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Sensitive dependence on initial conditions
created by
Li Kao
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Li Kao
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Mon Dec 04 2000 at 0:54:38
Sensitive dependence on initial conditions (or SDIC as my
lazy
fingers prefer) is an
important
and often
misunderstood
concept
that is relevant to both
chaos theory
and
dynamical systems
. In
Golem
's well crafted
writeup
regarding
chaos theory
, the author ascribes properties to SDIC which are really more the
purview
of a feature known as
topological transitivity
. This is a very common
misunderstanding
(or
oversimplification
). I will first provide a technical definition for SDIC and will limit the discussion to the domain of real numbers.
Definition
A function
f
which maps a set
V
onto itself has
sensitive dependence on initial conditions
if there exists
d
> 0 such that, for any
x
which is an element of
V
and any neighborhood
N
of
x
, there exists
y
(which is a member of
N
) and
n
>= 0 such that |
f
n
(x) - f
n
(y)|
>
d
.
Now, some of you may be asking yourself, what the hell does all that mean? More or less, it means that if you start off with a number
x
there is a number
y
that can be arbitrarily close to
x
, such that we could iterate the function
f
some number of times for both values and have the difference (|
f
n
(x) - f
n
(y)|
) be arbitrarily large. Note that this does not mean that all values in the neighborhood of
x
will behave in this manner, just that there is at least one.
So, in other words, even if you
measure
something with an arbitrarily small but non-zero
error
, you have the possibility of having a value that could diverge
scad
s from what the "real" value would have done.
Using
f(x)=x
3
as an example, we see a function can exhibit SDIC without exhibiting the
wonky
behavior that characterizes a
chaotic
dynamical system
.
Throw in a dash of
topological transitivity
and that is when you start
getting crazy with the cheese whiz
. We have a
property
( SDIC ) which insures that you will move arbitrarily far from at least one
person
in your
neighborhood
(or they will move away from you, depending on who was using the
cheese whiz
). In
topological transitivity
we will find a property that insures that the
people
in your
neighborhood
will have, over time, moved through every other
neighborhood
.
Note: I have created this writeup with only the
symbols
that would display on my
browser
. Every code above 255 in the
html symbol reference
displays as '?' regardless of whether or not I use the numerical of alpha representation. Therefore, please have mercy on my
crappy
looking
symbols
. Once I figure this out, I will do the
writeup
for
topological transitivity
.
printable version
chaos
Chaos theory
A line allows progress, a circle does not
Great Attractor
Sensitivity to initial conditions
Butterfly effect
A case study in genetic ideation
Chaos
August 4, 1907
Feigenbaum number
dynamical system
Beck
An attempted topological eversion of self in regular homotopy
transitivity
Population growth model
bifurcation
Autobiographical Statement: Indexed by Rivers
Vinland the Dream
Topological Space
locally compact
Wilkinson's polynomial
Russell's paradox
homogeneous differential equation
Lorenz Attractor
language games
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