For problem 1-8, given the eigenpairs, \((\lambda, \vBld)\), sketch the corresponding phase portrait. (The matrix \(\ABld\) is provided when necessary.)
\(\left(1, \begin{pmatrix} 2 \\ 1 \end{pmatrix} \right) \hspace{0.9 in} \left(2, \begin{pmatrix} -1 \\ 1 \end{pmatrix} \right)\)
\(\left(-1, \begin{pmatrix} 2 \\ 1 \end{pmatrix} \right) \hspace{0.8 in} \left(-3, \begin{pmatrix} -1 \\ 4 \end{pmatrix} \right)\)
\(\left(5, \begin{pmatrix} 1 \\ 2 \end{pmatrix} \right) \hspace{0.9 in} \left(3, \begin{pmatrix} 1 \\- 1 \end{pmatrix} \right)\)
\(\left(0, \begin{pmatrix} 1 \\ 1 \end{pmatrix} \right) \hspace{0.9 in} \left(-2, \begin{pmatrix} -5 \\ 1 \end{pmatrix} \right)\)
\(\left(2, \begin{pmatrix} 3 \\ 1 \end{pmatrix} \right) \hspace{0.9 in} \left(-1, \begin{pmatrix} 1 \\ -1 \end{pmatrix} \right)\)
\(\left(-2, \begin{pmatrix} -1 \\ 1 \end{pmatrix} \right) \hspace{0.7 in} \left(-4, \begin{pmatrix} 1 \\ 1 \end{pmatrix} \right)\)
\(\left(-1 + i2, \begin{pmatrix} 2i \\ 1 \end{pmatrix} \right) \hspace{0.4 in} \left(-1 - i2, \begin{pmatrix} -2i \\ 1 \end{pmatrix} \right) \hspace{0.4 in} {\bf A} = \begin{pmatrix}-1 & -4 \\ 1 & -1 \end{pmatrix}\)
\(\left(i, \begin{pmatrix} 1 \\ 2 - i \end{pmatrix} \right) \hspace{0.7 in} \left(-i, \begin{pmatrix} 1 \\ 2 + i \end{pmatrix} \right) \hspace{0.7 in} {\bf A} = \begin{pmatrix}2 & -1 \\ 5 & -2 \end{pmatrix} \)
For problems 9-20 sketch the phase portraits of the following system of differential equations. Determine the stability of each system (whether the system is stable or unstable and whether or not it is attracting or repelling)
\({\bf x}' = \begin{pmatrix} -2 & 0 \\ 0 & -2 \end{pmatrix} {\bf x}\)
\(\displaystyle \frac{\dee}{\dt} \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix} \)
\({\bf x}' = \begin{pmatrix} -1 & -1 \\ 1 & -3 \end{pmatrix} {\bf x}\)
\({\bf x}' = \begin{pmatrix} 2 & \frac{3}{2} \\ -\frac{3}{2} & -1 \end{pmatrix} {\bf x}\)
\(\displaystyle \frac{\dee}{\dt} \begin{pmatrix} x\\y \end{pmatrix} = \begin{pmatrix} 1 & \frac{1}{2} \\ 4 & 2 \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix} \)
\({\bf x}' = \begin{pmatrix} 2 & 2 \\ -1 & 4 \end{pmatrix} {\bf x}\)
\(\displaystyle \frac{\dee}{\dt} \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} 3 & 0 \\ 0 & 3 \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix} \)
\({\bf x}' = \begin{pmatrix} 4 & -1 \\ 1 & 2 \end{pmatrix} {\bf x}\)
\({\bf x}' = \begin{pmatrix} -3 & 2 \\ -2 & 1 \end{pmatrix} {\bf x}\)
\({\bf x}' = \begin{pmatrix} 1 & 2 \\ -5 & -1 \end{pmatrix} {\bf x}\)
\(\displaystyle \frac{\dee}{\dt} \begin{pmatrix} x\\ y \end{pmatrix} = \begin{pmatrix} -2 & 1\\ -1 & -2 \end{pmatrix} \begin{pmatrix} x\\ y \end{pmatrix}\)
\( \displaystyle \frac{\dee}{\dt} \begin{pmatrix} x\\ y \end{pmatrix} = \begin{pmatrix} 2 & -1\\ 0 & 3 \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix} \)
For problems 21-26 sketch the phase portrait as well as the trajectory associated with the initial condition. Determine the stability of each system. (Make sure to clearly label the trajectory)
\( \displaystyle \frac{\dee}{\dt} \begin{pmatrix} x\\ y \end{pmatrix} = \begin{pmatrix} 0 & -2\\ 2 & 0 \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix}, \hspace{.5in} \begin{pmatrix} x(0) \\ y(0) \end{pmatrix} = \begin{pmatrix} 2 \\ -1 \end{pmatrix} \)
\({\bf x}' = \begin{pmatrix} -3 & 2 \\ -1 & -1 \end{pmatrix} {\bf x}, \hspace{.5 in} {\bf x}(0) = \begin{pmatrix} 1 \\ -2 \end{pmatrix}\)
\({\bf x}' = \begin{pmatrix}3 & -2 \\ 3 & -1 \end{pmatrix}{\bf x}, \hspace{.65 in} {\bf x}(0) = \begin{pmatrix} 3 \\ 1 \end{pmatrix}\)
\(\displaystyle \frac{\dee}{\dt} \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix}2 & 4 \\-1 & -2 \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix} , \hspace{.5 in} \begin{pmatrix} x(0) \\ y(0) \end{pmatrix} = \begin{pmatrix} 1 \\ 1 \end{pmatrix}\)
\(\displaystyle \frac{\dee}{\dt} \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} 1 & 1 \\ 4 & -2 \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix} , \hspace{.65 in} \begin{pmatrix} x(0) \\ y(0) \end{pmatrix} = \begin{pmatrix} 0 \\ 2 \end{pmatrix}\)
\({\bf x}' = \begin{pmatrix} 5 & -1 \\ 3 & 1 \end{pmatrix} {\bf x}, \hspace{.65 in} {\bf x}(0) = \begin{pmatrix} 1 \\ \frac{1}{2} \end{pmatrix}\)
Consider the linear system given by \[\xBld^\prime = \begin{pmatrix} 1 & \alpha \\ 0 & 1 \end{pmatrix} \xBld.\] Sketch the phase portrait, and classify the behavior and stability near the origin when \(\alpha = 1\), \(\alpha = -1\) and \(\alpha = 0\). Can you see how the phase portrait might transform as \(\alpha\) moves from \(1\) to \(0\) to \(-1\)?
Consider the linear system given by \[\xBld^\prime = \begin{pmatrix} 1 & \alpha \\ -\alpha & 1 \end{pmatrix} \xBld.\] Sketch the phase portrait and classify the behavior and stability near the origin when \(\alpha =1\), \(\alpha = -1\) and \(\alpha = 0\). Can you see how the phase portrait might transform as \(\alpha\) moves from \(1\) to \(0\) to \(-1\)?
Consider the linear system given by \[\xBld^\prime = \begin{pmatrix} 2\alpha & 1\\ -1 & 0 \end{pmatrix} \xBld.\] Sketch the phase portrait and classify the stability near the origin when \(\alpha = 2\), \(\alpha =1\), \(\alpha=.5\), \(\alpha = 0\), \(\alpha = -.5\), \(\alpha=-1\), \(\alpha = -2\). There is a critical value \(\alpha_0\) such that \(\alpha > \alpha_0\) and \(\alpha< \alpha_0\) have different stability (i.e. the system changes from unstable to stable). What is \(\alpha_0\)?
Problems 30-33 apply to the planar system \(\xBld^\prime = \ABld\xBld\).
Show that the origin is attracting if and only if \(\text{tr}\left(\ABld\right)<0\) and \(\det\left(\ABld\right) > 0\).
Show that the origin is repelling if and only if \(\text{tr}\left(\ABld\right) >0\) and \(\det\left(\ABld\right) >0\).
Is it true that the origin is stable if and only if \(\text{tr}\left(\ABld\right) \leq 0\) and \(\det\left(\ABld\right) \geq 0\)? State why or why not.
Is it true that the origin is unstable if and only if \(\det\left(\ABld\right) \leq 0\)? State why or why not.