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 = 1.9. The diagram continues to unfold. More combinations of bins occur, but note that although square 11 is occupied (two successive iterates can fall in bin 1), square 44 is not occupied. s = 1.999. Fully developed chaos, yet with a clear pattern of forbidden combinations. These are some of the most effective situations in which to use driven IFS, to distinguish chaos from randomness.

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

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