In Strogatz's book, Nonlinear Dynamics and Chaos, he gives the following working definition of attractors (page 690 of second edition):
"More precisely, we define an attractor to be a closed set $Α$ with the following properties:
- $A$ is an invariant set: any trajectory $x(t)$ that starts in $A$ stays in $A$ for all time.
- $A$ attracts an open set of initial conditions: there is an open set $U$ containing $A$ such that if $x(0) \in U$, then the distance from $x(t)$ to $A$ tends to zero as $t \to \infty$. This means that $A$ attracts all trajectories that start sufficiently close to it. The largest such $U$ is called the basin of attraction of $A$.
- $A$ is minimal: there is no proper subset of $A$ that satisfies conditions 1 and 2."
It seems like his working definition of attractors has a significant issue, even in the example (9.3.3), which he gave to nuance out what is or isn’t an attractor:
"Consider the system $\dot{x}=x-x^3, \dot{y}=-y$. Let $I$ denote the interval $–1 \leq x \leq 1$, $y = 0$. Is $I$ an invariant set? Does it attract an open set of initial conditions? Is it an attractor?
Solution: The phase portrait is shown in Figure 9.3.7. There are stable fixed points at the endpoints $(\pm 1,0)$ of $I$ and a saddle point at the origin. Figure 9.3.7 shows that I is an invariant set; any trajectory that starts in $I$ stays in $I$ forever. (In fact the whole $x$-axis is an invariant set, since if $y(0) = 0$, then $y(t) = 0$ for all $t$.) So condition 1 is satisfied.
Moreover, $I$ certainly attracts an open set of initial conditions—it attracts all trajectories in the $(x,y)$-plane. So condition 2 is also satisfied.
But $I$ is not an attractor because it is not minimal. The stable fixed points $(\pm l,0)$ are proper subsets of $I$ that also satisfy properties 1 and 2. These points are the only attractors for the system."
Aren’t the intervals $\{(a,0) \mid -1<a<0\}$ and $\{(a,0) \mid 0<a<1\}$ each attractors by his definition?
Condition 1 is satisfied since trajectories get closer to the stable fixed points asymptotically.
Condition 2 is fulfilled since the removal of the fixed points doesn't change the measure of zero of the distance between trajectories and the fixed points/these intervals. In other words, all circles of radius greater than zero centered at any fixed point contain both some part of the adjacent interval(s) and that of the trajectories being attracted.
Condition 3 is fulfilled since the fixed points have been removed, making it minimal. Any subset (I just realized I'm wrong lol, but I'll continue in case this helps anyone else) of points missing will be in trajectories that start closer to the origin.
(Any edits are welcome to improve quality.)
All intervals (a,0), where -1<a<0, and (0,a), where 0<a<1, satisfy the conditions 1 and 2. Therefore none of these intervals are minimal (and other subsets fail condition 1) and the issue is resolved.