8 - Lorenz System

The goal of this program is to illustrate graphically the orbits of a three-dimensional system. For historical reasons we propose the system of Lorenz, with a=10,b=8/3 and r=28. Here are the three coordinate functions governing the system.

\(f1[x_, y_] := x + .01 \((\(-10\) x + 10  y)\); 


    f2[x_, y_, u_] := y + .01 \((\(-10\) x - x\ u - y)\); 
    


    f3[x_, y_, u_] := u + 0.01 \((x\ y - 8  u/3 - 304/3)\)\)

It is known that the system of Lorenz is of the form F(x)=x+hG(x) and there exist r,k>0 such that 
G(x)•x≤–k for every ||x||≥r (dissipative system).  Here is the ellipsoid defined by  G(x)•x=0.

a1 = 38/Sqrt[15]; a2 = 38/Sqrt[1.5]; a3 = 19;

p1 = 
      ParametricPlot3D[{a1 Cos[t1] Sin[t2], a2 Sin[t1] Sin[t2], 
          -19 + 19  Cos[t2]}, {t1, Pi/2, 13  Pi/12}, {t2, 0, Pi}]

Let us choose an initial condition

x = .1; y = .1; u = .1;

q = 
      Table[{v = x, w = y, x = f1[x, y], y = f2[v, y, u], u = f3[v, w, u]}, {
          1500}];

p = q[[Range[1, 1500], Range[3, 5]]];

p2 = Show[Graphics3D[{Line[p]}, Axes -> Automatic]];

p3 = 
      Graphics3D[{RGBColor[0, 0, 1], PointSize[0.03], Point[{0, 0, -38}], 
          Point[{Sqrt[72], Sqrt[72], -11}], 
          Point[{-Sqrt[72], -Sqrt[72], -11}]}];

Show[p1, p2, p3];

We see that the orbit goes in and out of the ellipsoid. The three fixed points of the system are on the surface of the ellispoid.