX teams have developed theorems and algorithms and deployed them in concrete problems. Thus, it is very timely to integrate them and take them to the next level. Having significant applications in mind adds an additional level of interest and opens lines of communications between different fields.