For differentiable dynamical systems defined on, say, an open ball in $\mathbb{R}^n$, when $n=2$ Poincaré-Bendixson tells us a lot about what can happen. In particular, P-B precludes chaos and strange attractors. When $n=3$ these things appear, as in the Lorenz attractor, and then it seems in all dimensions at least that large they are quite common.
Are there other features/properties of differentiable dynamical systems that (a) first appear in dimension 4? 5? 17? and (b) also turn out to be quite common, once at or above the dimension in which they first appear?
The point of condition (b) is to rule out things like "This particular system doesn't appear in dimension $< 5$"; that would be more an example of a property of that particular system (I'd call it it's embedding dimension, though I don't know if that's the term used in dynamical systems) rather than a property that applies more broadly to dynamics in dimension 5.