Attractors in Chaos
Build-up
Let’s say the vector field and map for space $X = \left( \mathbb{R}^{n} , \left\| \cdot \right\| \right)$ and function $f,g : X \to X$ are represented as follows. $$ \dot{x} = f(x) \\ x \mapsto g(x) $$
$\phi (t, \cdot)$ is the flow of the vector field $\dot{x} = f(x)$, $g^{n}$ represents the map $g$ taken $n$ times, and assume that $\Lambda \subset X$ is an invariant compact set under $\phi (t, \cdot)$ or $g(\cdot)$.
That $\phi (t,x)$ or $g(x)$ is Sensitive dependence on initial Conditions in $\Lambda$ means that for all $x \in \Lambda$, there exists $\varepsilon > 0$ satisfying the following condition and for all neighborhoods $U$ of $x$, there exists $y \in U$ and $t > 0$ satisfying the following condition. $$ \begin{align*} \left\| \phi (t,x) - \phi (t,y) \right\| > \varepsilon \text{ or } \left\| g^{n} (x) - g^{n} (y) \right\| > \varepsilon \end{align*} $$
This formula exactly describes how, depending on the change of initial values, a significantly large difference can occur in as short a time as we want, perfectly encapsulating the expression ‘Sensitive dependence on initial Conditions’.
In addition to this, the following two concepts are further needed.
Definition of Attractors: A closed invariant set $A$ is said to be Topologically Transitive if it satisfies the following for all open sets $V_{1},V_{2} \subset A$:
- (V): There exists a $t \in \mathbb{R}$ such that $\phi \left( t, V_{1} \right) \cap V_{2} \ne \emptyset$.
- (M): There exists a $n \in \mathbb{Z}$ such that $g^{n} \left( V_{1} \right) \cap V_{2} \ne \emptyset$.
If the attracting set is topologically transitive, it is called an attractor.
Density in Metric Spaces: Let $\left( X , d \right)$ be a metric space and let’s say $A \subset X$.
- When $\overline{A} = X$, $A$ is said to be dense in $X$.
Definition
$\Lambda$ is said to be Chaotic if it satisfies the following two conditions:
- (i): $\phi (t,x)$ or $g(x)$ is sensitive to the initial values in $\Lambda$.
- (ii): $\phi (t,x)$ or $g(x)$ is topologically transitive in $\Lambda$.
- (iii): The periodic orbits of $\phi (t,x)$ or $g(x)$ are dense in $\Lambda$.
Intuitive Explanation
In chaos theory, the chaotic attractor is a concept of such importance and interest that it could become the name of the field itself. However, it might be difficult to intuitively understand since it has moved far into the realm of mathematical and theoretical abstraction. Have a look at the following gif:
This gif is drawing the trajectory of the Lorenz Attractor. Initially, it seems to grow and rotate only on the left side, then suddenly it starts moving left and right without any apparent rule, not passing the same place twice but also not drifting off to somewhere distant. This is a quintessential example of Chaos, which is of interest in dynamics.
The fact that it doesn’t represent a periodic orbit means although the points in space seem to be looping around a similar area, they never return to a point once visited. This ‘butterfly-shaped trajectory’ is exactly what is referred to as a Strange Attractor.
Mathematical Explanation
- (0): Without the condition of being compact, there are many counterexamples where the future is quite evident while still meeting the below condition. Naively, if there’s a system like $\dot{x} = ax$ for $a>0$, its solution would be $\phi (t,x) = e^{at} x$, hence sensitive to initial values. However, calling it chaos when it diverges infinitely seems nonsensical.
- (i): Since dynamical systems are deterministic, being sensitive to initial values means ‘uncertainty’ about the future. Among non-specialists, chaos theory is often fantasized as the ‘Butterfly Effect’, meaning ‘a butterfly flapping its wings could cause a storm on the other side of the Earth’, and this is written straightforwardly and succinctly through equations.
- (ii): Being topologically transitive means, simply put, any point in $\Lambda$ can come from anywhere and go anywhere within $\Lambda$ due to the system. Without this condition, a system that diverges or converges at a point following some consistent flow could also be considered chaotic.
- (iii): Depending on the author, this condition may not be strictly necessary. This might seem odd, but it’s because chaos itself is hard to express in words. ‘Periodic orbit’ might not seem chaotic with the first two conditions met, but being dense in $\Lambda$ means, at least within $\Lambda$, there should be ’no gaps’, essentially rummaging through every nook and cranny of $\Lambda$. Conversely, this condition means that by excluding fixed points within $\Lambda$, it’s possible to eliminate counterexamples.
Strange Attractor?
An attractor $\mathcal{A} \subset X$ is defined as a Strange Attractor if it is chaotic. A widely adopted Korean euphemism is ‘strange dragger’, which unfortunately doesn’t capture the original meaning at all.