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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.

Fahrenheit997
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  • 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 01 '18 at 09:59
  • 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 01 '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 01 '18 at 10:16

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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.

Albert
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  • Okay that makes a little more sense. So would you say it's analogous to connectedness? In that disconnected attracting sets won't be attractors but connected attracting sets are probably going to be attractors? (Not rigourously but just for the purposes of intuition) – Fahrenheit997 Feb 01 '18 at 10:34
  • ok, i think this is one way to think of some examples of attracting sets that are not attractors. there are surely more elaborate examples – Albert Feb 01 '18 at 10:36
  • That's brilliant, thank you! – Fahrenheit997 Feb 01 '18 at 10:42
  • @Glougloubarbaki +1 I am having a similar problem but I do not see clear how fits in your explanation. In my case the discrete dynamical system, never reaches an attractor (sink or orbit) but seems to go around the same region always... just in case if you are interested still nobody answered: https://math.stackexchange.com/q/2626063/189215 – iadvd Feb 02 '18 at 00:07
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    @iadvd thanks, I just posted an aswer to your question – Albert Feb 02 '18 at 05:42