📂Dynamics

Attractors in Dynamical Systems

Buildup

Let’s assume that the vector field and maps are represented as follows for the space $X$ and function $f,g : X \to X$. $$ \dot{x} = f(x) \\ x \mapsto g(x) $$

$\phi (t, \cdot)$ denotes the flow of vector field $\dot{x} = f(x)$, and $g^{n}$ denotes the map obtained by taking map $g$ $n$ times.

Definition of Nonwandering¹

A point $x_{0} \in X$ is called a nonwandering point, and the set of such points is called the nonwandering set if it satisfies the following conditions:

• (V): For all neighborhoods $U$ of $x_{0}$ and for all $T > 0$, there exists $t > T$ satisfying: $$ \phi (t,U) \cap U \ne \emptyset $$
• (M): For all neighborhoods $U$ of $x_{0}$ and for all $T > 0$, there exists $n \in \mathbb{N}$ satisfying: $$ g^{n} (U) \cap U \ne \emptyset $$

Considering up to the past time, the above definitions change to $|t| > T$ and $n \in \mathbb{Z}^{ \ast }$, respectively.

Explanation of Nonwandering

The nonwandering set, as its name suggests, comprises points that eventually return, even if they leave. Fixed points and periodic orbits are trivial nonwandering sets as they don’t leave in the first place. The nonwandering set implies a weak condition, not specifically where to go but must return at some point.

Definition of Attracting²

A closed invariant set $A \subset \mathbb{R}^{n}$ is called an attracting set, and the open set $U$ at that time is called the trapping region if it has a neighborhood $U$ of $A$ satisfying the following conditions:

• (V): For $\forall t \ge 0$, it holds that $\phi (t , U) \subset U$, and $$ \bigcap_{t > 0} \phi (t, U) = A $$
• (M): For $\forall n \ge 0$, it holds that $g^{n} (U) \subset U$, and $$ \bigcap_{t > 0} g^{n} (U) = A $$

Explanation of Attracting

The first thing to understand from the definition of attracting set is the difference from the nonwandering set, primarily the existence of a specific boundary, the trapping region $U$, from which it cannot escape, and eventually, it must exactly become $A$ after an infinite amount of time has passed. This implies that while $A$ is confined within $U$, it draws everything to itself, justifying the name attracting set.

Definition of Attractor³

A closed invariant set $A$ is called topologically transitive if for every open set $V_{1},V_{2} \subset A$, it satisfies the following:

• (V): There exists $\phi \left( t, V_{1} \right) \cap V_{2} \ne \emptyset$ such that $t \in \mathbb{R}$.
• (M): There exists $g^{n} \left( V_{1} \right) \cap V_{2} \ne \emptyset$ such that $n \in \mathbb{Z}$.

If the attracting set is topologically transitive, it is called an attractor.

Explanation of Attractor

The definition of topological transitivity, ’every’ open set $V_{1} , V_{2}$, means that no matter how small the set is chosen, the flow or map given enough time, can precisely send $V_{1}$ to $V_{2}$. However, if two points actually meet, it is just a simple periodic orbit, and the concept of an open set is used to express ‘passing through’. The addition of topological transitivity allows the attracting set to acquire various properties that were insufficient with just collecting trapping regions $U$.