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Section 2.6 Parametrized 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 basic examples that we'll spend some time with 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 Figure 2.3.1 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 Figure 2.6.1 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 2.9.

Figure 2.6.1 Some cobweb plots for the quadratic family

These pictures beg the question - how can we find values of \(c\) that generate these specific behaviors? What other types of behavior are possible? Given some desired behavior, how can we find functions that yield that behavior?

Our classification of fixed points Definition 2.5.1 is a nice algebraic tool that will help, particularly when combined with appropriate geometric tools.