More on Logistic Map

Here are several "slices" of the output vs. time from the orbit diagram. They show the output as it changes from iteration to iteration (time). The plot title is the corresponding r value, so you can compare how many different "lines" there are going across each plot (after about midway through the x-axis), to the number of points for each r value in the orbit diagram. It may be hard to see unless you enlarge these images, but you should see one, two, four, eight, numerous lines on the time series plots (points on the orbit diagram) for r=2.95,3.01,3.52,3.555,3.57, respectively.

Logistic Map

This is an orbit diagram for the logistic map. It shows how the period doubles for various r values along with windows where the period reduces. Approaching these windows, the output is aperiodic (chaotic)...I think I can say that and it's true. The growth parameter (x-axis), r ranged from 2.9 to 4 and is incremented by 0.001. For each r value, there are ~200 output values (y-axis). I only took outputs after the system reached steady state (eyeballing - half of total iterations). The initial input to the map is 0.5 (so the initial output is based on this...output = r*input*(1-input) ). Since this is a nonlinear map, the input generates an output which then becomes the new input, and repeat for many iterations. For some reason after r = 4, the system blows up after a few iterations...I'm not sure what to say about that right now.

Edit: The system blows up for r > 4 because input/output is bounded by 0 and 1. The map is a parabola with it's maximum at output = r/4...this assuming 0<=input<=1.