I have a system of very high-dimensions (1000s of independent variables), but I could show that the dynamics is attracted to a 1D limit cycle or a 2D torus (with commensurate frequencies, so still periodic), via either plotting (all variables on x-axis, time on y-axis, and color representing values of variables) or measuring the correlation dimension. What I would like to do is how do I visualize these 1D or 2D tori?
I have done some searching but most I have come across are machine-learning-related non-linear dimensionality reduction method such as isomap. I do not think these are relevant because their results depend on choosing parameters (such as number of neighbors to consider) and some methods are even stochastic so different runs will give different answers. I think the low-dimensional structure I am after is a fundamental and concrete property of the dynamical system so shouldn't vary like these.
I guess what I am looking for is an example like this: take the 4D Henon-Heiles system when it has a 2D torus as an attractor. How do I visualize this 2D torus using all 4 variables instead of just choosing 3 out of the 4 variables and then plot, which is what I have seen in literatures.