Basin of attraction
If a fixed point p of f is attracting, then all points near p are attracted
toward p , in the sense that their iterates converge to p. the collection of all
points whose iterates converge to p is called the basin of attraction of p .
Definition
Suppose that p is a fixed point of f. Then the basin of attraction of p consists of all
x such that 𝑓 𝑛 (𝑥) → 𝑝 as → ∞ , and is denoted by 𝐵𝑝
i.e : 𝐵𝑝 = {𝑥 ∈ 𝑑𝑜𝑚(𝑓): 𝑓 𝑛 (𝑥) → 𝑝 𝑎𝑠 𝑛 → ∞}
Examples:
1- Find the basin of attraction 𝐵0 , 𝐵1 of the fixed points of 𝑓(𝑥) = 𝑥 2
Solution:
Since 𝑓(𝑥) = 𝑥 2 then 𝑓 2 (𝑥) = 𝑥 4 = 𝑥 2
2
𝑛
By induction 𝑓 𝑛 (𝑥) = 𝑥 2
𝑛
If |𝑥| < 1 then 𝑓 𝑛 (𝑥) = 𝑥 2 → 0 𝑎𝑠 𝑛 → ∞
So 𝑥 ∈ 𝐵0 that is 𝐵0 = (−1,1)
𝑛
By contrast if |𝑥| > 1 then 𝑓 𝑛 (𝑥) = 𝑥 2 → ∞ 𝑎𝑠 𝑛 → ∞ and 𝑓(1) =
1 , 𝑓(−1) = 1
then 𝐵1 = {−1,1}
2- Find the basin of attraction 𝐵0 , 𝐵1 , 𝐵−1 of the fixed points of 𝑓(𝑥) = 𝑥 3
3- Find the basin of attraction 𝐵0 of the fixed points of 𝑓(𝑥) = sin 𝑥
Remark
1- If p is an attracting fixed point then its basin of attraction interval.
2- If p is a repelling fixed point then its basin of attraction can consist of the
finite points.
3- If p is a saddle fixed point then its basin of attraction closed or semi open
interval.
Example :
let f(x)=2x the x=0 is the repelling fixed point 𝐵0 = {0}
Exercises:
1- Find the fixed points and determine whether each is attracting or repelling
or saddle:
a- 𝑓(𝑥) = 4𝑥(1 − 𝑥)
b- 𝑓(𝑥) = √𝑥
𝑥
c- 𝑓(𝑥) = 𝑥 3 −
d- 𝑓(𝑥) =
1
3
𝑥
e- 𝑓(𝑥) = 𝑒 𝑥−1
1
1
2- Let (𝑥) = 𝑥 2 + . Show that if |𝑥| > 𝑡ℎ𝑒𝑛 |𝑔𝑛 (𝑥)| → ∞ 𝑎𝑠 𝑛 → ∞
4
2
3- Let 𝑓(𝑥) = cos 𝑥:
a- Show that there is exactly one fixed point p , and that it is attracting
b- Find the basin of attraction of p.