We are going to see how we can use mathematical software to draw a phase portrait and solution curves. But you can also perform a qualitative analysis in which you just roughly outline in what direction a solution curve would go at a point in the phase plane.

As an example we consider the system $$\left\{\begin{aligned} \frac{\dd x}{\dd t} &= -2x+ y\\[0.25cm] \frac{\dd y}{\dd t} &= x -2 y\end{aligned}\right.$$ First you look at the $x$-nullclines and $y$-nullclines, that is, at the points in the $y$-$x$ plane where $\frac{\dd x}{\dd t}=0$ or $\frac{\dd y}{\dd t}=0$, respectively . This isoclines are straight lines: $$\frac{\dd x}{\dd t} =0\iff y=2x\qquad \text{and}\qquad \frac{\dd y}{\dd t}=0\iff y=\frac{1}{2}x$$ These straight lines separates the phase plane into areas in which solution curves go qualitatively in a different direction: for instance, in this example, curves move to the right on the left-hand side of the $x$-nullclines $(\frac{\dd x}{\dd t}>0)$ and curves move to the left on the right-hand side of the $x$-nullclines $(\frac{\dd x}{\dd t}<0)$; curves above the $y$-nullclines go downward $(\frac{\dd y}{\dd t}<0)$ and curves below the $y$-nullclines go upward $(\frac{\dd y}{\dd t}>0)$. This is what you see in the computer-generated phase portrait below, in which the red line forms the$x$-nullcline and the green line forms the $y$-nullcline.

In the diagram below, we have again drawn the nullclines, but no solution curves. Instead, we just indicated with arrows the directions in which a solution curve at a point would go in terms of change in $x$ and $y$. The resultant vectors would give the direction field.

Let us look in more detail at the above diagram with separate $x$ - and $y$ directions, and not at resultant vectors. Then we can distinguish four cases.

The four cases are:

1. $x$ and $y$ increase both
2. $x$ decreases and $y$ increases
3. $x$ increases and $y$ decreases
4. $x$ and $y$ decrease both

The first two cases I and II are separated by the $x$-nullcline and the cases I and III are separated by the $y$-nullcline. But the $x$-nullcline also separates cases III and IV. Similarly, the $y$-nullcline also separates cases II and IV.

In addition it is true that the direction on the line of the $x$-nullcline is vertical, and on the line of the $y$-nullcline is horizontal.

Finally we note that currently all directions in the diagram point toward the equilibrium $(0,0)$: so, all solutions go towards the origin. In other words $(0,0)$ is an attracting equilibrium in this example.