11 - Plane Orbits

We woud like to study the orbits of the two-dimensional system governed by the maps defined below.

g1[x_, y_] := -2 x + y - 2 x y + 6; 
    g2[x_, y_] := x - 2  y + 2  x^2 - 3;

f1[x_, y_] := x + 0.3  g1[x, y]; f2[x_, y_] := y + 0.3 g2[x, y];

The system is dissipative, just like the Lorenz system (see Prog. 8), i.e. there exist r,k>0 such that G(x)•x≤–k for every ||x||≥r. The set G(x)•x=0 is an ellipse.

Expand[Apply[Plus, Flatten[{g1[x, y], g2[x, y]} *{x, y}]]]

The easiest way to plot the ellipse is to use the Implicit Plot command.

<<  Graphics`ImplicitPlot`

Clear[x]; Clear[y]

q0 = 
      ImplicitPlot[-2 x^2  -  2 y^2 + 2  x y + 6  x - 3  y  ==  0, 
         {x,  -1,  4}];

The system has the fixed point (1.5,1.5). As expected the point belong to the ellipse.

q1 = Graphics[{PointSize[0.02], Point[{1.5, 1.5}]}];

Show[q0, q1];

We now study the orbit starting from the point x=1, y=1. We are looking at the long range behavior of this orbit.

x = 1; y = 1;

q2 = Table[{u = x, y, x = f1[x, y], y = f2[u, y]}, {1000}];

q3 = q2[[Range[800, 1000], Range[1, 2]]];

q4 = ListPlot[q3, PlotStyle -> PointSize[0.01]];

Show[q0, q1, q4];