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 = 1.2. Here the iterates of the tent map hop between two ranges of values, yet within these ranges the map is chaotic. We see structure superimposed on a disordered array of colors. Click the picture to enlarge. s = 1.32. Still two bands, but they are wider and the chaos within is more fully expressed. 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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