---
tag: 05VDGH129F
kind: concept
name: Phase plane
msc: [34C05]
uses: [07BNRFQ0PJ, 0BVT6VSB3S, 0A6QZ367GX]
sources:
  - url: https://en.wikipedia.org/wiki/Phase_plane
    locator: lead section
  - url: https://en.wikipedia.org/wiki/Phase_plane
    locator: Example of a linear system
same_as:
  wikidata: Q2088081
  wikipedia: Phase plane
---

## Summary

The phase plane of a planar [[autonomous system|07BNRFQ0PJ]] $x' = f(x, y)$, $y' = g(x, y)$ is the plane with coordinates $(x, y)$, the two state variables.
Each solution traces a curve in this plane, its [[trajectory|0BVT6VSB3S]]. The vector $(f(x, y), g(x, y))$ is tangent to the trajectory through $(x, y)$.
It is the two-dimensional case of the phase space.

## Remarks

A second-order equation $x'' = h(x, x')$ gives a planar system with $y = x'$. Its phase plane has the position and the velocity as coordinates.
For a linear system $(x, y)' = A (x, y)$ with a $2 \times 2$ real matrix $A$, the eigenvalues of $A$ determine the shape of the trajectories. See the [[classification of planar linear systems|0N7ATVF2X8]].

## Explanations

### L1 human

The phase plane shows the two state variables at once, without the time axis.
Each point is a state, and the system moves from state to state along a curve.