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Section2.5Parametrized families of functions

Rather than explore the behavior of a single function at a time, we can introduce a parameter and explore the range of behavior that arises in a whole family of functions. Two important examples are

  1. The quadratic family: \(f_c(x)=x^2+c\)
  2. The logistic family: \(f_{\lambda}(x)=\lambda x(1-x)\)

The cobweb plots shown back in Figure 2.2.1 are all chosen from the logistic family with \(\lambda=2.8\text{,}\) \(\lambda=3.2\text{,}\) and \(\lambda=4\text{.}\) Even in those three pictures with graphs that look so very similar, we see three different types of behavior: an attractive fixed point, an attractive orbit of period two, and chaos (which can be given a very technical meaning.

Figure 1 shows some cobweb plots for the quadratic family of functions. Note that the behavior we see is very similar to the behavior we see for the logistic family - a fact that will become more understandable once we study conjugacy in section Section 6

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Figure2.5.1Some cobweb plots for the quadratic family