📂Dynamics

Attracting Set's Basin

Definition ¹

Let’s say the vector field and map for the space $X$ and function $f,g : X \to X$ are represented as follows. $$ \dot{x} = f(x) \\ x \mapsto g(x) $$ Let $\phi (t, \cdot)$ be the flow of the vector field $\dot{x} = f(x)$ and $g^{n}$ be the map $g$ applied $n$ times. The following defined sets are called the Basin of attracting set $A$:

• Vector Field $$\displaystyle \bigcup_{t \le 0} \phi ( t, U )$$
• Map $$\displaystyle \bigcup_{n \le 0} g^{n} ( U )$$

Explanation

Basin is a word not familiar to us, but it’s important to understand the concept more than the definition itself.

In other fields than mathematics, a basin refers to something with a concave shape or a terrain surrounded by mountains as shown in the picture above. Such expressions, indeed, help us understand what a basin is.

Imagine placing a red marble on top of a concave basin as shown above. The marble will move towards the drain due to gravity. Even if there’s kinetic energy left and it overshoots the drain, it will eventually stop at the location of the drain. That’s because the basin is designed to naturally lead water to the drain without any intervention. In this sense, the drain is the fixed point of the basin, and the basin is the basin of the drain.

Approaching a bit more abstractly, there are exactly three fixed points when a ball is randomly dropped on such a surface. If it’s dropped precisely at $\color{red}{b}$, there’s no reason to move either left or right, and if it drops at $\color{green}{a}$ and $\color{green}{c}$, there’s no further place to fall.

Thus, the basin of $\color{red}{b}$ precisely contains only the singleton set $\left\{ b \right\}$. [ NOTE: Mathematically, the periodic point itself is always included in its basin.] The basin of $\color{green}{a}$ is on the left of the red line, and the basin of $\color{green}{c}$ is on the right.

In other words, knowing which periodic point’s basin the initial value belongs to allows us to predict the outcome. From the perspective of periodic points, it becomes a collection of initial values that eventually converge to them, and from the system’s perspective, it creates a partition made by periodic points.