Bring the chapter together: write a system as a matrix, find its eigenvalues and straight-line solutions, fit a solution to its start, and tell a saddle, a node, a spiral and a centre apart from the trace and the determinant. No worked example sits above the answers.

A radiologist looking at an X-ray sees at once which of a few known pictures it is. The trace and the determinant are the X-ray of a matrix.

Decide first what is asked. For $\mathbf{x}'=A\mathbf{x}$ find $\operatorname{tr}A$ and $\det A$. $\det A<0$: a saddle. $\det A>0$ and $(\operatorname{tr}A)^{2}>4\det A$: a node. $(\operatorname{tr}A)^{2}<4\det A$: a spiral, or a centre when $\operatorname{tr}A=0$. The sign of the trace says stable or unstable. For a solution, find the eigenvectors and fit the constants to $\mathbf{x}(0)$.

• Real, different eigenvalues: $\mathbf{x}=c_{1}e^{\lambda_{1}t}\mathbf{v}_{1}+c_{2}e^{\lambda_{2}t}\mathbf{v}_{2}$.
• A complex pair $\alpha\pm i\beta$: one turn every $\tfrac{2\pi}{\beta}$, winding in when $\alpha<0$.
• A repeated eigenvalue with one eigenvector: a second solution $e^{\lambda t}(t\mathbf{v}+\mathbf{w})$.
• Stable: nearby solutions stay near. Asymptotically stable: they also arrive.