The Logistic Equation

where the "<=" is self-explanatory. Let's call this the Logistic function. The parameter "a" (read as "alpha") can take on any real positive value, although the theory looks at the cases where 0 < a <= 4 as we will see.

We define the iterates of f as follows:

• First, fix a value of "a", say, a = 3.1
  Let x_o > 0, be a given number smaller than 1 (e.g., 0.21432).

• Define a new number x₁ by setting
  x₁ = a x_o( 1 - x_o)
  This means that x₁ = f(x_o).

• OK, now we have x₁, so we can define x₂ in a similar way, that is, we set
  x₂ = a x₁( 1 - x₁)
  (since we know x₁ and "a" is given). Now, in terms of f, this definition of x₂ says that x₂ = f(x₁). But we know that x₁ = f(x_o). It follows that x₂ = f(f(x_o)).

• We can define x₃ in the same fashion. Indeed, we can set x₃ = f(f(f(x_o))).
  Click and hold the box below for some examples...
  Examples with a = 3.1, xo = 0.21432 x1 = f(xo) = 0.5219995066 x2 = f(x1) = 0.7734996671 x3 = f(x2) = 0.5431135895

Continuing this process indefinitely we obtain the sequence of iterates of f, namely the set of points

also called the orbit of x_o under f.
Here are the first few terms of a typical orbit:

The basic question is:

The study of these questions and the answers we obtain lead one naturally to the theory of chaotic maps.

In these notes, a map will refer to a function defined on a closed and bounded interval whose values are within that same interval. So, for example, using basic techniques in freshman Calculus, one can derive this