📂Dynamics

Stability of Invariant Manifolds

Definition

Stability and Instability of Invariant Sets ¹

Let’s define the two invariant sets with respect to a fixed point $\overline{x}$ of a dynamical system $\left( T, X , \varphi^{t} \right)$ as follows. $$\begin{align*} W^{s} \left( \overline{x} \right) :=& \left\{ x : \varphi^{t} x \to \overline{x} , t \to + \infty \right\} \\ W^{u} \left( \overline{x} \right) :=& \left\{ x : \varphi^{t} x \to \overline{x} , t \to - \infty \right\} \end{align*}$$ We define $W^{s} \left( \overline{x} \right)$ as the stable set of $\overline{x}$ and $W^{u} \left( \overline{x} \right)$ as the unstable set of $\overline{x}$.

Invariant Manifolds ²

Given a space $\mathbb{R}^{n}$ and a function $f : \mathbb{R}^{n} \to \mathbb{R}^{n}$, let’s assume that the following vector field is given by a differential equation. $$\dot{x} = f(x)$$ Given a fixed point of the system represented by such a differential equation $\overline{x}$, we classify the eigenvectors $e$ corresponding to each eigenvalue $\lambda$ of the linearization matrix $A := D f \left( \overline{x} \right)$ based on the real part $\operatorname{Re} (\lambda)$, and represent their generation as follows. $$E^{s} := \text{span} \left\{ e : \lambda e = A e , \operatorname{Re} (\lambda) < 0 \right\} \\ E^{u} := \text{span} \left\{ e : \lambda e = A e , \operatorname{Re} (\lambda) > 0 \right\} \\ E^{c} := \text{span} \left\{ e : \lambda e = A e , \operatorname{Re} (\lambda) = 0 \right\}$$ We refer to $E^{s}$ as the stable manifold, $E^{u}$ as the unstable manifold, and $E^{c}$ as the center manifold.

Given a space $\mathbb{R}^{n}$ and a function $g : \mathbb{R}^{n} \to \mathbb{R}^{n}$, let’s assume that the following vector field is given by a differential equation. $$x \mapsto g(x)$$ Given a fixed point $\overline{x}$ of such a system, we classify the eigenvectors $e$ corresponding to each eigenvalue $\lambda$ of the linearization matrix $B := D g \left( \overline{x} \right)$ based on the absolute value $\left| \lambda \right|$, and represent their generation $\text{span}$ as follows. $$E^{s} := \text{span} \left\{ e : \lambda e = B e , \left| \lambda \right| < 1 \right\} \\ E^{u} := \text{span} \left\{ e : \lambda e = B e , \left| \lambda \right| > 1 \right\} \\ E^{c} := \text{span} \left\{ e : \lambda e = B e , \left| \lambda \right| = 1 \right\}$$ We refer to $E^{s}$ as the stable manifold, $E^{u}$ as the unstable manifold, and $E^{c}$ as the center manifold.

Explanation

The subscript of $E$, $s,u,c$, derives from the initials of stable, unstable, and center, respectively, and the following holds true. $$\mathbb{R}^{n} = E^{s} \oplus E^{u} \oplus E^{c}$$

In dimensions $1$ or $2$, one might imagine getting closer or farther away, or approaching from one direction and leaving in another, but considering a general Euclidean space $\mathbb{R}^{n}$, the concept of ‘direction’ becomes meaningless. Thus, we simply distinguish between entering or leaving, using the word manifold.

On the other hand, some textbooks use a concise definition³. When $\overline{x}$ is a periodic point of the map $g$, the definitions are given as follows: $\mathcal{S}(\overline{x})$ is called the stable manifold of $\overline{x}$, and $\mathcal{U}(\overline{x})$ is called the unstable manifold of $\overline{x}$. $$\begin{align*} \mathcal{S} (\overline{x}) :=& \left\{ x \in \mathbb{R}^{n} : \lim_{k \to \infty} \left| f^{k} ( x ) - f^{k} ( \overline{x} ) \right| = 0 \right\} \\ \mathcal{U} (\overline{x}) :=& \left\{ x \in \mathbb{R}^{n} : \lim_{k \to \infty} \left| f^{-k} ( x ) - f^{-k} ( \overline{x} ) \right| = 0 \right\} \end{align*}$$