📂Dynamics

Invariant Sets in Dynamics

Definition¹

Consider a space $X$ and a function $f,g : X \to X$, the vector field, and map are expressed as follows. $$ \dot{x} = f(x) \\ x \mapsto g(x) $$ Let $S \subset X$.

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

Invariant Sets can also be called as follows depending on the condition:

1. If the invariant set $S$ is considered up to time $t \ge 0$ or $n \ge 0$, it is called a Positively Invariant Set; Conversely, if it is considered up to time $t \le 0$ or $n \le 0$, it is called a Negatively Invariant Set.
2. If the invariant set $S$ forms a structure of a differentiable manifold in $C^{r}$, it is called a $C^{r}$ Invariant Manifold.

Explanation

An invariant set refers to a set that cannot escape, whether in the past or in the future. Not being able to escape to the past means, in other words, that it is not allowed to come in from outside of the invariant set. Because all times $\mathbb{R}$ are considered, it’s more appropriate to imagine it as an already determined ‘space’ rather than a ‘movement’ which is dynamic.

Not only are manifolds mentioned, but also the exploration of the space itself, which makes many people think of the connection with Topological Mathematics, as historically, dynamics and Topological Mathematics come from the same root, so it’s inevitable for familiar things to frequently appear from both sides. A scholar who left significant achievements in both areas is Henri Poincaré who is also known for the ‘Poincaré Conjecture’. Since at that time, they were not distinguished, to say that he left achievements in both fields might be inappropriate. The early 1900s was the beginning era of Topological Mathematics and dynamics, and the theories developed by scholars like Poincaré should be seen as diverging according to their interests.

The major methods for finding the existence of invariant sets in a given system include the Hadamard’s Method and the Liapunov-Perron Method, and there is also a lot of interest regarding their stability and differentiability.