Exercise Group 6.2.5.9–14. One-Parameter Families and Bifurcations.

Each of the following matrices in Exercise Group 6.2.5.9–14 describes a family of differential equations \(\mathbf x' = A \mathbf x\) that depends on the parameter \(\alpha\text{.}\) For each one-parameter family sketch the curve in the trace-determinant plane determined by \(\alpha\text{.}\) Identify any values of \(\alpha\) where the type of system changes. These values are bifurcation values of \(\alpha\text{.}\)
9.
\(A = \begin{pmatrix} \alpha \amp 3 \\ -1 \amp 0 \end{pmatrix}\)
10.
\(A = \begin{pmatrix} \alpha \amp 3 \\ \alpha \amp 0 \end{pmatrix}\)
11.
\(A = \begin{pmatrix} \alpha \amp 2 \\ \alpha \amp \alpha \end{pmatrix}\)
12.
\(A = \begin{pmatrix} 1 \amp 2 \\ \alpha \amp 0 \end{pmatrix}\)
13.
\(A = \begin{pmatrix} \alpha \amp 1 \\ 1 \amp \alpha - 1 \end{pmatrix}\)
14.
\(A = \begin{pmatrix} 0 \amp 1 \\ \alpha \amp \sqrt{1 - \alpha^2} \end{pmatrix}\)
in-context