Ch05 Eigenvalues and Eigenvectors

5.6 Discrete Dynamical Systems

Graphical Description of Solutions

When $\textbf{A}$ is $2 \times 2$, algebraic calculations can be supplemented by a geometric description of a system’s evolution.

We can view the equation $\textbf{x}_{k+1} = A \textbf{x}_k$ as a description of what happens to an initial point $\textbf{x}_0 \in \mathbb{R}^2$ as it is transformed repeatedly by the mapping $\textbf{x} \mapsto A\textbf{x}$

The graph of $\textbf{x}_0, \textbf{x}_1, \cdots,$ is called a trajectory of the dynamical system.

Example 2

Plot several trajectories of the dynamical system $\textbf{x}_{k+1} = A \textbf{x}_k$, when

$$A=\begin{bmatrix} .8 & 0 \\ 0 & .64 \end{bmatrix}$$

Solution

The eigenvalues of $A$ are .8 and .64, with eigenvectors $\textbf{v}_1 = \begin{bmatrix} 1 \\ 0 \end{bmatrix}$ and $\textbf{v}_2 = \begin{bmatrix} 0 \\ 1 \end{bmatrix}$. if $\textbf{x}_0 = c_1 \textbf{v}_1 +c_2 \textbf{v}_2$, then $\textbf{x}_k = c_1 (.8)^k \begin{bmatrix} 1 \\ 0 \end{bmatrix} + c_2 (.64)^k \begin{bmatrix} 0 \\ 1 \end{bmatrix}$.

Of course, $\textbf{x}_k$ tends to 0 because $(.8)^k$ and $(.64)^k$ both approach 0 as $k \leftarrow \infty$. But the way $\textbf{x}_k$ goes toward 0 is interesting. See Fig. 1 below.

Figure 1 shows the first few terms of several trajectories that begin at points on the boundary of the box with corners at $(\pm 3,\pm 3)$. The points on each trajectory are connected by a thin curve, to make the trajectory easier to see.

In example 2, the origin is called an attractor of the dynamical system because all trajectories tend toward 0.

In the next example, both eigenvalues of $A$ are larger than 1 in magnitude, and 0 is called a repeller of the dynamical system.

Example 3

Plot several typical solutions of the equation $\textbf{x}_{k+1} = A \textbf{x}_k$, where

$$A=\begin{bmatrix} 1.44 & 0 \\ 0 & 1.2 \end{bmatrix}$$

Solution

The eigenvalues of $A$ are 1.44 and 1.2. If $\textbf{x}_0 = \begin{bmatrix} c_1 \\ c_2 \end{bmatrix}$, then $\textbf{x}_k = c_1 (1.44)^k \begin{bmatrix} 1 \\ 0 \end{bmatrix} + c_2 (1.2)^k \begin{bmatrix} 0 \\ 1 \end{bmatrix}$.

Both terms grow in size, but the first term grows faster.

So the direction of greatest repulsion is the line through 0 and the eigenvector for the eigenvalue of larger magnitude.

Fig. 2 below shows several trajectories that begin at points quite close to 0.