Phase Portraits of Linear Systems

Consider a $2 \times 2$ linear homogeneous system ${\bf x}' = A {\bf x}$. We think of this as describing the motion of a point in the $xy$ plane (which in this context is called the phase plane), with the independent variable $t$ as time. The path travelled by the point in a solution is called a trajectory of the system. A picture of the trajectories is called a phase portrait of the system. In the animated version of this page, you can see the moving points as well as the trajectories. But on paper, the best we can do is to use arrows to indicate the direction of motion.

In this section we study the qualitative features of the phase portraits, obtaining a classification of the different possibilities that can arise. One reason that this is important is because, as we will see shortly, it will be very useful in the study of nonlinear systems. The classification will not be quite complete, because we'll leave out the cases where 0 is an eigenvalue of $A$.

The first step in the classification is to find the characteristic polynomial, $\mbox{det}\,(A - r I)$, which will be a quadratic: we write it as $r^2 + p r + q$ where $p$ and $q$ are real numbers (assuming as usual that our matrix $A$ has real entries). The classification will depend mainly on $p$ and $q$, and we make a chart of the possibilities in the $pq$ plane.

Now we look at the discriminant of this quadratic, $p^2 - 4 q$. The sign of this determines what type of eigenvalues our matrix has:

Image stabil.gif

Each of these cases has subcases, depending on the signs (or in the complex case, the sign of the real part) of the eigenvalues. Note that $q$ is the product of the eigenvalues (since $r^2 + p r + q = (r - r_1)(r - r_2)$), so for $p^2 - 4 q > 0$ the sign of $q$ determines whether the eigenvalues have the same sign or opposite sign. We will ignore the possibility of $q=0$, as that would mean 0 is an eigenvalue.

The sum of the eigenvalues is $-p$, so if they have the same sign this is opposite to the sign of $p$. If the eigenvalues are complex, their real part is $-p/2$.

Another important tool for sketching the phase portrait is the following: an eigenvector ${\bf u}$ for a real eigenvalue $r$ corresponds to a solution $\displaystyle {\bf x}= {\rm e}^{r t} {\bf u}$ that is always on the ray from the origin in the direction of the eigenvector ${\bf u}$. The solution $\displaystyle {\bf x}= - {\rm e}^{rt} {\bf u}$ is on the ray in the opposite direction. If $r > 0$ the motion is outward, while if $r < 0$ it is inward. As $t \to -\infty$ (if $r > 0$) or $+\infty$ (if $r < 0$), these trajectories approach the origin, while as $t \to +\infty$ (if $r > 0$) or $-\infty$ (if $r < 0$) they go off to $\infty$. For complex eigenvalues, on the other hand, the eigenvector is not so useful.

In addition to a classification on the basis of what the curves look like, we will want to discuss the stability of the origin as an equilibrium point.

Here, then, is the classification of the phase portraits of $2 \times 2$ linear systems.





Robert Israel
2002-03-24