Iterating f(x)=ax(1-x)


Introduction

Iterating the quadratic functions f(x)=ax(1-x) is a famous example of chaos. The coefficient a is a parameter that determines the specific function of this class. When x0 is defined, x1,x2,x3,.... is calculated by the following iteration.

x1=a x0 (1-x0)
x2=a x1 (1-x1)
x3=a x2 (1-x2)

Observe the behavior of xn when n gets bigger for various values of the coefficient a.

Problem

Let's try next Activity and observe with this applet.

(1)a=1.0 x0=0.1, 0.3
(2)When a=1.0 and 0<x0<1 , forecast xn.
(3)a=1.6 x0=0.1
(4)a=2.8 x0=0.1
(5)a=3.2 x0=0.1
(6)a=4 x0=0.1

Applet


How to use this applet

  • Press "+" button to increase a or x0.
  • Press "-" button to decrease a or x0.
  • Press "Define" button.
  • Press "Next" button.