I come from a background in statistics, where we often visualize high dimensional data sets by projecting them onto lower dimensional subspaces. The most common example of this approach is Principal Components Analysis, but there is also Multi-dimensional Scaling, and other similar tools. The "Manifold Assumption" we make is that the data exist on some lower dimensional manifold--where the dimensionality of the manifold is much lower than the dimensionality of the dataset. This assumption does hold pretty well for real-world data sets where only a few dimensions exhibit substantial variation, and the remaining dimensions are highly correlated with these key dimensions.
Lately I have been looking more into the dynamical systems literature to round out some tools in my toolkit. While visualizing phase spaces and direction fields up to 3 dimensions is fine for textbooks, I was wondering what visualization methods work for higher dimensional dynamical systems? So if we have a large compartment model or such, what are the good ways to visualize the dynamics of such systems?
Hence to bring it back to my question, does this manifold assumption make sense for visualizing high dimensional dynamical systems? Can we project the dynamics of the system down to say 3 dimensions and then visualize the system's behavior there? Or are there other better ways to visualize the orbits and phase spaces of high-dimensional dynamical systems?