I have heard many times about this system (
http://en.wikipedia.org/wiki/Lorenz_system#Equations) and its properties, attractors and stuff, recently I have had an opportunity to play with it a little, using Runge-Kutta's method of the 4th order(
http://en.wikipedia.org/wiki/Runge-Kutta) and tangent linear solution of the linearized model(This is the description of the numerical solution of the linearized model: As soon as we calculate dx/dt, we use x + dx to calculate dy/dt, and then x + dx and y + dy to calculate dz/dt). The curves in red below are calculated using this method.
a) dr0 = (0.1,0.1,0.1)
dt = 0.01 s
the same thing but with dt = dt / 10 = 0.001 s
b) dr0 = (10.0,10.0,10.0)
c) Just for the sake of having a beautiful picture here, 100 seconds of simulation of the system, give the following image:
Sensitivity to the initial conditions(2 solutions were obtained using RK4):
