📂Dynamics

Invariant Sets in Dynamics

Definition

Rigorous Definition ¹

In a dynamical system $\left( T, X , \varphi^{t} \right)$, if a subset $S \subset X$ satisfies the following conditions $S$, it is called an invariant set. $$ x_{0} \in S \implies \varphi^{t} x_{0} \in S \qquad , \forall t \in T $$

Vector Fields and Maps ²

Let us denote a space $X$ and a function $f,g : X \to X$, and consider that a vector field or a map is expressed as follows. $$ \dot{x} = f(x) \\ x \mapsto g(x) $$ Let it be $S \subset X$.

• (V): If $\forall x_{0} \in S$ satisfies the following conditions for all $t \in \mathbb{R}$, it is called an invariant set under the vector field $\dot{x}=f(x)$. $$ x(t,x_{0}) \in S $$
• (M): If $\forall x_{0} \in S$ satisfies the following conditions for all $n \in \mathbb{Z}$, it is called an invariant set under the map $x \mapsto g(x)$. $$ g^{n} (x_{0}) \in S $$

Invariant sets may also be referred to as follows based on the conditions:

1. If the time of the invariant set $S$ is considered only until $t \ge 0$ or $n \ge 0$, it is called a positively invariant set, and conversely, if it is considered only until $t \le 0$ or $n \le 0$, it is called a negatively invariant set.
2. If the invariant set $S$ forms a structure of a $C^{r}$ differentiable manifold, it is called an $C^{r}$ invariant manifold.

Explanation

An invariant set refers to a set from which one cannot escape, whether in past time or future time. Saying that one cannot escape to past time also means that entry from outside the invariant set is not allowed. Since all time $\mathbb{R}$ is considered, it is better to imagine it as a predetermined ‘space’ rather than as a dynamic shape like ‘movement’.

Given that manifolds are mentioned but also that space itself is explored, many people may think of the association with topology. Historically, dynamical systems and topology have emerged from the same roots, so it is natural that familiar concepts appear frequently. One notable scholar on both sides is Henri Poincaré, famous for the ‘Poincaré conjecture’. However, distinguishing between achievements in topology and dynamical systems may not be appropriate since these disciplines were not separated back then. In the early 1900s, the theories advanced by scholars like Poincaré led to the differentiation based on their interests.

Major methods for finding the existence of invariant sets in a given system include the Hadamard’s method and the Liapunov-Perron method, with considerable interest in their stability and differentiability.