I have a system of odes where the state vector has 6 elements. The system is a population biology model, where I am tracking the evolution of some competing species over time.
Now I was trying to think about how to look at the phase space for this system? Since this is a 6 dimensional system, that is like $\binom{6}{2}$ combinations.
My sense is that I should apply some dimensional reduction method. So I could do this in a couple of ways. First, I could use Principal Components Analysis (PCA) and then plot the first 2 or 3 components--assuming that they capture a suitably large percentage of the variation. The second choice would be to use something like an autoencoder to essentially learn a good transformation of the data into 2 or 3 coordinates, and then map those. There are of course some variations on these methods, like robust PCA, or kernel PCA, etc., which I can experiment with.
I just wanted to make sure that my approach was correct here. Given that I can't look at the full dynamics in 6 dimensions at once, is this projection method the best approach? Should I be concerned about missing some dynamics or multiscale time-scale information when I do this kind of reduction?