Orbit Diagram of the Logistic Map

magnification

3

Plot the points

(λ,

x

n

)

, where

x

0

=0

and the

x

n

are the iterates of the quadratic map

x↦

2

x

+λ

. Each

λ

leads to a distinct orbit; for example, for

λ=1.5

, the orbit is 0, -1.5, 0.75, -0.9375, …. For constant λ, the set of

x

n

is infinite, but in this picture only a finite sample is shown. The whole plot is called the orbit diagram for the map. The set of points on the right that looks like a curve consists of stable points. In the middle the points orbit between two values. Both of these cases are examples of attractors.