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So we've seen two examples of the butterfly effect
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or sensitive dependence on initial conditions.
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The first example was the logistic equation with r equals 4
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and the second example was the Lorenz equations, a differential equation.
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So, now let me define Sensitive Dependence on Initial Conditions (SDIC)
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little more carefully and then I’ll be in a position
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to define chaos more carefully.
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Ok, so the idea behind sensitive dependence on intial conditions is this
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we say, a dynamical system has sensitive dependence on initial conditions
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if very small differences in initial conditions,
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so we got two small orbits that start off very close together,
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that after a little while the difference between those initial conditions become large.
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And we saw that perhaps most clearly for the logistic equation
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where actually was plotting the difference between those two orbits.
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So there is a more formal definition and I don’t want to get bogged down in it too much.
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But on the other hand, some of you might like to see this formalism.
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And the notion of the butterfly effect is perhaps this central thing, or one of the central things,
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and maybe the most popularized idea from chaos.
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so it’s worth and I think seeing what the formal definition looks like.
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Ok, so here is the formal definition, I’ll state this and then try to translate this
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into something a little bit more understandable.
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Ok, so we have a function f, or more generally dynamical system
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and let x-naught and y-naught be two possible seeds, two different initial conditions.
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And so we would say that this dynamical system or function
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has sensitive dependence on initial conditions if the following conditions are met.
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There needs to be some number delta such that
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for any initial condition x-naught there is some other initial condition
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y-naught, that's within epsilon of delta, it's not,excuse me, that's within epsilon of x-naught.
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It's not more than epsilon away from this first initial condition.
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Moreover, this initial condition y-naught has a property that
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there is some number n so that after n iterates the orbit of y-naught is more
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than delta away from the orbit of x-naught.
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So in other words, xn the nth iterate of x minus
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yn the nth iterate of y is greater than this number delta.
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Ok, so this is certainly abstract and
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I think the best way to illustrate this is to imagine a game of sorts.
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So, here is how to think how I imagined thinking about this game.
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So, you’re going to give me a puzzle and you get to choose the following
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so you get to choose an initial condition x-naught
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And let’s say, you just choose whatever initial condition you want
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Let's think about the logistic equation, so it has to be a number between 0 and 1.
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So let's say you choose 0.7
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Ok, then you get to choose some error threshold, that’s delta.
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So this is a lower case Greek delta and again this has to be between 0 and 1.
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let’s say you choose 0.2
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and then the last thing you get to choose is the initial error epsilon,
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and let's say you choose 0.1
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Ok, so you make these choices of these three numbers for the logistic equation,
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and then the puzzle is I have to come up with the following
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so the puzzle the task for me is can I find a y-naught
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some other initial condition that is within 0.1 of 0.7
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that's within epsilon of the x-naught you chose.
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And this y-naught that I find has to have the following property
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there is some n number of iterates such that after that
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or add that iterate the nth iterate of xn is now more than 0.2 away from the nth iterate of yn
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So, the puzzle for me is I need to find a y-naught that has this property.
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Can I do it? I am pretty sure I can.
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I’ll use the program that we were using earlier on the web
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to see if I can find a y-naught that meets this criteria.
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Ok, so I am looking for a y-naught that's within 0.1 of 0.7
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So let's see, so I should have mentioned that this is for the logistic equation with r equals 4.0
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since we’re looking at the aperiodic chaotic case.
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and so for x-naught I choose 0.7
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well, I mean I type that in, but you chose that for me in this task you are given me.
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and I need to choose some other initial condition that's close to 0.7
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how close, it can’t be more than epsilon away, the epsilon you gave me is 0.1
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I’ll try 0.72 that meets the criteria.
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and then I am going to make the time series plots.
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and now here we have in green and in purple the two orbits.
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The orbits of x-naught equals 0.7
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and the orbit of the seed y-naught equals 0.72.
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And now I want to see if I can find some n such that
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the orbits at that n are more than 0.2 apart.
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so let’s see, well, we can see right here by the time
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we get to 5 the difference between the two,
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or the absolute value of the difference between the two looks to be about is 0.6.
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and if I go down further and I look in the table of numbers here on the right
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I can see indeed that by the fifth iterate the difference between
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or the absolute value of the difference between
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the two orbits is greater than the delta you gave me 0.2
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so yes I can find the y-naught within 0.1 of the initial condition 0.7
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such that after some amount of iterates it's more that
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the difference between the two orbits is greater than 0.2
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so the answer to this question is yes
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and in particular I can find that y-naught is 0.72 then
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so if y-naught, if I choose a y-naught of 0.72
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then I iterated my y-naught and my x-naught,
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they started off with an epsilon of each other
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and then after 5 iterates the difference between these two the absolute value
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is about 0.63 and 0.63 is the key thing is that this should say
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this is the larger than delta.
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so you said I had to make it larger than 0.2
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no problem in fact I got it a lot larger than 0.2
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Ok, so this is just one particular case,
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one particular choice of x-naught, delta, and epsilon
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so if a function has sensitive dependence on initial conditions,
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I have to be able to answer this question with yes,
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for whatever you choose for these three things.
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Again as long as you don’t do something crazy like choose a delta greater than 1
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So for talking about the logistics equation everything has to be between 0 and 1.
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So as long as you choose a reasonable, and not crazy numbers here
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for any reasonable choice of these three numbers
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I can always answer this question yes.
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So what this means is that any initial condition you choose
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and you can choose a really big error threshold 0.9 0.99
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if you want it and you could choose a really small epsilon
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I can make even draw this so, here’s the initial condition x-naught
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and you choose that and then there’s going to be some little thing
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this will be x-naught plus epsilon x-naught minus epsilon
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so, somewhere in here I can find an initial condition y-naught
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such that after some amount of iterates
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these two points are arbitrarly far apart any delta you choose
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so if a function has some has sensitive dependence on initial conditions
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any initial condition very close to that initial condition
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is another initial condition which eventually gets very far away
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so that’s the formal definition for sensitive dependence on initial conditions
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here’s that long wordy thing
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it’s an awful run-on sentence but I didn’t really know another way to write it.
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This is a standard definition you can find it
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I’m sure on wikipedia, or in any text if you want it
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and it’s quite general we need to have a f just needs to be a function
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that map's space to itself the space needs to be a metric space
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because we have to have some notion of distance
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but that’s really the only requirement for being able to use this definition.
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so it applies to a whole lot of dynamical systems not just iterated functions.