Deterministic Chaos

6.J. Bifurcation Diagrams for the

Tent Map and the Logistic Map

Recall the bifurcation diagram is a graph of s-values on the horizontal axis, and the eventual x-values for that s plotted on the vertical line through each s. That is, for each s, iterate x₁, ..., x₂₀₀ say, and plot x₁₀₀, ..., x₂₀₀. We are interested in the long-term behavior of the iterates, so usually we drop some initial values.

In dust in the tent we see most points iterate to -infinity for s > 2. Consequently, for the bifurcation diagram we restrict our attention to s in the range 0 <= s <= 2.

Here are some examples of tent map histograms.

Putting all this and many more histograms, we obtain the full bifurcation diagram for the tent map.

How much of this structure can be understood from the first few iterates?

Just as for the tent map, we can compute the bifurcation diagram of the logistic map.

We shall discover other structures in the logistic bifurcation diagram. But for now point out how, just as with the tent map, the first few iterates determine the main features of the logistic bifurcation diagram.

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