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 = 1.7. More chaos. Note the horizontal blue bands. This signifies that the iterates have landed near the (unstable) fixed point, but of course eventually wander away from it. The green-brown-black 3-cycle pattern also appears. Click the picture to reducee. s = 1.8. The chaos is spread over a larger range, but some short-lived cycles still are visible. 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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