Intuition behind dense orbits

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.

• How did you define attracting set? Also, how did you define dense orbit? I can only find this notion in the context of chaotic attractors or the definition of chaos, which requires dense periodic orbits. – Wrzlprmft Feb 1 '18 at 9:59
• We defined an attractive set as a closed, invariant set $A$ such that there exists some neighborhood $U$ of $A$ with the properties that $\phi_t(x) \in U \forall t \geq 0$ and $\phi_t(x) \to A$ as $t \to \infty \forall X \in U$. Then a dense orbit was defined as a trajectory $\tilde{\Gamma}$ such that $\forall \epsilon \geq 0 \forall \Gamma \in A \exists x, \tilde{x} \in\Gamma , \tilde{\Gamma} with |x-\tilde{x} | \leq \epsilon$ – Fahrenheit997 Feb 1 '18 at 10:15
• I'm trying to type latex on my phone so I'm sorry if that doesn't come out right, I can't tell if it does or not – Fahrenheit997 Feb 1 '18 at 10:16

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.