**Out line **

** Chaos:
Chapter 10**

Definitions

Aside: Decimal expansion of Rational Numbers

Summary

**Definition of Chaos**

**Definition**
A dynamical system F is chaotic if
Periodic points of F are dense
F is transitive
F depends sensitively on initial
conditions |

We will define and then examine each of these conditions
and

verify they hold for the shift map on sequence space.

**Density of Periodic Points**

**Definition**
A subset D of a metric space, (X, d_{X} ) is dense if there is a point
of D in every open ball. |

An statement is equivalent to : For every point x∈X there
is a

sequence of points d_{n} ∈ D which converges to x.

**Proof of Density of Q**

Recall that a decimal number, i.e. real number in R is the

decimal expansion of a quotient of integers (rational number

from Q) if and only if the decimal expansion is eventially

periodic.

Proof: If the eventual period is m, then 10^{m}x - x is a finite

decimal (hence a quotient of integers) and so is
)

**Density of Rational Numbers**

Proof: Viewing the real numbers as (equivalence classes,

decimal expansions, we need to find a
rational number

which is ε > 0 close. Choose a positive integer N with
.

Truncating the decimal expansion of x after the N decimal

place (or repeating zeroes ) gives a finite decimal expansion

which is the decimal expansion of a rational number.

**Proof: Density of Periodic Points in ∑**

Proof: In sequence space, ∑, given a sequence s and the

radius, ε of an open ball at s, choose an integer n with
.

Take the initial n + 1 terms of the sequence s and define a

periodic sequence which begins with these n + 1 terms and

then repeats them periodically. By the proximity theorem
is

distance less than ε from s. Note also, by the proximity

theorem, converges to s.

**Transitivity**

**Definition**
A dynamical system F is transitive if for every pair of points x, y
and any ε > 0 there is a third point z with d[x, z] < ε and with
the orbit of z intersecting the ball of radius ε at y, that is, the
orbit of z passes at distance less than ε from y. |

A sequence s which embeds all finite sequences has an
orbit

which is dense. To construct such an s, enumerate all the finite

sequences of symbols , and concatenate:

**Sensitivity on Initial Conditions**

**Definition**
A dynamical system F depends sensitively on initial conditions
if there is a number β > 0 such that for any x and any ε > 0
there is a y ε-close to x, i.e. d[y, x] < ε, and an integer k such
that d[F^{k} (x), F^{k} (y)] > β. |

**Proof: Sensitivity of shift map **
**on Initial Conditions**

For the shift map , select β = 1. As usual, given a
sequence s

and ε > 0, choose an integer n with .

Suppose T ∈∑ satisfies d[s, t] < 1/2^{n} but t ≠ s. Then, by

proximity, for i = 0, 1, . . . n and there is
an integer k > 0

with . Then:

This completes the demonstration of sensitivity of shift
map

on initial conditions.

** Summary of Chaos **

A chaotic map possesses three properties :

Indecomposability: A transitive system cannot be

decomposed into two subsystems that do not interact.

Un predictability : Sensitive dependence on initial conditions

Regularity: Periodic points are dense