natural symmetry of the Lorenz equation

The Lorenz equationMathworldPlanetmath has a natural symmetry defined by

(x,y,z)(-x,-y,z). (1)

To verify that (1) is a symmetry of an ordinary differential equationMathworldPlanetmath (Lorenz equation) there must exist a 3×3 matrix which commutes with the differential equation. This can be easily verified by observing that the symmetry is associated with the matrix R defined as

R=[-1000-10001]. (2)


𝐱˙=f(𝐱)=[σ(y-x)x(τ-z)-yxy-βz] (3)

where f(𝐱) is the Lorenz equation and 𝐱T=(x,y,z). We proceed by showing that Rf(𝐱)=f(R𝐱). Looking at the left hand side

Rf(𝐱) = [-1000-10001][σ(y-x)x(τ-z)-yxy-βz]
= [σ(x-y)x(z-τ)+yxy-βz]

and now looking at the right hand side

f(R𝐱) = f([-1000-10001][xyz])
= f([-x-yz])
= [σ(x-y)x(z-τ)+yxy-βz].

Since the left hand side is equal to the right hand side then (1) is a symmetry of the Lorenz equation.

Title natural symmetry of the Lorenz equation
Canonical name NaturalSymmetryOfTheLorenzEquation
Date of creation 2013-03-22 13:44:12
Last modified on 2013-03-22 13:44:12
Owner Daume (40)
Last modified by Daume (40)
Numerical id 5
Author Daume (40)
Entry type Result
Classification msc 34-00
Classification msc 65P20
Classification msc 65P30
Classification msc 65P40
Classification msc 65P99