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 = 1.5. The nonzero fixed point is stable, so why do the iterates run down toward (0,0)? Because here we started with x = 0.5, so the iterates decrease to the fixed point, which consequently lies in bin 1. Here we take the same value of s, but start from x = 0.01 so the iterates increase toward the fixed point. Several iterations are needed to get out of bin 1, a few are in bin 2, a few in bin 3, and the bulk in bin 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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