In many dynamical processes in natural sciences, understanding qualitative changes of a system, i.e. how, where and when reorganization of the dynamics takes place, provides the key to the understanding of the mechanisms at play and to the global understanding of the dynamics. This applies to processes as diverse as chemical reactions, the rearrangement of clusters, the ionization of atoms, the capture of asteroids by large celestial bodies, phase transitions in cosmology, and other systems. The most important problems in the study of reorganization processes are to predict and, where possible, to control whether the reorganization will happen.
Many reorganization processes share a common formal structure that can be exploited in their study: The qualitative structure of the dynamics is determined by invariant geometric objects in phase space, called invariant manifolds. These manifolds act as barriers that channel the dynamics of typical trajectories. Once the invariant manifolds are known, it can be predicted which initial conditions/states of the system will or will not lead to a qualitative reorganization. This structural description also yields quantitative information like, for example, chemical reaction rates, ionization yields and other transport properties.
Unfortunately, in a realistic situation the invariant manifolds are often quite difficult to compute, in particular in a high-dimensional system. It is also not always as easy to see how to use the manifolds once they are known. In TraX, we tackle both these problems: We develop computational methods of wide applicability along with the mathematical theory that underlies them, and to demonstrate the applicability of these methods in various fields of science. The research results of the network include applications to celestial mechanics, chemistry and atomic physics.
