Question

Sketch rough phase portraits for the dynamical systems given in Exercise 32 through 39.

$\overrightarrow{\mathbf{x}}\left(t+1\right)\mathbf{=}\left[\begin{array}{cc}1& 0.3\\ -0.2& 1.7\end{array}\right]\overrightarrow{\mathbf{x}}\left(t\right)$

Short answer

Thus, the rough sketch of the system in the explanation.

Step-by-step solution

Given in the question.

The given systems are:

$\overrightarrow{x}\left(t+1\right)=\left[\begin{array}{cc}1& 0.3\\ -0.2& 1.7\end{array}\right]\overrightarrow{x}\left(t\right)$

Solve the given system.

Solve the given system as follows:

$$\begin{gathered} \left[\begin{array}{c}{t}_1+1\\ t_2+1\end{array}\right]=\left[\begin{array}{cc}1& 0.3\\ -0.2& 1.7\end{array}\right]\left[\begin{array}{c}{t}_1\\ t_2\end{array}\right] \\ \left[\begin{array}{c}{t}_1+1\\ t_2+1\end{array}\right]=\left[\begin{array}{c}{t}_1+0.3t_2\\ -0.2t_1+1.7t_2\end{array}\right] \end{gathered}$$

Now, equating both sides gives:

$$\begin{gathered} \{\begin{array}{c}{t}_1+1=t_1+0.3t_2\\ t_2+1=-0.2t_1+1.7t_2\end{array} \\ \{\begin{array}{l}1=0.3t_2\\ 1=-0.2t_1+1.7t_2\end{array} \end{gathered}$$

Solve the above two equations gives:

$$\begin{gathered} t_2=3.33 \\ {t}_1=6.66 \end{gathered}$$

Thus the solutions is the span of$\left[\begin{array}{c}1\\ 2\end{array}\right]$

Graph of the given system.

The graph is plotted below: