The Lorenz attractor is the attractor of the dynamical system defined by the following system of differential equations.
sigma = 3; rho = 26.5; beta = 1;
x0 = 0; y0 = 1; z0 = 1;
lorenzEqs = {x'[t] == sigma (y[t] - x[t]),
y'[t] == rho x[t] - x[t] z[t] - y[t],
z'[t] == x[t] y[t] - beta z[t],
x[0] == 0, y[0] == 1, z[0] == 1};
Although, there is no closed form solution for this system, Mathematica can
approximate a solution with NDSolve and then plot the
resulting points as a function of t. Here is the necessary code:
tmin = 100; duration = 100;
sol = NDSolve[lorenzEqs, {x[t], y[t], z[t]}, {t,tmin,tmin+duration},
MaxSteps -> 20000];
ParametricPlot3D[{x[t], y[t], z[t],
RGBColor[(t-tmin)/duration,0,1-(t-tmin)/duration]} /. sol // Evaluate,
{t,tmin,tmin+duration}, PlotPoints -> 1000, Boxed -> False,
Axes -> False, ViewPoint -> {4.5, 2.8, 14}]
Check out
Patrick Warfolk's Lorenz attractor applet for a more dynamical
view.