Deterministic Chaos

6.C. Test Functions

To learn tests of chaos in experimental data, we first analyze the dynamics of iterates of some simple functions. In general, given a function f(x), we are interested in predicting the long-term behavior of the orbit

x₀, x₁ = f(x₀), x₂ = f(x₁) = f²(x₀), x₃ = f(x₂) = f³(x₀), ...

of the point x₀.

Can we predict any features of this orbit without computing the x_i?

How does the long-term behavior of the orbit depend on the choice of initial value x₀?

We shall study these and related questions for these test functions

The Tent Map is one of the simplest of nonlinear maps, its graph is just two straight lines.
The Logistic Map is a model for a population growing in an environment with limited resources. Amazingly, there are questions about these parabolas that cannot be answered.

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