Drawing curves in the trace-determinant plane Consider the spring-mass differential equation with $m = 1, k = 3$, and $0 \leq b < \infty$:
$$\dfrac{d^2y}{dt^2}+b\dfrac{dy}{dt}+3y = 0.$$
(a) Rewrite this one-parameter family of second-order linear scalar equations as a one-parameter family of first-order linear systems (with a matrix $A$ that depends on the parameter $b$).
(b) Draw the curve/line in the trace-determinant plane obtained by varying the parameter $b$ (as $b$ varies from $0$ to $\infty$). Identify where along this curve the system is underdamped, critically damped, and overdamped. Draw an arrow indicating the direction to travel along this curve/line as $b$ increases. Give
rough sketches of typical phase portraits of the underdamped and overdamped cases. Finally, find the value of $b$ where the system is critically damped (that is, find the bifurcation value of $b$).
Part (a) is quite straightforward and I was able to find the first-order linear systems very easily as well as the matrix $A$ that is $A=\begin{pmatrix}0&1\\-3&-b\end{pmatrix}$. However I'm confused about part (b) of the question. I found the trace to be $-b$ and the determinant to be $3$ but I don't know how to go about drawing the curve using that information.
 A: Your calculations seem to be correct; let us be more explicit about the parameter $b$ and write
$$A_b=\begin{pmatrix} 0&1\\-3&-b\end{pmatrix}.$$
To draw the curve on the trace-determinant plane, we need to interpret the trace as the horizontal axis and the determinant as the vertical axis. Since $\operatorname{tr}(A_b)=-b$ and $\det(A_b)=3$, this means that the curve on the trace-determinant plane that forms by varying $0\leq b<\infty$ has to be horizontal. At $b=0$, we are at the point $(0,-3)$ on the trace-determinant plane, and in general the point $A_b$ is represented by on the trace-determinant plane has coordinates $(-b,3)$. In particular, as $b$ increases the point moves (horizontally) to the left (the trace-determinant plane is from Hirsch, Smale and Devaney's Differential Equations, Dynamical Systems, and an Introduction to Chaos (3e, p.64)):

Here I drew the underdamped part in red, overdamped part in blue, and the critically damped point in green. Note that critical damping is exactly when $\dfrac{\operatorname{tr}(A_b)^2}{4}=\det(A_b)$, i.e. when $b= 2\sqrt{3}$.

As a further exercise it might be interesting to consider a general harmonic oscillator
$$my''+by'+ky=0$$
and see what the effect of fixing any two parameters and varying the third one is in terms of curves on the trace-determinant plane.
