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 = 1.2. Here the iterates of the tent map hop between two ranges of values, yet within these ranges the map is chaotic. The driven IFS strcture is dominated by the alternate hopping between these two bands, the lower in bin 1, the higher in bin 4. s = 1.32. Still two bands, but they are wider and have points lying in bins 2 and 3, as well as 1 and 4.

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

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