Nullcline

Definition

Let $U$ be an open subset of $\mathbb{R}^n$, let $f = (f_1, \ldots, f_n) : U \to \mathbb{R}^n$ be a function, and consider the autonomous system $x_j' = f_j(x_1, \ldots, x_n)$ for $j = 1, \ldots, n$. For $j \in \{1, \ldots, n\}$, the $j$-th nullcline of the system is the set $$N_j = \{x \in U : f_j(x) = 0\}.$$

Remarks

On the $j$-th nullcline, the velocity has no component in the direction $x_j$. In the plane, trajectories cross the $x$-nullcline vertically and the $y$-nullcline horizontally. The equilibrium points are the points that lie on all the nullclines. The nullclines divide the phase plane into regions where each component of the velocity has a constant sign. For a planar linear system, the nullclines are lines through the origin, or the whole plane when a row of the matrix is zero.

Explanations

L1 human

A nullcline is the set of states where one of the variables stops changing for an instant. Drawing the nullclines is a quick first step to sketch a phase portrait.