Lec 11 MIT 18.085 Computational Science and Engineering I

missed out on one that's quite fun uh and uh it leads to uh these fantastic pictures of fractals you may have seen I intended to bring in a book by pipin that has uh sort of the beauty of fractals is his title has pictures of the mandal Bro Set uh just amazing um and just it's interesting mathematics and it's mathematics of recent years so I'm going to put off the wave equation and heat equation to the second lecture today lecture 12 and discuss this one because it's sort of it's so here's the general plan you have some function and you start at some point u0 and you compute F of u0 and that gives the next point then you compute F of U1 and that gives U2 and then you keep going so so u n is so to speak F of f of f of it's where're we're uh we're taking well the mathematical word is composition so so let me just put up U2 so you see U2 U2 is f of f of U uo because that's U1 so we're we're repeating the same operation F and it's simply amazing what you get and and it's a topic that uh was not very much studied until we could actually execute it so I guess what I'm hoping this is a sort of partly a fun lecture uh but you could experiment with some uh FS they don't have to be very complicated here's a linear one and and we'll quickly see all the possibilities there but it's really nonlinear functions F you see if suppose f is a quadratic which is you you might think okay quadratics can't be too hard but the best Minds in mathematics have gone in recent years to understanding what if f is a quadratic so it's a so we have a quadratic function of un not then when I do another one it's a fourth deegree function because it's a quadratic of a quadratic we we're squaring squares just already at u3 I'm up to eth degree and and and uh just you need new ideas to to get anywhere with it and uh just to be sure I mention it uh the one famous picture associated with that uh um with the quadratic case is this one that's on the back of the book so we'll make some connection to that picture on the back of the book which is uh one of the that's not the mandal Bro set that's the picture of period doubling and Chaos stuff that people sort of it's like a third world that had been um uh invisible for years people imagine that uh things either converged to some answer or they diverged things blew up but what so hadn't been seen was the possibility that they bounce around in a in a in a chaot if I can use that word in a chaotic way a way that's very hard to uh give any formula for but uh nevertheless you can understand something about the about the average Behavior or the or the the the limiting points uh what can happen uh it it's this if I mention a few of the other key ideas it's this sensitive dependence on You Know It uh that comes up that that that is quite typical and somehow it just nobody had really caught on to that for such a long time uh I mean the example one here is is you know if a if a butterfly flaps Wings then uh that you that creates a little change in the air uh in the circulation of the air in some tiny amount but can that have a big effect uh weeks later I don't know to tell the truth if it can that it seems like pushing it but that's the point that a slight change can produce um um big changes later and actually can I indulge in a little philosophy because uh I'm which I'm absolutely unqualified to do uh but I I naturally think the same about um the history of uh man uh do is our is our sort of human history um sensitive to small changes so that a a little if I change you notd a little bit then after 100 steps I'm I'm just indescribably far away I'm I'm not near the two the two 00s are not close anymore and so I don't know whether we uh uh so it's somehow it's it's it's it's a it's deterministic officially speaking I mean it's a formula completely determined but yet so sensitive that even though it is determined you can't predict that's that's basically it and I guess with weather probably predictions now are maybe you can predict pretty well for a week but Beyond two weeks it's it's uh it's probably the sensitive dependen is uh is taking over so that's the that's it's just a whole area of interesting mathematics which actually began with weather um a guy named ed lorence at MIT was uh used to computer or this is years ago so it would have been really slow uh to solve some differential equations his model of the atmosphere three equations and three unknowns and what he found was that they bounced around and then they would stay in one Loop for a while and then it in some erratic way it would go over to another loop so it's a picture of the attractor is sort of a loop loop loop loop for maybe a 100 maybe a thousand times and then it'll jump over maybe go 4177 times and then jump back it's uh quite uh striking and you can see it on the computer uh quite readily um and now if I connect it well I I will write something on the board in a minute if I connect it to something classical I would think of Newton's method for solving nonlinear equations that's an iteration that's a famous iteration if I have a nonlinear system and I want to solve it I use uh Newton's method I could I'll write that down and before the lecture ends just just you know Newton's idea is uh linearize wherever you are and that and solve the linear problem and that gives you the next guess well that's an iteration and where Newton's method will go which answers it converges to which not well let's let me start with the simplest case which will make will will exhibit some of the possibilities but not the but it'll be but it'll be simple okay so what happens if I iterate that function so that's a linear function I'm just multiplying by a every time so now in that one unique case I can produce a formula what is what do I have after n steps if I started from u0 and I multiplied by a every time just a to the nth right so now you can tell me what all the possibilities are when will it uh so what are the possibilities it could converge to zero maybe I'll call that U infinity or something equals zero and when would that happen when would this limit these this iteration multiplying by a every time when would that go to zero when a is less than one so there so for some cases it it'll less than one when would it blow up when will we off go off to Infinity if we allow ourselves to use that language uh that would happen if a is greater than one right a is greater than one and notice I'm doing absolute value because a could be a negative number here but still uh that would what I've said would be still true now now what about um are there any other possibilities I I could have just a steady I could have a fixed point so it just stays um or yeah fixed a fixed Point what when would it just stay put with a equal 1 and we even have one more possibility which is what it it could so the only remaining a is minus one and what happens if a is minus one it would go U minus U plus u minus U it would oscillate so it would be uh like a it would have period two can I use that word period two it would oscillate between uh a pair of values plus and minus you know okay so those are some possibilities that show up even in this simple case I I guess I can't resist always asking about the Matrix case uh suppose uh so this is like time out to practice linear algebra uh suppose this is a vector everything is a vector and a is a matrix so we're sort of doing a power method here we're taking this formula would still be true start with any Vector apply a matrix a apply it again and again and again and again and that's what you're doing in Matrix iteration so what we're about to say answer is the most fundamental little bit of theory in iterative methods in linear algebra iterative methods in linear algebra when do the things converge okay so so what takes the place of a here that was just a number and we could see that the test for convergence was the number should be smaller than one now what's the test for the Matrix case so let me let me just uh use a little space here for the Matrix case tell me the coresponding if I have a matrix I Tak powers and powers and powers and powers of that Matrix how do I know when it when those Powers get smaller and smaller eventually I mean they might grow a little while that's possible but then when do they go to zero the igen values are the test the igen values are the test so the Matrix case we would need all igen values below one so that's quite an important fact uh and it's one of course powers of matrices that's exactly what igen values are just right for because the igen values of the hundredth power are Lambda to the 100s so we're the igen values like bring us back to the scalar case that's what values do uh I'll just comment that in recent years another there's always something new uh so some some people are paying attention to the fact that well it's true if the igen values are smaller than one for example let me just write an example here 1 12 one2 zero a million okay okay or maybe I'll make it 34s just to pep it up a little okay now do the I does that the powers of that Matrix this is my Matrix a what happens to the powers of that Matrix do they go to zero or not well you can see the igen values of that Matrix because it's a triangular Matrix uh that means that the ion values are sitting right there on the diagonal so the I values are both it happens to have a double value it's certainly smaller than one so the answer is yes we're we get we're in this convergence to zero for all the powers of that converge to zero but those Powers might grow for a while and then Decay and uh so so there's some information not the igen values don't give you total information is what people are saying and the idea of something called the pseudo Spectrum has been uh uh or pushed by some uh analysts okay but as far as eventually that will go to zero okay and blow up when would when would the powers of the Matrix blow up well if any if any ion value is bigger than one if any ion value is bigger than one then the hundreds power will have this igen value to the hundreds getting bigger and bigger and the Matrix has to get bigger and bigger right when would we have fixed points let's see H when would did well the identity Matrix would certainly do the job let me just output a equal identity I didn't really think through this see as soon as you get the matrices are all some new questions uh what what about well you can see what's the marginal case here I mean what the marginal case uh which which is all these and other and whatever POS whatever possibilities might arise is what the marginal case would be when some igen value has absolute value one the largest no if anyone is bigger than one then we we take off but if some if one IG value or several igen values have absolute value exactly one like an orthogonal Matrix an orthogonal Matrix all the igen values have absolute value exactly one and you can see suppose The orthogonal Matrix is a rotation typical orthogonal Matrix just rotates the space then I rotate and rotate and rotate and and I sort of wander around uh we have an attractor that that maybe that's a good example where you'd have an attractor all all the U INF Infinities would kind of be a you see would kind of be a set that it would wander around in this attractor it would uh it wouldn't blow up it wouldn't go to zero it would sort of approach this attractor anyway even the The Matrix case already uh so so U so I'll just put some Lambda equal one uh down there as a just to kind of complete that list but not a very specific I didn't fill in all the details there okay now ready for nonlinear which is the reality here okay so um can we get an idea of if I iterate let me let me take an example then second example so my example is going to be a unu n minus so this has a linear part as before but now I'm going to put in a quadratic part that's this is a great example where a is a number and I'm interested in what happens for different A's so a is a parameter here and the question is what happens it I hope everybody sees that you could um just uh set up a write a little code three lines would do it uh input a input u0o and let it run and then you could decide does it blow up what does it do and you could input different U Zer and different A's and we'll see that as a changes so a is this critical parameter which we already saw in this case that as a changed we got different Behavior now it becomes more interesting uh again if a so let me take a typical a somewhere somewhere between one and three is a is a is a typical a what will happen to this iteration can I try to draw it I guess if it converged what would it converge to if so if it converges if the UN approach something what equation has to be solved by that limiting point so if so this is the idea of iteration I can remember actually this came up in my first Consulting at Lincoln lab when I was an instructor at MIT came out and was involved with a with the scientist about who would established an iteration and the question was first question was well if it converged which it seemed to do what's the limit okay sorry sorry the left side would match the right side if if if as n goes to Infinity this thing goes to should I call the limit let me call the limit rather than U Infinity needing a double symbol can I can I use capital u I'll use capital u for the limit if it if it exists so if here's the here's the important point if un approaches capital u and of course then un n plus one does too then the left side is approaching capital u and the right side is approaching well whatever it is then Au U minus Au u 2 good so there's an if here but if it converges then it's not hard to figure out that we can identify what the limit is actually what what's up here so what could capital u be we got a second deegree equation now so we got two possible limits uh let's see we could be have U equals z right that would that that's one possible approach it could approach that or what's the other U that it might be if I if I divide through by U to to to uh me divide through by U1 = a minus Au U above that's so I I've knocked out the U equals zero root and the other root of this quadratic is let's see if I put this yeah is it one yeah if I put this do I get a minus one over a I think that's what I get yeah yeah yeah actually bring the one over there so I have a minus one bring the Au over here and then divide by a so that's okay which does it approach or or neither that's the that's that's the that's our simple question does it approach zero or this thing or does it blow up it it can't approach anything else because if I approach something else that then that's something you would solve our equation but we've got the only solutions so how to tell let me uh let me let me uh think about um uh let me let me draw a picture oh so here's the crucial picture in the whole subject okay here's U and here's F of U okay which is au minus u^2 so I'm just going to graph Au minus Au u 2 start start with that all right so can that's a parabola right and it equals it starts at zero and where does it end where it comes back to zero at uh one right at U equal one I'm back to zero so it's a parabola oh it's a perfect it's just a parabola that sits on S nice symmetrically between zero and one okay so that's graphing U that's f f of U that's just my that's just my thing okay now um let me graph this iteration so let me start at a u not a point U not the nice thing is we can stay inside this interval between zero and one for this example okay so let me start at some you note and follow the iteration okay so what's the where's U1 how how do I find U1 I take F of U not so I what do I do in the graph I go up to I there's F of U1 right is that right yes okay and then I have to think what do I do next I'm taking an A okay so so now that's U1 what do I do now do I have a feeling I want to I want to go over yep I should have practiced this uh so so somehow if that height is U1 then my new G let me let me somewhere there is U1 I I'll I'll I'll figure out in a minute what these Dopey lines I'm supposed to draw actually I'll cheat I'll look to because I'm sure I did it yes okay so why did I oh yeah yeah my the good figure has a 45 degree line in it you're you're absolutely right here's the line 45° okay that's 45 degrees okay so so now let me remove that ridiculous U1 all right let me here get this figure right okay so why did I draw that 45 5° because if I go over to that 45° line say to there where it hits the 45 degree line then that should be U2 U1 is that right that that height is U1 so when I go along here that's the point U1 U1 right it's on the 45 degree line so this is the point let's just stick in a few the coordinates of this are U not U1 because it went along you not and up up the height up was U1 coordinates of this guy are U1 U1 okay so I'm at U1 now what if I if you'll take one more iteration for me and get U2 in this picture I'll be happyy so how do I get to U2 up to the curve right it's across up across up so so here I'm at U2 U1 so I evaluate F of U2 U2 is f of U1 that's the rule so I go up to where that is good and now what now I know U2 so I go along the 45 degree line to and then I'm at U2 and then there is u3 are you am I okay I'm I keep so I don't know what's happening here it looks to to me like maybe this isn't going to converge did you do you see what's happening here it's like we have a little spiderweb the rule is start wherever you like up across to 45 deg up across to 45° up or down to the curve across to 45° up to the curve across to 45° and the question is does our little spiral a little of square spiral converge in to now what's so special about that point wh why is why is that point that's the point where the 45 degree line intersects the curve that I should that deserves a little bit of yellow right that where what's what's so great about that point that's the point where the 45 degree line hits the curve it must be capital u because at that that's the point where the 45 Dee line is capital u and the curve is au U minus u^2 and that's the equation that gives gives you so so it's quite fun uh to see uh does this thing spiral in or does it spiral out in other words what we now want to decide is is this a stable point where iteration goes into it or is it an unstable point where iteration goes out of it let me just and this was another this was another capital u right that was our other candidate that was an the other U was it zero cuz that's the other place where the 45° line and the curve meet so they meet at two points there and there and uh what do you think about the stability or instability of this one uh at zero what's what's going to control the instability if I start near zero yeah let's let's let's draw a blowup picture of this guy because that's that that we certainly ought to be able to pin down so so here's here's my zero point here's my 45 degree line and here's my curve and it seemed to be starting off like that going somewhere now is that point that would be one possibility another possibility would have been for the 40 for the curve to stay under the 45 Dee line this would be I think this is an a probably bigger than one and this would be an a smaller than one oh yeah can you see which of those pictures is going to give me stable and which is going to give unstable uh converging to that um to this capital u the U happen to be at zero where we can spot it so what do we what happens here we start close we go up to the curve we go over the line up to the curve we're leaving it unstable right this would be this is this is unstable we're not going to settle in on that because every time we get close we get further away what about this one we start here we go up to the curve over to the 45 degree line down to the curve over to the 45 degree line what's happening there stable and what's the difference between the two the difference is that the derivative the slope is bigger than one there here the slope is less than one that's what's happening unstable is when the slope is bigger than one and here is stable is when the slope is less than one I'm I'm I'm really I'm already here on top of the main ideas and I can't resist bringing it back to the linear case what was the slope when this was F of un n what's the derivative when when F of un was just a * un the derivative is just a right the function is au and the derivative is a so there we saw the same fact that if the slope was smaller than one at the zero point it went to it if the slope was larger than one it took off so that's just what we're seeing here in a in a picture that has that's locally the same picture if if I'm if I microscope in zoom in around here or here I'm seeing just a totally linear picture but the nonlinear part the fun starts when it gets away from this point and the question is where does it go to so so finally I'll I'll put in absolute values there because the the same would be true if the function was going downwards if if if if in well as it is there here we here our slope when at this point we were interested in so the big question is what is what is f at fpr Prime the derivative at U that's that's the decider F Prime bigger than one decides unstable it won't settle in fime less than one decides stable it will settle in okay so we have to find fime of U for this quadratic case now okay so that that we can do we have this formula for this here's our U this is our U so now can I plug no now I have to find F Prime at that point so what's F Prime guess we could even figure it out so here's my function Au minus Au u^ 2 so the derivative of that is Aus 2 Au U right that's the derivative and now I'm plugging in I want to know at this particular U so this is a - 2 a * a - 1/ A and fortunately the A's cancel and I what do I have all together maybe there's a two there so forget that so I so I have minus 2 a is a two plus an a I think it's just 2 minus a I think that's what we got okay that's the slope which controls stability or instability at this at this one the slope here was just a the slope here was just a the the at U equals zero this this just boils down to a okay so tell me now what's up tell me the the possibilities so we might have a between 0 and one what then what will happen to the unu NS next I'm going to say one between a and a between one and three and I want to ask what happens to the UNS okay okay so suppose first of all a is small suppose first of all a is small then what's our picture what's our little Parabola Loop when a is small okay if a is small the parabola is small it just is down it's creeping down here and what so what's the limit point then zero right so unu n goes to zero which is a nice stable point because the slope is because F Prime is exactly a and it's less than one and that's all great now what about if a is a little bigger suppose a moves up suppose a moves up to two you could you could execute this iteration with a equal to two and what would happen or a equal 2 and A2 suppose a was 2 and A2 all right suppose a is 2 and A2 what will happen will it will it converge to the origin to this solution where that is a solution the curve meets the 45° line right there will it converge when a is two and a half to this guy yeah so so it'll go to this one why won't it go to this one because the derivative here is a and if a is two and a half that's unstable why will it go to this one if a is 2 and a half because we figured out the derivative and if a is 2 and a half the derivative is minus a half so it's going down like this but slowly and it's stable you see everything's like critical is the question is is that is that slope at that crucial Point steep or flat if it's if a is 2 and a half it's still pretty flat the slope is only minus a half that's that's okay it can zoom in but it looks like the way I've drawn it it looks like I've got a slope bigger than one and it's not doing what we want so what will it do so so so the to complete this for a between 1 and three we got this F Prime is 2 - A and it's smaller than 1 okay so it goes to this limit a minus one/ a this this winner okay and now comes the real question what if a passes above three what what what does this iteration do if a goes up above three you see you can say Okay Professor Strang has just cooked up this problem and now he's making a big lecture out of it but uh it's the it's the perfect model for iteration where there's some nonlinear term in it we've got now instead of just a very limited range of possibilities we're we're going to see more possibilities and right now with a bigger than three so three between a and there'll be some next limit it's it's happens to be about 3.45 five uh what's up what could it do it can't approach that and it can't approach that because it it's the case of my figure here uh when it's steeper than one it's it it it moves away from that so right so oscillation okay it's it it's going to bounce somehow and now what do we mean by bounce this this case bounced right when a was minus one it bounced between two points so it had it it it ended up it had period two that could happen here and I think the answer is yes that is what happens here I I think oh I think it bounces between two points and how would you plan to figure out what those two points are I believe it has period two so I believe that it that it in the limit of of course it so what does period two mean so it means that of course it's not going to in this case it bounced exactly from U to minus un to plus unu to minus un it there wasn't any limiting process it it did it right away up here it's going to you know it's not doing it right away it's it's gradually this web is unfolding until it comes over to some other guy probably and it's trying something there well I don't know what's happening there this point has become unstable it's not going to that's not going to be one of the two points oh yeah because I guess what I my wording made it sound like maybe maybe that could be one of the two points but no it it leaves there once it leaves there it doesn't like that point that's a that's uh what's the opposite of of attractive repulsive right repulsive that's that point is a repelling point it's that's not going to be a limit but what will be a limit will be two attractive points but what are they well if we've got period two we're looking what are we those a period two we're looking at a limit point of what here here's the key idea we're looking at a limit point if if if we have a period two limit point what are we looking at we're looking at F of f f of f if we do two FS then it's a regular period for two FS so could I that's the thing I for period two I would have to draw I would look at the iteration F of f of u n would be unu n + 2 and I would look at that itation just the even ends right U not U2 u4 U6 and so forth and then the U1 u3 u5 U7 they're going off to some other guy so those are the two that it's bouncing back and forth so I have to draw F of F and can I draw that well I had enough trouble drawing F of U but that was a parabola now what will F of f of U look like well we know what roughly what it looks like let's see can I draw F of f of U tell me what sort of a function I've got here so it's F of f of U what do I mean by f of f of U it's there's F of U and then I should plug in F of U so it would be sort of a * Au u- Au u 2 - A * Au u - Au u 2 squ that's F of f of U right because there's F of U and I plugged it into F so I got it's fourth degree so it will look something like this it'll go it'll it'll go up down up down it'll have a like a potential well so to speak let me try to improve that picture a little a little more so that it does what I hope it does okay now this is our function f of f of U and how do I decide does it converge what to all that stuff well the same way here's my function I draw the 45 degree line oops try to make it straight without cheating too much so it's a fourth degree thing so it's got four roots and would you like to tell me two of them two of the roots of this F of f of you are well one is clear zero and what's another one another one will be this guy right because that solved why yeah it's kind of neat to see why it solved it was a fixed Point F of U was U so if if I do F again I'm still at U so this was these two guys and that's this one here those are the two that we already found U equals z or a - 1 / a those are the two but now it's a fourth degree say and we've got two new ones which we could figure out and here's the point here's the now I'm I'm ready to give the whole picture uh this guy is now unstable a has got too big these guys are stable that's there is the the uh those are the two that had balances between those two values if I start if I start uh yeah is that right yes if I if I start here if I start near here oh can you what happens if I start near this point I start near this point where do where is U1 well this was the this was the one U1 will be near this point and then if I apply F to U1 that's applying F of f to U not so I'm back here so U2 is here and anyway I'm I'm converging to this period two so this is this idea of period doubling period doubling and uh uh now I let me let me give you the whole picture for this famous this is sometimes called the logistic map because it came into uh for the for differential equations it's a highly important differential equation D if if it was DT equal this that would be that's a predator prey that's the nonlinear problem that people study like mad if it's dudt equal this uh then uh depending how much this this is like this is like growth this is like yeah you get food you you do good stuff this is a negative term so that like if there's too many people yeah actually maybe that's for since we just have one variable we don't it's not Predator prey it's just population this is the actually this predicts the population of the Earth not that badly actually so so there's some magic number a that has to do with the with you know if When U is small then un squar is extremely small so when there you know like 10 people or just Adam and Eve or whatever uh there wasn't much there so the population grew it grows the population of the Earth grows for a while and then at some point it this this stuff begins to slow it down and it approach so the solution to that is this like famous S curve so I I'm I'm off under the differential equation and the whole point is that the differential equation is doesn't have any of this funny stuff so people who classically study differential equations it you know the the population increases for a while with a slope a and then it has a pretty steep increase when when uh and then it slows down and and I think right the limmit is is estimated to be what about 10 billion or something I think the the limiting population of the earth is somewhere around 10 billion and we're somewhere perhaps above are we above five yeah yeah so we're near the end right uh well you and I won't live to see it but uh we're we're gonna we're we're somehow going to have to slow down which everybody knows okay so that's like the differential equation case and contrast that with the with the iteration the discreet case where so many new possibilities appear and that's that's sort of what people missed because you know in calculus they could solve differential equations 50 or 100 years ago but this you really have to uh execute so what happens I promise to say in a few words what does happen as a increases further these become unstable as a increases the slopes get steeper and steeper as a increases everything gets steeper so these go unstable too so there's no stable Period 2 cycle to converge to well what can happen well you think okay it probably changes to period four and you're right so at a certain point around AAL 3.45 it goes to period four if I graphed F of which I don't plan to do if I graphed F of f of f of f of U I would um I'd have it you know it go like this and my four points would still be on it but they would would be all unstable and there would be four stable guys and that would then those that would be the period four and then I keep going increase a and that's what you're seeing on the back of this book here uh so period two is coming out of the bottom of the book and it splits into period four so so across here is is this is the is the is the is the well at the lowest part of the book see I didn't I didn't have a big enough book to start at AAL zero so the book really picked up uh around a equal 3.3 or 3.4 where there's a double period and now as a increases I go up as a increases and then you get period four and then period 8 but then quite amazing stuff happens so that people don't really yet know everything in this picture uh so so what's plotted a white Point means a stable um stable um attractor part of a period so you get 2 4 8 16 but the thing is you get to 2 to the 100th 2 to the thousandth and a is only increasing by a little bit so that after you've got all the powers of two you still don't know the whole story because a is only up around I've forgotten where all the powers of two are in this picture but maybe just probably says here on the on the side uh anyway we've got more A's to go yeah oh yeah okay oh all the way down here See's period 1632 2 to the infinity just quit there but now that a go some more and something else has to happen and there's a period I don't know if you can see there's there's a particular point where there would be exactly six points along and then 12 and then 244 and then 48 and so forth and then uh up then we start up and then a period odd period showing up period five and then here way up here for a near for a near up near Four a period three shows up and they follow a particular pattern of course so that even though so the details of the picture are completely uh difficult to pin down the overall picture the of the order in which periods appear is uh kind of universal but there's there's more to it there's values of a that don't have any period at all when we get up here there are values of a for which there isn't a period and that then that's the chaos that that's where chaos is that's where uh there's no limit it just jumps around forever in a totally erratic pseudo random way well I I could should say deterministic but but uh pseudo random so that's uh that's this logistic map and sorry all those period guys so this this was this was stable this this period to for that range of A's I don't think our eyes are able to spot it but uh um I think it's hidden [Music] in in some of those lines see we uh I've identified yeah I the text would add more to what I'm able to say but the there will be lines there which we can't pick out by I where the there isn't a period where it's chaotic and uh uh and that's typical it's just uh so this very simple map has exhibited period doubling in some Universal way and then also has exhibited chaos in some Universal way and there's somebody there's a guy named Fen Bal who was at Los Alamos and he figured out some Fen Bal constant which which uh again was Universal and controlled what was happening now I I don't want to see the time run out before I mention fractals and mandal bro set you you know mandal bro's name uh so he he realized that uh that most that a lot of Curves a lot of boundaries in reality were not just nice smooth curves not just like one one dimensional curves like like a curve like that but were were higher had higher Dimensions not Dimension two Dimension two is like the dimension of the plane but had some fractional dimension so he christened those fractals so he's um originally French um uh came to IBM spent some time at Yale some time at Harvard he's a difficult guy actually um I remember having dinner with him once and and was just in a group of course and uh he he totally dominated the conversation you know it's okay for I mean he's a wonderful lots to say too much to say I'll just I'll leave it there okay and uh but I have to but but what he did uh revolutionized uh this theory of iteration so I'm I'm going to leave the logistic map with a real variable and go to XY to complex variables because we've just been using complex numbers so so the iteration now is going to be ZN + one is zn^ 2 the quadratic plus a constant well you say boy if you can't figure that out uh you know what do you pay mathematicians to do but uh it's incredibly complicated so we could ask two questions we could ask for now C is the is like our a before that we vary so for a particular C just like a particular a we could solve this equation find we could see what happens for a particular C for particular C we could see what happens to this iteration right if C is if C is well if we took C to be zero we could probably figure out real fast what happened yeah what will will happen if C is zero well it'll depend on the starting point right if C is zero so forget the C then I'm just squaring every time so it depends on the starting point right so for each c i could graph I could plot the plot I could draw all start all the zzer that that that uh for which for which ZN stay bounded for which the Z's stay bounded okay just tell me what that would be for C equals z if if C is zero which complex numbers starting with z0 would would this iteration stay under control it would be the unit circle right if if if z0 because we're squaring every time so if the original start is in inside the unit circle that we're taking powers and powers and Powers okay and this would be called the Julia set so these were actually studied uh Again by a Frenchman uh that's his last name not his first name uh the quite a while ago you know classically 100 years ago or 80 years ago so that's quite interesting and these sets are quite remarkable and that's the sort of thing we're doing now mandal bro's idea with which I'll close so the mandal Bro set was let me show you what the mandal Bro set looks like how how did the so mandal bro just looked at it a little differently so so now now so mandle bro's idea is start at z0 equals 0 always and he plotted the C's you see that little twist this one had a different for each C we have a Julia set okay now over here C is the parameter we start let's say at zero and we plot the c's for which the iterations stay bounded same same idea for which the Z's for which the the zns that follow stay bound so we we plotting the the c's for which our iteration stays reasonable and the plot of those C's C could be a complex number of course this is we're really in the complex uh so it's I could write in terms of X and Y it's so we have Z is you know has a real and an imaginary part so so this is a quick way to get two equations in two unknowns and uh and this plot well I guess I'm hoping because I'm feeling bad that I didn't bring the picture of the mandal Bro set I'll bring it next time uh this book of mandle bro sets but maybe many of you have seen it you it it should I try to draw it I mean oh I'll totally fail right it had but it might remind you it sort of has funny uh did I oh perhaps I put it in the oh it's in the book yes good good good so there is the famous mandero set looked at you know at a distance um and uh so these black the black the origin is somewhere here and the black ones are the points are the c's for which um it stays bounded but the point is that the boundary of this that it's so delicate if I just if I'm near the edge of this mandal bro set and I just change see by tiny tiny amount it may be out of the mandal Roo set the boundary between in and out is total is a fractal it's it's it's like a in incredibly complicated Coastline and uh people can compute the uh dimension of that boundary and it's somewhere between 1 and two and uh and this is the kind of so this figure is the kind of figure you get by zooming in you see a little bit of just incredibly comp and if I zoom in on that I see even more complications so it's this zoom and zoom and zoom sort of it's it's sort like self similar no matter how much I zoom in I'm seeing new stuff or seeing uh uh complications and um and that a feature of the fractal business well okay so that's uh um uh taking our lecture 11 uh uh a sort of Blue Sky lecture if you like uh but it certainly makes this point that iteration is more unpredictable than you would think and that uh of course most numerical methods are are iterative to compute igen values we iterate we we'll discuss maybe the QR iteration for computing igen values for solving linear systems for solving nonlinear systems we use Newton's method and there I'll put in the notes some suggest one or two of the exercises that that uh uh have produced terrific pictures I mean you you can produce these pictures just uh testing an iteration to see whether it uh what limit it approaches so we have uh yeah there's a whole world here of of iterations and uh and we see the in the discret case we see the the complexity already in one variable then we see mandalo level complexity in two variables and in the differential equations lorence had to go up to three variables his his was a nonlinear differential system of three variables and it was just an extremely original observation he must have you know just let his computer run and plotted the thing and started thinking about what does this mean and it led to uh uh a whole new branch of Applied Mathematics okay so that's lecture 11 lecture 12 comes back to partial differential equations the Heat and wave equation uh just to see their um how they compare with lass's equation then then we will have the three big types of of part of pdes Lassa equation elliptic heat equation parabolic wave equation hyperbolic yeah so one you should really see how those three differ uh and that'll be elector 12 okay