The Lorenz Attractor

Clear["Global'*"]

<< Programm`RungeKutta`

?RungeKutta

Lorenz Equation:
  
      x'  =  -16.0 (x-y)
      y'  =  - x z  + 45.92 x  - y 
      z'  =  x y - 4.0 z                                 ` = d/dt

\(\(RungeKutta[
      \ {\ \(-16.0\)\ \((x - y)\), 

        		\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 
        \(-x\)\ z\  + \ 45.92\ x\  - \ y, 

        		\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 
        x\ y\  - \ 4.0\ z\ }, 

      	\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {x, y, z}, 
      \ {0, 0, 0}, \ {50, 0.01}\ ]; \)\)

Show[Graphics3D[Line[%]],  Axes  ->  Automatic ];

\(\(NDSolve[
      \ {\(x'\)[t] == \(-16.0\) \((x[t] - y[t])\), 

        		\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 
        \(y'\)[t] == \(-x[t]\) z[t] + 45.92\ x[t]\  - \ y[t], 

        		\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 
        \(z'\)[t] == \ x[t]\ y[t]\  - 4.0\ z[t], 

        		\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ x[0] == \(-5\), \ 
        y[0] == \(-10\), \ z[0] == 45}, 

      	\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {x, y, z}, \ {t, 0, 40}, \ 
      MaxSteps -> 10000, 
	\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 
      Method\  -> \ RungeKutta\ ]\ \)\)

\(\(ParametricPlot3D[Evaluate[{x[t], y[t], z[t]} /. \ %], 

      	\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {t, 0, 40}, \ PlotPoints -> \ 4000]; \)\)

?ParametricPlot3D