I am looking at some introductory material on nonlinear odes, and systems of nonlinear odes. The material is simple enough, but I was trying to figure out what would happen in very high dimensional situations. Most of the pedagogy on nonlinear odes stops at around 3 dimensions with the introduction of chaos, but many practical applications of odes are very high dimensional, and everyone reading this post is no doubt aware.
So my question is both analytic and numerical. I am trying to understand if there is an upper bound or limit on the number of equilibrium points that a system of nonlinear ode may possess. I have to be a little more precise in my question. So if I have an N-dimensional system, then can I have at most N equilibrium points? And, as N becomes very big, like say a 1000 dimensional or 10,000 dimensional, then would I still have at most 10,000 equilibrium points? My reason for asking this question is that I am trying to figure out whether I can use a system of nonlinear ODEs to track a hold a sort of "memory," where the equilibria define the memory states. But I would need the ability to hold a high dimensional memory state, hence my question.
I am trying to figure out how to think about this problem. Now as system of nonlinear ODEs would probably not be square, so I would have to look at the behavior of the singular values. So I could borrow from random matrix theory and determine that as long as there are 10,000 distinct singular values in my system, then there would be 10,000 equilibrium points? Is that accurate or am I missing something.
Note that there are also numerical issues here. So in a 10,000 dimensional system I imagine that the singular values may not be evenly dispersed. So there could be basis functions that are colinear, so the effective rank of the matrix would be less than 10,000. Also, some of the basis vectors could be very close to each other meaning that they were nearly colinear and hence the equilibria from the nearly colinear vectors might be highly related or even proximate to each other?
Hence, I was hoping to understand under what conditions, could I obtain the maximum number of equilibria from an N-dimensional system of nonlinear ODEs?
There is no correlation between dimension and number of equilibria. Let us take a look at a very general formulation of an ode: $$x'=f(x)$$ for some function $f$. Here the number of equilibria (i.e. constant solutions) is the number of zeros of $f$. For example in $\mathbb{R}^1$ $$f(x)=\sin \frac{1}{x},\ x>0$$ has infinitely many zeros, hence the ode $x'=f(x)$ has infinitely many equilibria.