Deterministic Chaos

6.R. Kelly Plots of the Tent Map and the Logistic Map

Recall the Kelly plot method for representing data. We take the tent and logistic maps as the generators of our examples.

Here is the color scheme for the bins: left to right for lowest to highest.

Here are some examples with the logisitc and tent maps, for the indicated s.

Logistic 0.9, 0.99 1.5, 1.5 2.4, 2.9 3.1, 3.5 3.55, 3.57 3.6, 3.7 3.8, 3.825 3.826, 3.827 3.828, 3.829 3.846, 3.999 Tent 1.2, 1.32 1.4, 1.5 1.7, 1.8 1.9, 1.999

s = 3.6. This is chaos, and yet there are hints of ordered behavior. Click the picture to enlarge. s = 3.7. Even more chaos, but do you see some orderly arrangements of colors? Click the picture to reduce.

Here is an illustration of sensitivity to initial conditions by Kelly plots.

Here are some animations of how the Kelly plot varies with the s parameter of the logistic and tent maps.

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