I'm having some difficulties discerning the difference between attracting sets an attractors in my nonlinear systems course. The definition we've been given is that attractors are attracting sets that contain a dense orbit, but I'm really struggling to see what this changes about them. Are there attracting sets that aren't attractors? How can I tell the difference between them? Any kind of intuition that I could get about this would be much appreciated.
2026-02-22 19:47:14.1771789634
Intuition behind dense orbits
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I think this is just an irreducibility issue (if an attracting set doesn't have a dense orbit, you probably want to split it into attracting sets with dense orbits and study them separately).
Ex: take your favorite function $f: \mathbb R \to \mathbb R$ with two attracting fixed points (eg $f(x)=x^3+0.1$); it has two attracting fixed points $\pm x_0$. The set $\{ \pm x_0\}$ is an attracting set that is not an attractor; but it can be written as a union of two attractors.
Edit: that's the discrete version, I'll leave it to you to write down a vector field with two sinks.