6.1 Iteration of Maps

Map

$f : A \to B$ is a map if for every $a \in A$ , there exists a unique $b \in B$ such that $f (a) = b$ .

Iteration (of a Map)

Is an evolution in a subset $U \subset R^{n}$ consisting of

an initial state $x_{0} \in U$
a map $f : U \to U$ determining the iteration:

x_{n} = f (x_{n - 1}), \forall n \in N

The $m^{t h}$ iteration of the map $f$ by:

f^{(0)} (x) = f (x), f^{(m)} (x) = f (f^{(m - 1)} (x))

Linear Map

$f : R^{n} \to R^{n}$ is linear if $A \in R^{n \times n}$ such that

f (x) = A \cdot x

Fixed Points and Periodic Points

fixed point, if

x^{*} = f (x^{*})

m-periodic point, if for some $m \in N, m \geq 2$ such that

f^{(m)} (x^{*}) = x^{*}, and f^{(k)} (x^{*}) \neq x^{*}, \forall 1 \leq m

Invariant Sets

Let $f : Ω \subseteq R \to Ω \subseteq R$ be a map.
$D \subseteq Ω$ is invariant if $f (D) \subseteq D$

\forall x \in D, f (x) \in D

Stable/Unstable Sets

Let $x^{*} \in Ω$ be a fixed point under the iteration of $f .$
Stable:

W^{s} = {x \in Ω | lim_{n \to \infty} f^{(n)} = x^{*}}

Unstable:

W^{u} = {x \in Ω | lim_{n \to - \infty} f^{(n)} (x) = x^{*}}

Intersection of Invariant Sets

D = ⋂_{n = 0}^{\infty} f^{n} (D) = D \cap f (D) \cap f (f (D)) \cap \dots

Trapping Set

A trapping region is a closed connected invariant set $D$ such that

f (D) \subseteq \overset{\circ}{D}

Attracting Set

If $f : Ω \to Ω$ be a map and $D$ be a trapping region under $f$ . Then

A = ⋂_{n = 0}^{\infty} f^{(n)} (D) is the attracting set D under f

If $\exists x_{0} \in A$ such that the orbit $O (x_{0})$ is dense in $A$ , then $A$ is called an attractor.

Contraction Maps

A map is a contraction if:

d (f (x), f (y)) \leq λ d (x, y)

Contraction Principle

Let $I \subset R$ be a closed interval, possibly infinite on one or both sides, and $f : I \to I$ a $λ$ -contraction. Then $f$ has a unique fixed point $x_{0}$ and $| f^{n} (x) - x_{0} | \leq λ^{n} | x - x_{0} |$ for every $x \in R$ , that is, every orbit of $f$ converges to $x_{0}$ exponentially.