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