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 = 1.5. The nonzero fixed point is stable, so why do the iterates run down toward black? Because here we started with x = 0.5, so the iterates decrease to the fixed point. Click the picture to reduce. Here we take the same value of s, but start from x = 0.01 so the iterates increase toward the fixed point. Click the picture to enlarge.

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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