Question

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

33.

$\frac{\mathbf{d}\overrightarrow{\mathbf{x}}}{\mathbf{d}\mathbf{t}}\mathbf{=}\left[\begin{array}{cc}-4& 3\\ 2& -3\end{array}\right]\overrightarrow{\mathbf{x}}$

Short answer

Sketch the lines that determent by eigenvectors and depend on the signs of Eigen values sketch the arrows that depict the future and past of the system.

Step-by-step solution

Determine the Eigen values and Eigen vectors

Consider that both ${\lambda }_1\ne 0$ and ${\lambda }_2\ne 0$. When${\lambda }_i$ corresponding to eigenvector${\overrightarrow{v}}_i$ would be greater than 0, then the arrows on the line determent by vector ${\overrightarrow{v}}_i$point away from the origin and when ${\lambda }_i<0$then they point towards to the origin.

Sketch the rough phase portraits for the dynamical systems

There are three cases: ${\lambda }_1<{\lambda }_2<0,{\lambda }_1>{\lambda }_2>0$and ${\lambda }_1<0<{\lambda }_2$.

When ${\lambda }_i=0$trajectories are rays parallel to ${\overrightarrow{v}}_j$and arrows on those trajectories converge into the line determent by ${\overrightarrow{v}}_j$ when ${\lambda }_j<0$and they diverge from it when ${\lambda }_j>0$ .

Sketch the rough phase portraits for the dynamical system as follows: