# pubdoc_12_15649_154

```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 ๐ ๐ (๐ฅ) → ๐ as → ∞ , and is denoted by ๐ต๐
i.e : ๐ต๐ = {๐ฅ ∈ ๐๐๐(๐): ๐ ๐ (๐ฅ) → ๐ ๐๐  ๐ → ∞}
Examples:
1- Find the basin of attraction ๐ต0 , ๐ต1 of the fixed points of ๐(๐ฅ) = ๐ฅ 2
Solution:
Since ๐(๐ฅ) = ๐ฅ 2 then ๐ 2 (๐ฅ) = ๐ฅ 4 = ๐ฅ 2
2
๐
By induction ๐ ๐ (๐ฅ) = ๐ฅ 2
๐
If |๐ฅ| &lt; 1 then ๐ ๐ (๐ฅ) = ๐ฅ 2 → 0 ๐๐  ๐ → ∞
So ๐ฅ ∈ ๐ต0 that is ๐ต0 = (−1,1)
๐
By contrast if |๐ฅ| &gt; 1 then ๐ ๐ (๐ฅ) = ๐ฅ 2 → ∞ ๐๐  ๐ → ∞ and ๐(1) =
1 , ๐(−1) = 1
then ๐ต1 = {−1,1}
2- Find the basin of attraction ๐ต0 , ๐ต1 , ๐ต−1 of the fixed points of ๐(๐ฅ) = ๐ฅ 3
3- Find the basin of attraction ๐ต0 of the fixed points of ๐(๐ฅ) = sin ๐ฅ
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 ๐ต0 = {0}
Exercises:
1- Find the fixed points and determine whether each is attracting or repelling
a- ๐(๐ฅ) = 4๐ฅ(1 − ๐ฅ)
b- ๐(๐ฅ) = √๐ฅ
๐ฅ
c- ๐(๐ฅ) = ๐ฅ 3 −
d- ๐(๐ฅ) =
1
3
๐ฅ
e- ๐(๐ฅ) = ๐ ๐ฅ−1
1
1
2- Let (๐ฅ) = ๐ฅ 2 + . Show that if |๐ฅ| &gt; ๐กโ๐๐ |๐๐ (๐ฅ)| → ∞ ๐๐  ๐ → ∞
4
2
3- Let ๐(๐ฅ) = cos ๐ฅ:
a- Show that there is exactly one fixed point p , and that it is attracting
b- Find the basin of attraction of p.
```