a nullcline

When you want to sketch a phase portrait by hand, nullclines are the scaffolding that makes it manageable. A nullcline is a curve in the phase plane where ONE of the two variables momentarily stops changing — where its rate of change is zero. There are two of them: the x-nullcline, where x' = 0 (x is instantaneously frozen), and the y-nullcline, where y' = 0 (y is instantaneously frozen). They carve the plane into tidy regions.

Here is how they help. On the x-nullcline, since x' = 0, the velocity arrow has no horizontal part, so it points purely up or down (vertically). On the y-nullcline, since y' = 0, the arrow points purely left or right (horizontally). And where the two nullclines INTERSECT, both x' and y' are zero at once — which is exactly an equilibrium point. So drawing the nullclines simultaneously locates every equilibrium (the crossings) and tells you the arrow direction along key curves, while in each region between them the signs of x' and g fix whether trajectories drift up-right, down-left, and so on. You assemble the portrait region by region.

Nullclines are purely a sketching aid, not a deep object — but a tremendously practical one, because they reduce 'where does everything go?' to checking the sign of f and g in a handful of regions. They are especially valuable for nonlinear systems too messy to classify by formula. A caution: a nullcline is NOT a trajectory. A trajectory only touches a nullcline and crosses it (vertically or horizontally); it does not run along it, except in the special case where a nullcline happens to coincide with a straight-line solution.