Many problems in the natural sciences are modeled by a system with a simple rule that gives the next state of the system as a function of the present one. It has become increasingly apparent that many of these rules lead to complicated behavior when applied repeatedly. This complicated behavior is often called "chaos". To understand the behavior of a system and make useful predictions is a deep mathematical problem. One possible approach is to find "landmarks". A landmark is a small subsystem that behaves in a simple manner and which anchors the behavior of the system. If enough of these landmarks can be found, they can provide a skeleton for the dynamics which gives a global understanding of the system itself.
The plan is to develop systematic and accurate methods for the calculation of landmarks. Another goal is to prove theorems which show that the calculations satisfying some conditions are correct. It is planned to obtain results which show that if certain landmarks are found in some configurations (which can be verified with finite accuracy calculations) then conclusions can be obtained for all times. The hope is to make contact with concrete problems motivated by questions in solid state physics and chemistry. The work in the project will also be used as a training ground for graduate students, postdocs and visitors. For over 50 years, the most commonly used landmarks in dynamical systems have been normally hyperbolic manifolds and quasi-periodic orbits as well as their stable and unstable manifolds. Here, it is proposed to develop a systematic way of computing these objects theoretically as well as numerically. Another goal is to prove constructive existence theorems that validate approximate solutions and also to develop and implement fast and accurate algorithms. The unifying principle is to look for functional equations that describe the invariance and then use a variety of methods to try to solve them. The methods will vary from geometry to functional analysis. A further part of the proposed work is to begin studying special solutions in some infinite dimensional problems including partial differential equations and delay differential equations with state dependent delays.
