Phase portraits for real eigenvalues Problem set A Answers

1. (a) \(\lambda=3\ \text{and}\ 6\)
2. (b) \(\lambda=-7\ \text{and}\ -2\)

Unstable node: eigenvalues \(2\text{ and }5\); eigenvector lines gradient −2 (\(\lambda=2\)) and gradient 1 (\(\lambda=5\)); arrows away from the origin on both lines

Stable node: eigenvalues \(-4\text{ and }-2\); eigenvector lines gradient −3 (\(\lambda=-4\)) and gradient −1 (\(\lambda=-2\)); arrows towards the origin on both lines

Saddle point: eigenvalues \(-2\text{ and }3\); eigenvector lines gradient −4 (\(\lambda=-2\)) and gradient 1 (\(\lambda=3\)); arrows towards the origin on the \(\lambda=-2\) line and away on the other

1. (a) Stable node: eigenvalues \(-5\text{ and }-2\); eigenvector lines gradient −1 (\(\lambda=-5\)) and gradient 2 (\(\lambda=-2\)); arrows towards the origin on both lines
2. (b) Draw the line through the origin with direction \(\begin{pmatrix}1\\-1\end{pmatrix}\) (gradient −1) for \(\lambda=-5\): arrows point towards the origin. Draw the line through the origin with direction \(\begin{pmatrix}1\\2\end{pmatrix}\) (gradient 2) for \(\lambda=-2\): arrows point towards the origin. Other trajectories come in from far away alongside the \(\lambda=-5\) line and reach the origin tangent to the \(\lambda=-2\) line.
3. (c) The point lies on the eigenvector line of \(\lambda=-2\), so the solution stays on that line: \(x=3e^{-2t},\ y=6e^{-2t}\). Both amounts decay to 0 and their ratio stays constant.

1. (a) Saddle point: eigenvalues \(-3\text{ and }2\); eigenvector lines gradient −1 (\(\lambda=-3\)) and gradient 3/2 (\(\lambda=2\)); arrows towards the origin on the \(\lambda=-3\) line and away on the other
2. (b) It moves in towards the origin along the line, getting closer for ever but never reaching it (\(x=2e^{-3t},\ y=-2e^{-3t}\)).
3. (c) It moves away from the origin: for large t it follows the line of gradient 3/2 (\(x\approx0.8e^{2t},\ y\approx1.2e^{2t}\)).

1. (a) \(\det\mathbf M=3-2k\)
2. (b) \(k>\tfrac32\)
3. (c) \(\lambda=1\text{ and }3\): unstable node