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.
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