12 - Intermittent Chaos

We will give an example of a dynamical system which displays intermittent chaos. For a while the system seems to settle down to a periodic orbit, then this behavior changes to a chaotic pattern, which in turn is again replaced by the same periodic orbit.

f[a_, x_] := a x (1 - x)

We first take a look at a strictly periodic orbit of period 2

a = 3.2;

x = .3; n = 0;

q1 = Table[{n = n + 1, x = f[a, x]}, {200}];

q2 = q1[[Range[1, 60], Range[1, 2]]];

q3 = ListPlot[q2, PlotStyle -> PointSize[0.02]];

q4 = ListPlot[q2, PlotJoined -> True];

Show[q3, q4]

We see that the periodic behavior emerges almost immediately. Let us now change the parameter of the system

a = 3.82842;

x = .3; n = 0;

p1 = Table[{n = n + 1, x = f[a, x]}, {6000}];

We examine the behavior of the orbit for the first 400 iterates

p2 = p1[[Range[1, 400], Range[1, 2]]];

ListPlot[p2, PlotStyle -> PointSize[0.02]]

The we look at their behavior for the iterates from 2500 to 3000.

p3 = p1[[Range[2500, 3000], Range[1, 2]]];

ListPlot[p3, PlotStyle -> PointSize[0.02]]

Finally, we analize the behavior for the iterates from  5500 to 6000. We see that after long periods of stability the coordinates behave in a chaotic manner.

p4 = p1[[Range[5500, 6000], Range[1, 2]]];

ListPlot[p4, PlotStyle -> PointSize[0.02]]