So we’ve seen that the logistic equation with r equals 4 shows the butterfly effect sensitive dependence on initial conditions two imperceptibly different starting points can lead a big differences in the orbit later on and we saw this by these of computers I used the computer to iterate a function or solve the differential equation and we saw these diverging trajectories so I argued that this makes these systems even though they’re deterministic essentially unpredictable because we need a possible accuracy in order to do long term where often even medium term prediction in addition to posing a challenge to our predictions of real processes it also posses a challenge or puzzle for how we think about the computer results themselves since computers don’t store numbers to infinite precision and that’s what we need in order to do an accurate long term prediction on the computer we need enormous precision in the starting point and in all the numbers along the way so given that computers make tiny round off errors due to how they represents decimals we might wonder can we trust the result of computer simulations at all. and the answer turns out to be yes and there is a nice result that illustrates this known as the shadowing lemma. so what I want to do in this video is present the basic idea behind the shadowing lemma partly so you feel little better about the computer results but also because it’s just a neat result and a fun way and another way to think about what’s sensitive dependence on initial conditions and chaotic dynamics mean. so here is the idea behind shadowing Let’s say we’re studying an iterated function like the logistic equation and we choose an initial condition and then we compute we use a computer to compute an orbit and we could plot it in a time series plot just like I was having the computer do and maybe it looks something like this so the problem is that the computer is making round off errors it doesn’t have infinite precision and so because of sensitive dependence on initial conditions this computed orbit is not actually the true orbit for the initial condition that we chose so initially I’ll drive in blue the true orbit with this initial condition I start here it would be close but then this is sensitive dependence on initial conditions the true orbit might depart from the computed orbit so I wanted the computer to tell me the orbit with this initial condition and what the computer told me is showing here in this black curve however this black curve the computed orbit is not the true orbit for this initial condition the computer makes small round off errors in how it is storing decimal numbers and so the computed orbit is not the same as a true orbit this is another manifestation on the butterfly effect so at this point we can ask if this black curve, this black time series is not the true orbit, what is it? does it have any meaning at all or is it just garbage just some sort of randomness and amazingly it turns out that this computed orbit does have some meaning it’s the true orbit for some other initial condition so let me draw, picture of what that might look like the risk of clouding this up ok, so the black time series is what we computed. the blue is the true orbit for this initial condition the red is some other true orbit so this black curve is indeed an orbit of the logistic equation or whatever it is we’re studying it just happens to not be the exact orbit or the true orbit for initial condition that I thought but it’s a true orbit for some other initial condition so I can still interpret this black time series as a trajectory of the logistic equation it’s not garbage, it’s not nonsense. it just happens to not be the exact trajectory I maybe thought I was initially studying so this says that numerical results or at least iterated maps like the logistic equation even though round off error precision in the computer finite precision in the computer and the butterfly effect means that this black computed orbit it’s not true a faithful representation of the orbit for this initial condition it’s still faithful representation of an orbit of the logistic equation so we would say that this computed orbit in black shadows this red orbit it might not be exactly this true orbit but it’s arbitrarily close to some true orbit and this result that this is true is know as the shadowing lemma and a lemma in mathematics is a result that’s used as an intermediate step to prove or demonstrate some other central or more important result in any event this is a pretty famous result and it’s known as the shadowing lemma and the phenomenon is shadowing shadowing is when a computed orbit which is in a sense wrong due to the butterfly effect and finite precision nevertheless shadows comes arbitrarily close to some other true orbit so shadowing is I think it’s a strange phenomena and it’s fun and interesting to think about Let me give an analogy to illustrate this idea behind shadowing so let’s say you asked me to draw a portrait of somebody and as you know because you’re seeing my draw I’m actually not very good at drawing at all so I try to draw a true accurate portrait of whomever you asked me to make a portrait of but since I’m not very good I make a little mistake around the eyes and I make a little mistake around the mouth and I make a little mistake around the nose I make little mistake actually kind of all the time because I’m just not very very good at drawing so the result is I hand you a portrait of that I’ve drawn and you would look at it and you’d say that looks nothing like the person you were supposed to be drawing and I would say, yeah you’re right sorry but in the shadowing lemma picture I’ve drawn yes what I’ve drawn is not an accurate picture of the person I was trying to draw or you wanted me to draw but nevertheless what I’ve drawn is an accurate portrait of somebody else so I haven’t drawn an accurate picture of your friend you wanted portrait of but of the six or seven billion people in the world I got this just very very close to right for somebody else so that’s the idea behind the shadowing lemma you ask the computer to not draw portrait but make time series of the particular initial condition and the computer makes little errors because it just has finite precision it’s certainly better at arithmetic than I am at drawing but it still has finite precision and so it hands you back not a portrait but a time series and this time series is not exactly what you wanted it’s not the true exact time series for the initial condition but it’s a it is true or arbitrarily close to true an arbitrarily close to true time series for some other initial condition so maybe you will be a little bit disappointed but nevertheless what the computer has given you does say something true about the dynamical system you’re studying it is arbitrarily close to a true orbit of the dynamical system I can’t prove the shadowing lemma in this course it’s a pretty technical result and we just don’t have the mathematical machinery to do it but let me say a few things that maybe will make it seem at least a little bit more plausible Ok, so let’s imagine that the dynamical system we’re interested in is not a deterministic iterated function but is a fair coin something that's random that has an element of chance in it. so that just every time you toss the coin with an equal probability it comes up heads or tails and then you asked me to say ok what are the next five outcomes going to be so that’s in a sense the orbit for this system and I might say oh I think it’s going to be heads heads tails tails tails and then the system ran and it turns out I was wrong not surprisingly it’s hard to predict perfectly random fair coin processes so anyway so I said heads heads tails tails tails and I’m wrong but then I could say yeah alright I’m wrong but I’m not totally wrong because if you wait long enough you’re going to see heads heads tails tails tails for sure because all possible outcomes of heads and tails are equally likely so yeah I told you a wrong result in this particular instance but it’s true in that you really will see heads heads tails tails tails so I’m not as wrong as you think I am ok so that’s not surprising or very deep statement about a random process like tossing coins what’s interesting is the same sort of argument holds for this deterministic system an iterated function so you asked me to predict the next five or the next fifth orbits and I and my calculator on the computer give you an answer and it turns out that that answer is wrong compared to the true orbit but I could say yeah ok sorry I got it wrong but I guarantee you that this if you wait long enough you would see this orbit so I’ve given you something that’s actually true so in a sense if a system is random or mixed up enough then errors made in trying to predict the system in this context can still produce results that could have been produced by the system and maybe another way to think about this is the things I’m calling errors are not errors in the sense that I forget the rules of arithmetic or think that you know I think that two plus two is five they’re just you know they are very small imprecisions and then it’s the butterfly effect that amplifies those imprecisions so in effect that we have these tiny errors but then the macroscopic manifestation of these errors is result of the dynamics of the system itself so in a sense that the errors or imprecisions I’m not even sure what word to use arise from the dynamics of the system not from some horrible blender or external source and so maybe it’s not surprising that the system induced errors or imprecisions are nevertheless in some way true to the system ok, so again this is certainly not a proof of the phenomena of shadowing but maybe these remarks help to make shadowing seem a little bit more plausible and as maybe you could tell in the last couple minutes there are different notions of randomness that might be appearing here is a process random if it’s made by a random process can a deterministic process have a random outcome so there’s some ideas that we want to unpack here and think more carefully about what one means when one says something is random so this will be the topic what is randomness and how do we think about it in dynamical systems of the next set of lectures.