7.1 Chaos

Symbolic Dynamics

Shift Space The Bernoulli Shift Map

Shift Space

We denote the shift space by $Σ_{+}$ which consists of all infinite sequences of symbols

s = s_{1} s_{2} \dots s_{n} \dots

Bernoulli Shift Map

σ : Σ_{+} \to Σ_{+}

σ (s_{1} s_{2} \dots s_{n \dots}) = s_{2} s_{3} \dots s_{n} \dots

Metric in Shift Space

$δ = {\begin{cases} 0 & x = y \\ 1 & x \neq y \end{cases}$

d (s, t) = \sum_{i = 1}^{\infty} \frac{δ (s_{i}, t_{i})}{2^{i}}

$d (s, t) \geq 0$
$d (s, t) = d (t, s)$
$d (s, t) + d (t, q) \geq d (s, w)$

So $(Σ_{+}, d)$ is a metric space

Upper Bound of Distance

$s, t \in Σ +_{}$ where the first $p$ terms are identical

s_{1} = t_{1}, s_{2} = t_{2}, \dots, s_{p} = t_{p}

Then

d (s, t) \leq \frac{1}{2^{p}}

Proof:

d (s, t) = \sum_{i = 1}^{\infty} \frac{δ (s_{i}, t_{i})}{2^{i}} = \sum_{i = 1}^{p} \frac{δ (s_{i}, t_{i})}{2^{i}} + \sum_{i = p + 1}^{\infty} \frac{δ (s_{i}, t_{i})}{2^{i}} = \sum_{i = p + 1}^{\infty} \frac{δ (s_{i}, t_{i})}{2^{i}} \leq \sum_{i = p + 1}^{\infty} \frac{1}{2^{i}} = \frac{1}{2^{p}}

Periodic Orbits

If $s = s_{1} s_{2} \dots s_{k} s_{1} s_{2} \dots s_{k} s_{1} \dots$

\begin{aligned} σ (s) & = s_{2} \dots s_{k} s_{1} s_{2} \dots s_{k} s_{1} \dots \\ σ^{(2)} (s) & = s_{3} \dots s_{k} s_{1} s_{2} \dots s_{k} s_{1} \dots \\ \dots \\ σ^{(k)} (s) & = s_{1} s_{2} \dots s_{k} s_{1} s_{2} \dots = s \end{aligned}

Properties:

Countably many periodic orbits. For each fixed period $k \in N$ there are only finitely many $k$ -periods
The periodic points are dense in the shift space:

\forall ε > 0, \exists n \in N, s.t 2^{- n} < ε

Bound:
For any $s = s_{1} s_{2} s_{3} \dots s_{n} s_{n + 1} \in Σ_{+}$ construct $t = s_{1} s_{2} s_{3} \dots s_{n} s_{1} s_{2} s_{3} \dots s_{n} s_{1} s_{2} s_{3} \dots$ where $d (s, t) < 2^{- n} < ε$

Dense Orbit

What to prove there exists an $s^{*} \in Σ_{+}$ whose orbit

O_{σ} (s^{*}) = {s_{k} = σ^{(k)} (s^{*}), k \in N}

is a dense subset of $Σ_{+}$ .

We construct $s^{*}$ explicitly:

$s_{1} = 1$ , $s_{2} = 1$
$s_{3} s_{4} = 00, s_{5} s_{6} = 01, s_{7} s_{8} = 10, s_{9} s_{10} = 11$
$s_{11} s_{12} s_{13} = 000, s_{14} s_{15} s_{16} = 001, s_{17} s_{18} s_{19} = 010$ , $s_{20} s_{21} s_{22} = 011$ , $s_{23} s_{24} s_{25} = 100$ , $s_{26} s_{27} s_{28} = 101, s_{29} s_{30} s_{31} = 110, s_{32} s_{33} s_{34} = 111$
$\dots$
keeping concatenating all string of length $k$ consisting of 0 and 1

Take any $t \in Σ_{+}$ s.t $t = t_{1} t_{2} t_{3} \dots .$ for any $p \in N$ there exists a
$σ^{(m)} (s^{*}) = t_{1} t_{2} t_{3} \dots t_{p}$ so that for any $ε > 0$ there exists an $m$ such that:

d (s^{*}, t) < \frac{1}{2^{p}} < ε

Dynamic Behaviours and Chaos

Devaney's Definition of Chaos

Let $f : X \to X$ on a metric space $(X, d)$ . $X$ is chaotic under $f$ we have all of the following:

note:

It can be shown that the transitivity, under some mild assumption, is equivalent to the existence of a dense orbit, by the Birkhoff transitivity theorem.
It can be shown that the density of periodic points is a generic phenomena, according to the Kupka-Smale theory.

Transitivity

Let $f$ be a map from a metric space $(X, d)$ to itself.
$f$ is topologically transitive on a subset $A \subseteq X$ if:

\forall y \in A, \forall ε > 0 and n \in N such that d (f^{(n)} (x^{*}), y) < ε

There is an orbit $O_{f} (x^{*})$ , $x^{*} \in A$ that is dense in $A$ .

Alternatively for any $y \in A$ there exists a subsequence $n_{k}$ such that:

lim_{k \to \infty} d (f^{(n_{k})} (x^{*}, x)) = 0

Or for any pair of non-empty open subsets $U, V \subseteq X$ there exists an $n \geq 0$ such that:

f^{n} (U) \cap V \neq \emptyset

Birkhoff transitivity theorem:

Topological Transitivity ⟺ Existence of a Dense Orbit

dense orbit example

Density of a Periodic Orbit

Let $f$ be a map on a metric space $(X, d)$
Periodic points of $f$

P (f) = {x \in X : f^{(n)} (x) = x for some n \in N}

Density of Periodic Orbits if:

For every point $x \in X$ and every $ε > 0$ , $\exists$ $p$ such that $d (x, p) < ε$ .

Sensitive Dependence on Initial Condition

Let $f$ be a map on a metric space $(X, d)$ , there is sensitive dependence on initial conditions at point of $A \subseteq X$ if there

\exists ε > 0, so that \forall x_{0} \in A and \forall δ > 0

$y_{0} \in X$ such that

| y_{0} - x_{0} | < δ

and

d (f^{(k)} (y_{0}), f^{(k)} (x_{0})) > ε

So put otherwise $\forall x_{0} \in A, \forall N_{ε} (x_{0}), \exists y_{0} \in N_{ε} (x_{0})$ such that $d (f^{(k)} (y_{0}), f^{(k)} (x_{0}))$ grows larger than $ε$