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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 (๐ฅ) → ๐ as → ∞ , and is denoted by ๐ต๐ i.e : ๐ต๐ = {๐ฅ ∈ ๐๐๐(๐): ๐ ๐ (๐ฅ) → ๐ ๐๐ ๐ → ∞} Examples: 1- Find the basin of attraction ๐ต0 , ๐ต1 of the fixed points of ๐(๐ฅ) = ๐ฅ 2 Solution: Since ๐(๐ฅ) = ๐ฅ 2 then ๐ 2 (๐ฅ) = ๐ฅ 4 = ๐ฅ 2 2 ๐ By induction ๐ ๐ (๐ฅ) = ๐ฅ 2 ๐ If |๐ฅ| < 1 then ๐ ๐ (๐ฅ) = ๐ฅ 2 → 0 ๐๐ ๐ → ∞ So ๐ฅ ∈ ๐ต0 that is ๐ต0 = (−1,1) ๐ By contrast if |๐ฅ| > 1 then ๐ ๐ (๐ฅ) = ๐ฅ 2 → ∞ ๐๐ ๐ → ∞ and ๐(1) = 1 , ๐(−1) = 1 then ๐ต1 = {−1,1} 2- Find the basin of attraction ๐ต0 , ๐ต1 , ๐ต−1 of the fixed points of ๐(๐ฅ) = ๐ฅ 3 3- Find the basin of attraction ๐ต0 of the fixed points of ๐(๐ฅ) = sin ๐ฅ 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 ๐ต0 = {0} Exercises: 1- Find the fixed points and determine whether each is attracting or repelling or saddle: a- ๐(๐ฅ) = 4๐ฅ(1 − ๐ฅ) b- ๐(๐ฅ) = √๐ฅ ๐ฅ c- ๐(๐ฅ) = ๐ฅ 3 − d- ๐(๐ฅ) = 1 3 ๐ฅ e- ๐(๐ฅ) = ๐ ๐ฅ−1 1 1 2- Let (๐ฅ) = ๐ฅ 2 + . Show that if |๐ฅ| > ๐กโ๐๐ |๐๐ (๐ฅ)| → ∞ ๐๐ ๐ → ∞ 4 2 3- Let ๐(๐ฅ) = cos ๐ฅ: a- Show that there is exactly one fixed point p , and that it is attracting b- Find the basin of attraction of p.