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Chaos on Sequence Space

Out line

  Chaos: Chapter 10

Aside: Decimal expansion of Rational Numbers

Definition of Chaos


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


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
expansion of a quotient of integers (rational number
from Q) if and only if the decimal expansion is eventially

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.



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


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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