Lagrangian Coherent Structures: detecting dynamical structures in space
One of the most challenging problems in dynamical systems theory in general and astrodynamics in particular is the identification of immutable structures in the often chaotic sea of possible trajectories.
In many so called autonomous problems, where the mathematical formulation of the motion is independent of the current time, there are well known structures, the invariant manifolds, which separate the phase space. In those cases, these invariant manifolds provide a deeper understanding of the behavior of other orbits in the system. This knowledge can then be leveraged to design better, more fuel-efficient trajectories for spacecraft that benefit from the natural dynamics of the system. The most important example of such a dynamical system in astrodynamics is the circular restricted three-body problem (CR3BP).
Unfortunately, in practice many systems of interest are non-autonomous, that is the equations of motion do depend on time. For such systems, the mathematical theory of invariant manifolds does not apply any more. Lagrangian Coherent Structures (LCS) attempt to define a new mathematical object that behaves similar to invariant manifolds in the sense that they partition the phase space and govern the motion of particles. Originally developed in the context of fluid dynamics, the ACT is studying the possibility of applying these techniques to astrodynamics problems.
There are several challenges with this. While invariant manifolds have a mathematically very well described theory and over 100 years of research into the topic, LCS are a relatively new concept. Furthermore, their definition is very phenomenological, leading to a computationally very intensive problem of identifying these structures. While well suited for 2D problems appearing in fluid dynamics, they very quickly become unfeasible in higher dimensional systems. Typical space problems are at least 6 dimensional, which currently is out of reach for these methods. But by applying clever techniques to reduce the dimensionality of the problem it may be possible to still leverage this new technique in the design of space trajectories.