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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.

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    $\begingroup$ Here's an example of interesting structure disappearing in higher dimensions: in 3D, closed orbits can be viewed as knots, whose properties are a fascinating area of study. All that becomes trivial in 4+ dimensions. $\endgroup$ Sep 1, 2021 at 11:22
  • $\begingroup$ A general answer can be given like this. The higher the dimension, the higher the variety of bifurcations and complexity of attractors. For example, $m$-dimensional minimal tori may appear only for n>m. It is also sufficient to consider typical bifurcations of stationary states in multi-parameter families of system to see how typical possible degeneracy grows (thus leading to new bifurcations) when the dimension increases. $\endgroup$
    – demolishka
    Sep 1, 2021 at 14:53
  • $\begingroup$ @demolishka: The thing about minimal tori is potentially interesting, but is still a little bit along the lines of "7-dimensional things can't happen until dimension 7". Are there specific interesting phenomena that occur b/c, say, a 3-dimensional minimal torus appears? Or 4-dim? $\endgroup$ Sep 1, 2021 at 18:10
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    $\begingroup$ @JoshuaGrochow, interesting phenomena appears when we proceed further and study bifurcations of such basic objects. See, for example, S. Newhouse, D. Ruelle, and F. Takens, “Occurence of strange axiom A attractors near quasiperiodic flows on T^{m} ($m = 3$ or more),” Commun. Math. Phys. 64, 35–40 (1978). $\endgroup$
    – demolishka
    Sep 2, 2021 at 3:19
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    $\begingroup$ @JoshuaGrochow, And one more thing. When you say "then it seems in all dimensions at least that large they are quite common". You should emphasize that you mean by "common". Attractors (chaotic or not) may have different dimension-like characteristics (dimension, entropy etc). On this level two attractors may be not even homeomorphic. If you mean only sensivity to initial conditions, then even on the line or the plane you may have some sensivity in the presence of multisatbility (the simplest example is 3 stationary points on the line there the middle one is unstable and two other are stable). $\endgroup$
    – demolishka
    Sep 2, 2021 at 3:38

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