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Chaos on Sequence Space

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, dX ) 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 dn ∈ 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 10mx - 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[Fk (x), Fk (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] < 1/2n 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

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