---
tag: 0WW060JR6P
kind: definition
name: Nullcline
aliases: [zero-growth-isocline]
msc: [34C05]
defines: 02HR4C0B04
uses: [07BNRFQ0PJ, 0G4X36AE0X]
sources:
  - url: https://en.wikipedia.org/wiki/Nullcline
    locator: lead section
---

## 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|07BNRFQ0PJ]] $x_j' = f_j(x_1, \ldots, x_n)$ 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|0G4X36AE0X]] are the points that lie on all the nullclines.
The nullclines divide the [[phase plane|05VDGH129F]] 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.