Stability of Invariant Manifolds
Definition
Manifold of Vector Fields1
Given a space $\mathbb{R}^{n}$ and a function $f : \mathbb{R}^{n} \to \mathbb{R}^{n}$, let’s say the following vector field is given by a differential equation. $$ \dot{x} = f(x) $$ When a fixed point of autonomous system $\overline{x}$ is given, classify the eigenvectors $e$ corresponding to each eigenvalue $\lambda$ of the linearization matrix $A := D f \left( \overline{x} \right)$ according to the real part $\operatorname{Re} (\lambda)$, and let’s represent its generation as follows. $$ E^{s} := \text{span} \left\{ e : \lambda e = A e , \operatorname{Re} (\lambda) < 0 \right\} \\ E^{u} := \text{span} \left\{ e : \lambda e = A e , \operatorname{Re} (\lambda) > 0 \right\} \\ E^{c} := \text{span} \left\{ e : \lambda e = A e , \operatorname{Re} (\lambda) = 0 \right\} $$ $E^{s}$ is called stable manifold, $E^{u}$ is unstable manifold, and $E^{c}$ is center manifold.
Manifold of Maps2
Given a space $\mathbb{R}^{n}$ and a function $g : \mathbb{R}^{n} \to \mathbb{R}^{n}$, let’s say the following vector field is given by a differential equation. $$ x \mapsto g(x) $$ When a fixed point $\overline{x}$ of such a system is given, classify the eigenvectors $e$ corresponding to each eigenvalue $\lambda$ of the linearization matrix $B := D g \left( \overline{x} \right)$ according to the absolute value $\left| \lambda \right|$, and let’s represent its generation $\text{span}$ as follows. $$ E^{s} := \text{span} \left\{ e : \lambda e = B e , \left| \lambda \right| < 1 \right\} \\ E^{u} := \text{span} \left\{ e : \lambda e = B e , \left| \lambda \right| > 1 \right\} \\ E^{c} := \text{span} \left\{ e : \lambda e = B e , \left| \lambda \right| = 1 \right\} $$ $E^{s}$ is called stable manifold, $E^{u}$ is unstable manifold, and $E^{c}$ is center manifold.
Explanation
The subscript of $E$, that is $s,u,c$, comes from the first letters of Stable, Unstable, and Center, respectively, and the following holds. $$\mathbb{R}^{n} = E^{s} \oplus E^{u} \oplus E^{c} $$
In $1$ dimension, one can imagine approaching or departing, and in $2$ dimension, one can imagine coming from and going to a direction, but when considering the general Euclidean space $\mathbb{R}^{n}$, the concept of ‘direction’ is meaningless. Therefore, it is simplified to just entering or exiting, and the word manifold is used.
Meanwhile, some textbooks might use a simpler definition as follows3. When $\overline{x}$ is a periodic point of the map $g$, the following defined $\mathcal{S}(\overline{x})$ is called the stable manifold of $\overline{x}$, and $\mathcal{U}(\overline{x})$ is called the unstable manifold of $\overline{x}$. $$ \mathcal{S} (\overline{x}) := \left\{ x \in \mathbb{R}^{n} : \lim_{k \to \infty} \left| f^{k} ( x ) - f^{k} ( \overline{x} ) \right| = 0 \right\} \\ \mathcal{U} (\overline{x}) := \left\{ x \in \mathbb{R}^{n} : \lim_{k \to \infty} \left| f^{-k} ( x ) - f^{-k} ( \overline{x} ) \right| = 0 \right\} $$