I don’t really know anything about chaos and dynamical systems, but I’m trying to understand what the escape rate of transient chaos is. For this I need to know what a non-attracting chaotic set is, and thus need to know what an attracting chaotic set is, but I’m stuck on what an attractor is.
Alligood’s, Sauer’s and Yorke’s text says:
An attractor is a forward limit set which attracts a set of initial values that has nonzero measure.
I didn’t understand what it means by attracts, but after some searching online, I found a definition that says:
Let S be the forward limit set (of some $x_0$). Then an orbit $\{ f^n(y_0)\}$ is attracted to S if $\omega(y_0)$ is contained in S.
From Alligood’s definition, if $\{f^n(x_0)\}$ is a chaotic orbit and $x_0 \in \omega(x_0)$, then $\omega(x_0)$ is a chaotic set.
But my issue is, that $\omega(x_0)$ is clearly contained in $\omega(x_0)$. So isn’t $\{ f^n(x_0)\}$ attracted to $\omega(x_0)$?
I think I have some misunderstanding with my definitions.
Going by the definition, yes it’s attracted to itself, but that doesn’t necessarily mean it’s an attractor – as $\omega(x_0)$ may have a zero measure. Yes, this is in conflict with the common use of language, but that’s academics for you.
To understand non-attracting chaotic sets, first consider fixed points. There are three important types of these:
Stable fixed points: The orbit from any initial condition from a sufficiently small neighbourhood of the fixed point will converge to the fixed point. A simple example would a ball in a potential well with friction; with the fixed point (in state space) corresponding to the ball being in rest at the bottom of the potential well. These fixed points are attractors.
Unstable fixed points: The orbit from any initial condition from the vicinity of the fixed point will move away from the fixed point. A simple example is a ball on a perfect dome; with the fixed point corresponding to the ball being in rest on the top of the dome. These fixed points are not attractors.
Saddle fixed points: The orbit from any initial condition from the vicinity of the fixed point will move away from the fixed point. A simple example is a ball with friction on a, well, saddle surface. There is a line of initial conditions, which will yield orbits converging to the point, but all others will move away from it. These fixed points are not attractors either, as the states attracted to it (the points on the line) have zero measure.
To understand attracting and non-attracting chaotic sets, just replace the fixed point with a chaotic set: These are a set of points describing a chaotic motion. In case of a chaotic attractor, any initial condition in a sufficiently close vicinity will yield an orbit moving towards this set (the attractor). In case of a chaotic saddle (a.k.a. chaotic repeller, non-attracting chaotic set), almost all initial conditions in the vicinity will yield orbits that move away from the set (the saddle). In either way, the transient motion towards or away from the set will be chaotic.