Sensitive dependence on initial conditions is also known as the butterfly effect.
You’ve probably heard this term before and know the story
but I should mention it briefly
so again the butterfly effect here is this illustration
we have two different initial conditions
this is for the x part of the Lorenz equations
two different initial conditions that differ by tiny tiny tiny amount
we can think of this is 15 meters and this is 15 meters plus 10 nanometers
imperceptible meaninglessly different
and here is the trajectory, the orbit of the two solutions
one we could think of as reality, one is prediction
and hereafter I don’t know about 18 minutes or 18 time steps
the two curves go apart
so a tiny change this is idea, like the effects of a butterfly flapping its wings
can make a large difference later on
so maybe speaking somewhat metaphorically
this is somehow the path of a hurricane
and in this version it hits New York city
and this version it hits North Carolina
so the difference, but maybe this is wind speed or something
20 hours ago or 20 days ago who knows
and so this tiny difference could be caused by the flapping of a butterfly’s wings
so very small changes in initial circumstances
can lead to very large changes later on
and this is what makes phenomena like this essentially unpredictable
because we can never measure something like this
in fact maybe values this accurate might not even be physically meaningful
James Gleick in his book Chaos has a nice way of thinking about
the butterfly effect in particular the influence or
the effect of butterfly flapping its wings
and he says the following
it’s like giving an extra shuffle to a deck of already well shuffled cards
you know that it will change your luck
but there is no way of knowing how it will change it
so that the sort of how I think about the butterfly here
the butterfly might flap its wings
and we know it’s going to change something in the future
but we don’t know how it will change it
Here is another way to think about the butterfly effect
in our lives we might be accustomed to the idea that
there were particular moments where a small change
in something that we did led to a really big change in the course of our life
so a simple example might be you're burning the toast
and so you are a little bit late leaving the house
and so you catch a train that’s one later than the thought,
than the train you thought you were going to take
but the train you had been planning on taking crashed
and a lot of people died
so ok, that’s a dramatic example
but we can perhaps think of a number of times in our life
where we were at sort of a key decision point and some very small thing
may be that we didn’t know about at the time made a big difference later on
It’s sort of like a ball perched on top of the hill
a tiny little change will determine whether or not
the ball rolls down to the right or to the left
So the idea that we have these sort of sudden decision points
is a fairly commonplace one.
But in a system that has sensitive dependence on initial conditions
it’s as if every moment is a decision point
at every moment the future trajectory is poised between possible different futures
and the difference between those is a tiny little flap of
gust of wind from a butterfly flapping its wings
that determines which path that will go on
so it’s this continuous sensitivity where every point along the journey
a tiny change can make a big change later on
and there is no way to know what those changes will be
finally before moving on to talk about chaos
I should mention that there is a really interesting history to this term the butterfly effect
And I strongly recommend there is a short article by Hilborn in American journal of physics
There is a link to it on the Complexity Explorer page
right down there underneath this video
it’s a three page non-technical piece,
if you’re interested at all the history of this term
it’s a great read I definitely recommend it
Ok, so let me conclude this subunit by defining chaos
We’re finally in a position where we can define the chaos in the mathematical sense
So a dynamical system we would say is chaotic
if it meets the following four criteria
First, the dynamical system has to be deterministic
The two types of dynamical systems we have studied so far
in which are the topic of this course
iterated functions and differential equations are most certainly deterministic
as I’ve stressed throughout the first two units
the dynamical system is just a deterministic rule
if you know the rule and you know the initial condition
that trajectory is unique, it’s determined
let me skip over two I’ll do these in
well , ok, so let me say this
The system’s orbits are bounded
meaning they can’t fly off to infinity
So for the logistic equation we’ve seen that orbits
that start between 0 and 1, remain between 0 and 1
for a system to be chaotic the orbits also have to be aperiodic
ie they never repeat they never follow the exact same path twice
it doesn’t go into a cycle
so if the orbits were not bounded
let’s say that we had just just the doubling function
then we could have an orbit that would be aperiodic
rigth if you double a number
it just keeps getting bigger and bigger and bigger forever
and it never repeats so that’s not so interesting
so this requirement that the orbits be bounded is sort of like fine print
that eliminates the possibility of orbits going off to infinity
so we have bounded orbits that are aperiodic
that are confined to stay in a unit interval
or a square or something and yet nevertheless never repeat
and then the last criteria for something to be chaotic
is that it has to have sensitive dependence on initial conditions
in the sense of we’ve just been talking about
so if a dynamical system has these four properties
then in physics or math would say it’s chaotic.
There’s lots more to say about the implications of sensitive dependence
on initial conditions and in general on chaotic dynamical systems
We’ll talk about this in the rest of this unit
and in many ways in the rest of the course.
But in the next set of lecture I’d like to explore this idea that
a dynamical system a deterministic dynamical system is capable of producing
random behavior and in order to dig into this idea
we’ll need to start by thinking about what even means to call something random
so this will be the topic of the next lecture