So what does all this mean?
What is the importance of this?
What are the lessons to be learned
from chaotic dynamics?
Well I should say I don't think there is a simple
or single answer for this.
There isn't uniformed agreement on how important chaos is
or what is important about it.
And for sure I think there are different lessons
or implications for chaos in different area of science
and social science and in life.
So, I'm not going to have a simple conclusion here.
So instead let me make a few more comments
and give a couple more ways
of thinking things.
And I want to start inverting how I've been talking about things
so far which is to begin with an equation
and see what it can do
and of course that is not really how life works
we don't get handed equations
We get handed life and sometimes we turn life into data.
So let's imagine going in the other direction.
So suppose we had data.
Maybe this is data for population of rabbits on an island.
Or who knows it could be anything
But I'll talk about the rabbits.
And these are 40 years of data.
And we can see that the population settles
into two year cycle very quickly.
and that appears to be stable.
And maybe you know some year--the rabbits
something happens to the rabbits--
There are a bunch more or a bunch less
We would see it returning to this behavior.
And we could tell ourselves a very simple story.
Of why it is the way it is.
Some years when there are alot of rabbits
They eat the grass.
The next year there isn't so much grass
The rabbits are hungry, the population decreases.
The grass comes back and we cycle around.
Ok, so then imagine on another island,
a very similar island,
we also had some data results of observations
and it looked like this.
And so here, we see the rabbit population
jumping all over doesn't appear to be regular
It almost crashes, they almost all die off.
Grow fast, almost die off again.
And how might we, what might we respond to this
Well what would we do with this data.
Well one thing that would be natural to say,
hey who is messing with the rabbits?
We learned from this
that rabbits left alone will do their own thing
and stabilize and go into this nice cycle.
So if we saw some behavior like this
it would be logical to think
that somebody was messing with the rabbits
Something--there must be some external influence
crazy weather or random bands of gangs of poachers
or you know or who knows what.
Bad things to rabbits.
People who do bad things to rabbits are coming by.
Something is happening to them.
But what we have learned in this unit,
is that we can get random looking irregular behavior like this
without any external influence
In fact, the exact same process just changing
one little growth parameter by a pretty small amount
Can give to rise to both of these situations
so I think experimentally
One might look at these two and think
I have to think about these sort of phenomena in the world
very differently
That the world is made up of things that are orderly
and things that are random.
and those are separate.
They might get mixed together but they are separate things.
And I would need a different type of explanation for this
than I would for that.
and that is a reasonable thing
But chaos says, "No that is actually not true."
That you can get disorder from an orderly system
deterministic randomness.
That randomness and order
are not these completely separate things
that need to be that we need to think of in a completely
different way
that in a sense there are two sides to the same coin.
So, this suggests that randomness and order
that the relationship between them
is a subtle than we might have thought.
They are not some complete opposites
that we need to think completely differently.
Here is another way to think about this.
So in the prechaos pre-chatoic-dynamic view of things
We might be faced with some choices
of how we might think about the world
or how we think about a different phenomenon.
And one choice is the system follows rules
We have some rule based system
and the presumption and it's a logical one
would be that would be if something is following rules
it would be orderly.
Then the other choice we have
is that sytems behavior randomly
and when we see randomness we wouldn't explain it
by thinking about rules
well that's due to chance, stochasticity,
or some large amount of external influences.
So we would have these two choices
and these are very different sorts of ways of thinking about things
or explanatory framework.
So if we saw something like this
We would say "Ah-Ha' it must be following a rule
and that's why we see orderliness.
Because rules are orderly and rules give--
--orderly rules give rise to orderly behavior.
And if we saw this we would say,
"Ah-Ha!" The systems behaving randomly
its not settling down, it's not cycling
So, these rabbits whatever they are doing
They are not following a rule
It's due to chance
Maybe the rabbits behave randomly
Some inheriance stochasticy in the affairs of rabbits
or there is external influences that are acting irregularly
and at random that is making this happen
But what chaos and dynamical systems says
is that is not necessarily the case.
So, we could have a system follow rules.
But it doesn't have to be orderly.
It could then--we could then have a rule based system
that behaves randomly
so we might thinks rules imply order
we've seen that is not the case
logistic equation with r = 4 can produce randomness
So rather than this binary choice we have news ways
of explaining random or apparently random behavior.
We could have a system following a chaotic rule
or a chaotic function.
So I think this is an important realization
for I would think any area of science.
So if you see something like this
it doesn't automatically mean rulelessness or stochasticity.
It could but it doesn't have it.
There are other ways we can explain phenomenon like this.
So, what does chaos and the butterfly effect
have to say about this idea of Laplace and determinism.
Well I think that's a really difficult question
There are alot of competing ideas
and there certainly isn't universal agreement on this.
At one level, maybe one of the central premises
of Laplace determinism or at least this utonian world view
Persists and endures.
So, the dynamical system we have been studying
in this view the world is still one
in sense of cause and effect
The world is following rules.
Those rules however, don't lead to order or predictability.
That sort of twists of chaos and sensitive dependence
on initial conditions gives us.
In order to do long term prediction of chaotic systems
let along come close to Laplace's demon
one would need to know the initial condition
of whatever it is that we are studying
to a degree of percision that is not just impractical
but I think is impossible
So, I don't know if it spells the end of Laplace's Determinism
but it does require us to reconsider those notions alot.
Some people argue
that the phenomenon on sensitive dependence
on initial conditions sort of frees us from the shackles of
determinism and gives us a place
for free to to exist again.
Personally, I'm not so sure.
Maybe it frees us from the shackles from determinism sort of
But it unclear the butterfly effect,
which in a sense is unpredictable
shuffling already well shuffled decks of cards and extra
time to change your luck so
I don't know how the butterfly effect
leaves room for free will,
but I'm not sure.
I'm don't know how to think about free will as I said before.
So this brings us to the end of unit 3.
This was a long unit.
Longer than what I actually intended it to be.
But we covered alot of ground but we have learned
about the core phenomenon of chaos
and the key element of that
is sensitive dependence on initial conditions or the butterfly effect.
We will explore further and think more
about the consequences of this and other phenomenon
from dynamical systems in the next several units.
See next week in Unit 4.