Let A have the properties described in Exercise 1.

(a) Is the origin an attractor, a repeller, or a saddle point of the dynamical system\({x_{k + 1}} = A{x_k}\)?

(b) Find the directions of greatest attraction and/or repulsion for this dynamical system.

(c) Make a graphical description of the system, showing the directions of greatest attraction or repulsion. Include a rough sketch of several typical trajectories (without computing specific points).

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a)

The eigenvalues of the matrix \(A\) in Exercise 1 are 3 and 1/3. The origin is a saddle point because \(\left| 3 \right| > 1\) and \(\left| {1/3} \right| < 1\).

b)

The direction of the greatest attraction is determined by the eigenvector.

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The eigenvector corresponds to the eigenvalue with an absolute value less than 1.

The direction of greatest repulsion is determined by \({{\rm{v}}_1} = \left( {\begin{aligned}{}1\\1\end{aligned}} \right)\).

The eigenvector corresponds to the eigenvalue greater than 1.

c)

Lines through the eigenvectors and the origin.

Arrows toward the origin (showing attraction) on the line through \({{\rm{v}}_2}\) and arrows away from the origin (showing repulsion) on the line through \({{\rm{v}}_1}\),

Several typical trajectories (with arrows) that show the general flow of points are shown in the diagram below. Other than \({{\rm{v}}_1}\) and \({{\rm{v}}_2}\), no precise points were computed.

This is the only sort of drawing that can be made without utilizing a computer to plot points.