📂Dynamics

Multidimensional Linear Maps

Definition ¹

1. If a map $T_{A} : \mathbb{R}^{m} \to \mathbb{R}^{m}$ satisfies $$T_{A} ( a \mathbf{x} + b \mathbf{y} ) = a T_{A} ( \mathbf{x} ) + b T_{A} ( \mathbf{y} )$$ for all $a,b \in \mathbb{R}$ and $\mathbf{x}, \mathbf{y} \in \mathbb{R}^{m}$, then $T_{A}$ is said to be linear.

Let’s refer to the eigenvalues of $A$ as $\lambda_{1} , \cdots , \lambda_{m}$.

2. If $| \lambda_{1} | \ne 1, \cdots , | \lambda_{m} | \ne 1$, then $A$ is hyperbolic.
3. If for a hyperbolic $A$, there exists at least one $i,j$ satisfying $\begin{cases} | \lambda_{i} | >1 \\ | \lambda_{j} | <1 \end{cases}$, then $\mathbb{0}$ is a saddle.

Explanation

Indeed, the matrix of size $m \times m$ corresponding to the map $T_{A}$ satisfies $\displaystyle A ( a \mathbf{x} + b \mathbf{y} ) = a A \mathbf{x} + b A \mathbf{y}$. Essentially, $T_{A}$ and $A$ are the same, so there’s no need to differentiate between them; it is appropriate to call $A$ itself a map.

Meanwhile, the origin $\mathbb{0}$ satisfies $A \mathbb{0} = \mathbb{0}$, making it a fixed point of $T_{A}$. Naturally, this leads to discussions about fixed points.

Criteria when not a saddle

When $\mathbb{0}$ is not a saddle, it can be classified as a sink or source according to the following theorem:

• [1]: If $| \lambda_{1} | < 1, \cdots , | \lambda_{m} | < 1$, then $\mathbb{0}$ is a sink.
• [2]: If $| \lambda_{1} | > 1, \cdots , | \lambda_{m} | > 1$, then $\mathbb{0}$ is a source.

Examples

Considering $\mathbb{R}^2$ to $A:= \begin{bmatrix} 1/2 & 0 \\ 0 & 2 \end{bmatrix}$ as an example of a hyperbolic linear map, the eigenvalues are $\lambda_{1} = 1/2, \lambda_{2} = 2$. Given an initial point $\mathbf{v}_{0} = (x,y)$ on the plane, applying the map $\mathbf{v}_{n+1} := A \mathbf{v}_{n}$ would decrease the value of $x$ and increase $y$ with each iteration. Particularly, the origin becomes a saddle.

$\displaystyle y = {{ 1 } \over { x }}$ is a typical example of a Hyperbola, and from its shape, one can understand why such maps are called hyperbolic. Of course, there’s no need to be constrained by the shape; the term hyperbolic encompasses a much broader concept.

Considering $\mathbb{R}^2$ to $B:= \begin{bmatrix} 1/2 & 0 \\ 0 & 1/2 \end{bmatrix}$ as an example of a linear map where the origin is a sink, the eigenvalues are $\lambda_{1} = 1/2, \lambda_{2} = 1/2$, and every point on the plane moves closer to the origin with each application of the map. Conversely, for a linear map like $C:= \begin{bmatrix} 2 & 0 \\ 0 & 2 \end{bmatrix}$, every point except $\mathbb{0}$ moves away from the origin with each application of the map, becoming a source.