Deterministic Chaos

6.Q. IFS Driven by the Tent Map and the Logistic Map

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

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, yet the differences between this picture and that of 3.57 are minute. Indeed, the chaos present at s = 3.6 is limited to a range of values ordered in groups close to the values of the cycles that occur for slightly smaller s. s = 3.7. Even more chaos, but some patterns are present. In particular, look at the points along the top of the unit square. From left to right, they have addresses 3infinity, 34(3infinity), 4(3infinity), and 434(3infinity)

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

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