---
tag: 081QSEAGEY
kind: concept
name: Phase line
msc: [34C05, 34A34]
uses: [07BNRFQ0PJ, 0G4X36AE0X]
sources:
  - url: https://en.wikipedia.org/wiki/Phase_line_(mathematics)
    locator: Diagram
  - url: https://en.wikipedia.org/wiki/Phase_line_(mathematics)
    locator: Classification of critical points
same_as:
  wikidata: Q7180953
  wikipedia: Phase line (mathematics)
---

## Summary

Let $f$ be a function on an interval of $\mathbb{R}$. The phase line of the scalar [[autonomous equation|07BNRFQ0PJ]] $x' = f(x)$ is a diagram of the line of states.
It marks the [[equilibrium points|0G4X36AE0X]], which are the zeros of $f$. It puts an arrow on each interval between them: up where $f > 0$ and down where $f < 0$.
The arrows show the direction in which each solution moves.

## Remarks

An equilibrium point with both arrows toward it is stable, or a sink. With both arrows away from it, it is unstable, or a source. With one arrow toward it and one away, it is semi-stable.
For $f$ locally Lipschitz, a solution stays in one interval between equilibrium points and is monotone there.
For the [[logistic equation|0CC4VC7KR3]] $x' = x(1 - x)$, the point $0$ is a source and the point $1$ is a sink.

## Explanations

### L1 human

The phase line is a one-dimensional map of where the state can go.
Arrows show whether the state increases or decreases, and the equilibrium points are the places where it stops.