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 = 3.846. This is a 12-cycle, the second period-doubling of the 3-cycle window. s = 3.999. Fully developed chaos. The picture should remind you of the IFS driven by the exclusions2→1, 3→1, 2→2, 3→2, 2→3, 3→3, 1→4, and 4→4 forbidden. In fact, these are the types of patterns driven IFS are better at detecting. For cycles, other methods are better.

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

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