##### Theorem3.49

If a \(2 \times 2\) matrix \(A\) has eigenvalues \(\lambda_1\) and \(\lambda_2\text{,}\) then the trace of \(A\) is \(\lambda_1 + \lambda_2\) and \(\det(A) = \lambda_1 \lambda_2\text{.}\)

The key to solving the system
\begin{equation*}
\begin{pmatrix}
x' \\ y'
\end{pmatrix}
=
\begin{pmatrix}
a & b \\
c & d
\end{pmatrix}
\begin{pmatrix}
x \\ y
\end{pmatrix}
=
A
\begin{pmatrix}
x \\ y
\end{pmatrix}
\end{equation*}
is determining the eigenvalues of \(A\text{.}\) To find this eigenvalues, we need to derive the characteristic polynomial of \(A\text{,}\)
\begin{equation*}
\det(A - \lambda I)
=
\det
\begin{pmatrix}
a - \lambda & b \\
c & d - \lambda
\end{pmatrix}
=
\lambda^2 - (a + d) \lambda + (ad - bc).
\end{equation*}
Of course, \(D = \det(A) = ad -bc\) is the determinant of \(A\text{.}\) The quantity \(T = a + d\) is the sum of the diagonal elements of the matrix \(A\text{.}\) We call this quantity the *trace* of \(A\) and write \(\trace(A)\text{.}\) Thus, we can rewrite the characteristic polynomial as
\begin{equation*}
\det(A - \lambda I) = \lambda^2 - T \lambda + D.
\end{equation*}
We can use the trace and determinant to establish the nature of a solution to a linear system.

If a \(2 \times 2\) matrix \(A\) has eigenvalues \(\lambda_1\) and \(\lambda_2\text{,}\) then the trace of \(A\) is \(\lambda_1 + \lambda_2\) and \(\det(A) = \lambda_1 \lambda_2\text{.}\)

Theorem 3.49 tells us that we can determine the determinant and trace of a \(2 \times 2\) matrix from its eigenvalues. Thus, we should be able to determine the phase portrait of a system \({\mathbf x}' = A {\mathbf x}\) by simply examining the trace and determinant of \(A\text{.}\) Since the eigenvalues of \(A\) are given by \begin{equation*} \lambda = \frac{T \pm \sqrt{T^2 - 4D}}{2}, \end{equation*} we can immediately see that the expression \(T^2 - 4D\) determines the nature of the eigenvalues of \(A\text{.}\)

- If \(T^2 - 4D > 0\text{,}\) we have two distinct real eigenvalues.
- If \(T^2 - 4D \lt 0\text{,}\) we have two complex eigenvalues, and these eigenvalues are complex conjugates.
- If \(T^2 - 4D = 0\text{,}\) we have repeated eigenvalues.

If \(T^2 - 4D = 0\) or equivalently if \(D = T^2/4\text{,}\) we have repeated eigenvalues. In fact, we can represent those systems with repeated eigenvalues by graphing the parabola \(D= T^2/4\) on the \(TD\)-plane or *trace-determinant plane* (Figure 3.50). Therefore, points on the parabola correspond to systems with repeated eigenvalues, points above the parabola (\(D \gt T^2/4\) or equivalently \(T^2 - 4D \lt 0\)) correspond to systems with complex eigenvalues, and points below the parabola (\(D \lt T^2/4\) or equivalently \(T^2 - 4D \gt 0\)) correspond to systems with real eigenvalues.

The trace and determinant of a \(2 \times 2\) matrix are invariant under a change of coordinates. That is, \(\det(T^{-1} A T) = \det(A)\) and \(\trace(T^{-1} A T) = \trace(A)\) for any \(2 \times 2\) matrix \(A\) and any invertible \(2 \times 2\) matrix \(T\text{.}\)

Furthermore, each of the expression \(T^2 - 4D\) is not affected by a change of coordinates by Theorem 3.51. That is, we only need to consider systems \({\mathbf x}' = A {\mathbf x}\text{,}\) where \(A\) is one of the following matrices: \begin{equation*} \begin{pmatrix} \alpha & \beta \\ -\beta & \alpha \end{pmatrix}, \begin{pmatrix} \lambda & 0 \\ 0 & \mu \end{pmatrix}, \begin{pmatrix} \lambda & 0 \\ 0 & \lambda \end{pmatrix}, \begin{pmatrix} \lambda & 1 \\ 0 & \lambda \end{pmatrix}. \end{equation*}

The system
\begin{equation*}
{\mathbf x}' = \begin{pmatrix} \alpha & \beta \\ - \beta & \alpha \end{pmatrix} {\mathbf x}
\end{equation*}
has eigenvalues \(\lambda = \alpha \pm i \beta\text{.}\) The general solution to this system is
\begin{equation*}
{\mathbf x}(t)
=
c_1
e^{\alpha t}
\begin{pmatrix}
\cos \beta t \\ - \sin \beta t
\end{pmatrix}
+
c_2 e^{\alpha t}
\begin{pmatrix}
\sin \beta t \\ \cos \beta t
\end{pmatrix}.
\end{equation*}
The \(e^{\alpha t}\) factor tells us that the solutions either spiral into the origin if \(\alpha \lt 0\text{,}\) spiral out to infinity if \(\alpha \gt 0\text{,}\) or stay in a closed orbit if \(\alpha = 0\text{.}\) The equilibrium points are *spiral sinks* and *spiral sources*, or *centers*, respectively.

The eigenvalues of \(A\) are given by \begin{equation*} \lambda = \frac{T \pm \sqrt{T^2 - 4D}}{2}. \end{equation*} If \(T^2 - 4D \lt 0\text{,}\) then we have a complex eigenvalues, and the type of equilibrium point depends on the real part of the eigenvalue. The sign of the real part is determined solely by \(T\text{.}\) If \(T \gt 0\) we have a source. If \(T \lt 0\text{,}\) we have a sink. If \(T = 0\text{,}\) we have a center. See Figure 3.52.

The situation for distinct real eigenvalues is a bit more complicated. Suppose that we have a system \begin{equation*} {\mathbf x}' = \begin{pmatrix} \lambda & 0 \\ 0 & \mu \end{pmatrix} {\mathbf x} \end{equation*} with distinct eigenvalues \(\lambda\) and \(\mu\text{.}\) We will have three cases to consider if none of our eigenvalues are zero:

- Both eigenvalues are positive (source).
- Both eigenvalues are negative (sink).
- One eigenvalue is negative and the other is positive (saddle).

Our two eigenvalues are given by \begin{equation*} \lambda = \frac{T \pm \sqrt{T^2 - 4D}}{2}. \end{equation*}

If \(T \gt 0\text{,}\) then the eigenvalue
\begin{equation*}
\frac{T + \sqrt{T^2 - 4D}}{2}
\end{equation*}
is positive and we need only determine the sign of the second eigenvalue
\begin{equation*}
\frac{T - \sqrt{T^2 - 4D}}{2}
\end{equation*}
If \(D \lt 0\text{,}\) we have one positive and one zero eigenvalue. That is, we have a *saddle* if \(T \gt 0\) and \(D \lt 0\text{.}\)

If \(D \gt 0\text{,}\) then
\begin{equation*}
T^2 - 4D \lt T^2.
\end{equation*}
Since we are considering the case \(T \gt 0\text{,}\) we have
\begin{equation*}
\sqrt{T^2 - 4D} \lt T
\end{equation*}
and the value of the second eigenvalue \((T - \sqrt{T^2 - 4D}\,)/2\) is postive. Therefore, any point in the first quadrant below the parabola corresponds to a system with two positive eigenvalues and must correspond to a *nodal source*.

One the other hand, suppose that \(T \lt 0\text{.}\) Then the eigenvalue \((T - \sqrt{T^2 - 4D}\,)/2\) is always negative, and we need to determine if other eigenvalue is positive or negative. If \(D \lt 0\text{,}\) then
\begin{equation*}
T^2 - 4D \gt T^2
\end{equation*}
and
\begin{equation*}
\sqrt{T^2 - 4D} \gt T
\end{equation*}
Therefore, the other eigenvalue \((T - \sqrt{T^2 - 4D}\,)/2\) is positive, telling us that any point in the fourth quadrant must correspond to a saddle. If \(D \gt 0\text{,}\) then \(\sqrt{T^2 - 4D} \lt T\) and the second eigenvalue is negative. In this case, we will have a *nodal sink*. We summarize our findings in Figure 3.53.

For repeated eigenvalues, the analysis depends only on \(T\text{.}\) Since \begin{equation*} T^2 - 4D = 0, \end{equation*} the only eigenvalue is \(T/2\text{.}\) Thus, we have sources if \(T > 0\) and sinks if \(T \lt 0\) (Figure 3.54).

The trace-determinant plane is an example of a *parameter plane*. We can adjust the entries of a matrix \(A\) and, thus, change the value of the trace and the determinant.

Consider the system \begin{equation*} \begin{pmatrix} x' \\ y' \end{pmatrix} = A \mathbf x \begin{pmatrix} -2 & a \\ -2 & 0 \end{pmatrix} {\mathbf x}. \end{equation*} The trace of \(A\) is always \(T = -2\text{,}\) but \(D = \det(A) = 2a\text{.}\) We are on the parabola if \begin{equation*} T^2 - 4D = 4 - 8a = 0 \qquad \text{or}\qquad a = \frac{1}{2}. \end{equation*} Thus, a bifurcation occurs at \(a = 1/2\text{.}\) If \(a \gt 1/2\text{,}\) we have a spiral sink. If \(a \lt 1/2\text{,}\) we have a sink with real eigenvalues. Further more, if \(a \lt 0\text{,}\) our sink becomes a saddle (Figure 3.56).

Recall that a harmonic oscillator can be modeled by the second-order equation \begin{equation*} m \frac{d^2 x}{dt^2} + b \frac{dx}{dt} + k x = 0, \end{equation*} where \(m > 0\) is the mass, \(b \geq 0\) is the damping coefficient, and \(k \gt 0\) is the spring constant. If we rewrite this equation as a first-order system, we have \begin{equation*} {\mathbf x}' = \begin{pmatrix} 0 & 1 \\ -k/m & - b/m \end{pmatrix} {\mathbf x}. \end{equation*} Thus, for the harmonic oscillator \(T = -b/m\) and \(D= k/m\text{.}\) If we use the trace-determinant plane to analyze the harmonic oscillator, we need only concern ourselves with the second quadrant (Figure 3.57).

If \((T, D) = (-b/m, k/m)\) lies above the parabola, we have an underdamped oscillator. If \((T, D) = (-b/m, k/m)\) lies below the parabola, we have an overdamped oscillator. If \((T, D) = (-b/m, k/m)\) lies on the parabola, we have a critically damped oscillator. If \(b = 0\text{,}\) we have an undamped oscillator.

Now let us see what happens to our harmonic oscillator when we fix \(m = 1\) and \(k = 3\) and let the damping \(b\) vary between zero and infinity. We can rewrite our system as \begin{align*} \frac{dx}{dt} & = y\\ \frac{dy}{dt} & = - 3x - by. \end{align*} Thus, \(T = -b\) and \(D = 3\text{.}\) We can see how the phase portrait varies with the parameter \(b\) in Figure 3.59.

The line \(D = 3\) in the trace-determinant plane crosses the repeated eigenvalue parabola, \(D = T^2/4\) if \(b^2 = 12\) or when \(b = 2 \sqrt{3}\text{.}\) If \(b = 0\text{,}\) we have purely imaginary eigenvalues. This is the undamped harmonic oscillator. If \(0 \lt b \lt 2 \sqrt{3}\text{,}\) the eigenvalues are complex with a nonzero real part—the underdamped case. If \(b = 2 \sqrt{3}\text{,}\) the eigenvalues are negative and repeated—the critically damped case. Finally, if \(b \gt 2 \sqrt{3}\text{,}\) we have the overdamped case. In this case, the eigenvalues are real, distinct, and negative.

Although the trace-determinant plane gives us a great deal of information about our system, we can not determine everything from this parameter plane. For example, the matrices \begin{equation*} A = \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix} \qquad\text{and}\qquad B = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix} \end{equation*} both have the same trace and determinant, but the solutions to \({\mathbf x}' = A {\mathbf x}\) wind around the origin in a clockwise direction while those of \({\mathbf x}' = B{\mathbf x}\) wind around in a counterclockwise direction.

- The characteristic polynomial of a \(2 \times 2\) matrix can be written as \begin{equation*} \lambda^2 - T \lambda + D, \end{equation*} where \(T = \trace(A)\) and \(D = \det(A)\text{.}\)
- If a \(2 \times 2\) matrix \(A\) has eigenvalues \(\lambda_1\) and \(\lambda_2\text{,}\) then \(\trace(A)\) is \(\lambda_1 + \lambda_2\) and \(\det(A) = \lambda_1 \lambda_2\text{.}\)
- The trace and determinant of a \(2 \times 2\) matrix are invariant under a change of coordinates.
- The trace-determinant plane is determined by the graph of the parabola \(D= T^2/4\) on the \(TD\)-plane. Points on the trace-determinant plane correspond to the trace and determinant of a linear system \({\mathbf x}' = A {\mathbf x}\text{.}\) Since the trace and the determinant of a matrix determine the eigenvalues of \(A\text{,}\) we can use the trace-determinant plane to parameterize the phase portraits of linear systems.
- The trace-determinant plane is useful for studying bifurcations.

Consider the one-parameter family of linear systems given by \begin{equation*} \begin{pmatrix} x' \\ y' \end{pmatrix} = \begin{pmatrix} a & \sqrt{2} + a/2 \\ \sqrt{2} - a/2 & 0 \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix}. \end{equation*}

- Sketch the path traced out by this family of linear systems in the trace-determinant plane as \(a\) varies.
- Discuss any bifurcations that occur along this path and compute the corresponding values of \(a\text{.}\)

Consider the two-parameter family of linear systems \begin{equation*} \begin{pmatrix} x' \\ y' \end{pmatrix} = \begin{pmatrix} a & b \\ b & a \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix}. \end{equation*} Identify all of the regions in the \(ab\)-plane where this system possesses a saddle, a sink, a spiral sink, and so on.