# Lorentz system

% !TEX lualatex
\documentclass[tikz]{standalone}
\usepackage{luacode,pgfplots}
\begin{luacode*}
-- Differential equation of Lorenz attractor
function f(x,y,z)
local sigma = 3
local rho = 26.5
local beta = 1
return {sigma*(y-x),-x*z + rho*x - y,x*y - beta*z}
end

-- Code to write PGFplots data as coordinates
function print_LorAttrWithEulerMethod(h,npoints,option)
-- The initial point (x0,y0,z0)
local x0 = 0.0
local y0 = 1.0
local z0 = 0.0
-- add random number between -0.25 and 0.25
local x = x0 + (math.random()-0.5)/2
local y = y0 + (math.random()-0.5)/2
local z = z0 + (math.random()-0.5)/2
if option ~= [[]] then
tex.sprint("\\addplot3[" .. option .. "] coordinates{")
else
end
-- dismiss the first 100 points
-- to go into the attractor
for i=1, 100 do
m = f(x,y,z)
x = x + h * m[1]
y = y + h * m[2]
z = z + h * m[3]
end
for i=1, npoints do
m = f(x,y,z)
x = x + h * m[1]
y = y + h * m[2]
z = z + h * m[3]
tex.sprint("("..x..","..y..","..z..")")
end
tex.sprint("}")
end
\end{luacode*}
\directlua{print_LorAttrWithEulerMethod(#2,#3,[[#1]])}%
}
\pgfplotsset{width=.9\hsize}
\begin{document}
\begin{tikzpicture}
\begin{axis}
% SYNTAX: Solution of the Lorenz system
% with step h=0.02 sampled at 1000 points.
\end{document}