Okay, everybody. Good morning. I hope those of you taking the course officially will turn in at the end of the lecture if you haven't already.
The assignment. And as it says here, the next one will be handed out on Thursday and do two weeks from today.
So to give you a little bit of the big picture, this is week two and we're talking about odds.
Today will be about chaos and related things and then Thursday about boundary value problems as opposed to initial value problems.
Then on next week, we're going to move towards Pdes partial differential equations.
But today is a lot of fun with this beautiful topic of chaos that you've all heard of and you all know something about.
And it's very hard not to enjoy this subject.
I wanted to begin with some discussion of planetary motions, not just planets, but asteroids and comets and so on.
And only two days ago, we had the eclipse of the moon in English weather.
One never sees these things. Really. Did anyone actually see an eclipse?
There were rumours that in Liverpool you could see something, but not here anyway.
Let's start by showing a demo you just did.
Those of you taking the course tested the assignment involving a chaotic orbit of a very artificial sort.
So we had three fixed suns or planets and then another body orbiting around them in a chaotic fashion.
Let me show you another example that's a favourite of mine. Which I got originally from James Binney in theoretical physics.
And this is can be found in a code called M 29 planets.
So if I run that code. You'll see that.
It shows you three planets coloured green, red and yellow, which of course are just point masses.
And the thing that's memorable about them is that they're starting motionless in the configuration of a three,
four or five triangle, so that you can always remember it gets the right angle triangle three, four or five.
And nevertheless, what happens is interesting. So let me show you.
This is a computation, of course, in MATLAB.
These are random dots in the back. That's really the genius of this code.
The whole cosmos is produced with a random number generator.
You can see the Times 72, 78, and now at time 86.
See what happens. So up to time, 86, it looks like chaos.
Well, it is chaos in some sense. But then at time, 86, something surprising happens, which could be called ionisation or self ionisation,
where two of the particles go off in one direction to infinity and the other in the other direction.
And if you think about it, with just two planets, that couldn't happen, but with three, that is possible.
You can satisfy conservation of momentum and energy.
Let's just run that again to see how it looks.
So one more time. So you see, it looks like standard chaotic motion.
There's the time up there. So at 86. The ionisation takes place.
I think that's beautiful. It's one of the remarkable things about chaos is that.
Without computers, of course, it was very hard to see chaos.
And for hundreds of years people did odds and the whole subject of chaos wasn't really there.
Now, Poincaré, the great French mathematician, did sort of figure it out.
But without computers, it just wasn't so conspicuous.
As soon as you get computers, it's everywhere.
It's very hard to write down a system of dynamical equations with more than a few variables, which doesn't give chaotic solutions.
As an illustration, I'm going to run the same code again, but perturbing the initial condition.
So this one is called M 29 B.
And it's exactly the same code, except that the particle that begins at position three, I guess it's the vertical one now begins at position 3.01.
So it's a perturbation by a 30th of a percent.
And one of the initial conditions. Why am I not seeing anything?
Interesting. That's better.
Okay, so it looks the same because you can't tell that one of them has been perturbed
slightly and for a while the motions are going to be essentially the same.
Now, of course, you don't remember the details, but if we get to time 86 for this one.
You'll notice that's nearly ionised, but not quite. And nothing special happens.
This code keeps going to run to time. 2200.
There's still another planet over there. They're going to come back together slowly.
Yeah. There they are. Does this one?
Eventually, I and I probably I don't know, maybe there's even a theorem that with a set of initial conditions of a dense set,
maybe they'd always I and I says, I don't know that you can't run a thing like this without having questions.
Here's a question. Notice I cooked it up so that the three planets come pretty close together at time.
200. Suppose one wanted that to happen.
Exactly. That's probably possible.
At least at a time. Near 200. It'd be a crazy thing to say, but, you know, maybe we want an actual impact at a time.
200 or time, close to 200. Could one find initial conditions that do that?
Who knows? So it's easy to have fun with planetary orbits.
I want to say a few things about that subject. So first of all, regarding numerical methods, I'm not going to go into what people actually do,
but some of the good codes for this sort of thing are often built for second order oddities rather than first order.
If you really care about accuracy,
you can design finite differences that democratise the second derivative rather than a system of equations with the first derivative.
And a famous name in this field are called Sturmer formulas or Sturmer valet formulas, and these work with second order equations.
Another thing I want to mention is so called geometric integration.
This is a hot topic in the academic and practical side of numerical solution of odds.
A problem is physical, as this has all kinds of interesting things going on,
like conservation of momentum and energy and other conserved quantities sometimes.
And there's a field of geometric integrators which make special formulas that conserve appropriate things.
Another word that comes up in this business is symplectic integrators and so too a dynamic system.
Problems like this have a lot of mathematical symplectic structure that one can take advantage of, and there is a lot of work on that.
Let me mention in particular, if you're interested in this, there's an outstanding book by, well, two names you've already seen.
I've passed around a book by Hira and Varner, but the book on this particular subject is by three of them, also Luby.
So high were Varner and Lubitsch. Three Austrians have a great book on numerical methods for geometric integration.
But I wanted to say a bit more about planets. So I got curious years ago when they tell you there's going to be an eclipse.
Where does that information come from? So of course, you'll get it from the newspaper, the television or the web or something.
Where do they get it from? Well, usually they get it from some national organisation, such as?
I've listed a few the Admiralty office, the Royal Greenwich Observatory, the British Astronomical Society,
NASA Goddard Space Centre, the U.S. Naval Observatory, the in France, the Bureau de Longitude.
The Indian Department of Meteorology. The Astronomy Astronomical Division of the Department of Hydrography in the University of Tokyo.
Places like this give data about eclipses. Now, where do they get that data?
It turns out almost all of them get it from Southern California.
It's all JPL, the Jet Propulsion Lab. So that is the answer to the question.
All these calculations, mostly more than 90% of them come from the SS.
D is the Solar System Dynamics Group. And this is at the Jet Propulsion Laboratory, JPL, in Pasadena, California.
That's where most of the calculations are done.
I believe the French actually do it themselves, but all the people who are not French rely on Southern California, so far as I know.
The government shutdown has not affected this.
I guess the Jet Propulsion Laboratory is clever enough to insulate its fundings from the whims of the President.
But I copied for you there Home Page. It's not a very beautiful and exciting page to look at, but what it links to is really quite incredible.
Through here you can get all the details. So just just look at this.
These numbers. Their standard system tracks 791,373.
Asteroids, 3557 comets, 190 planetary satellites, eight planets.
So that answers our question about Pluto, the Sun, various spacecraft and so on.
It's an amazing system using these stern more fairly formulas.
I believe it does this to remarkable accuracy.
They can make predictions thousands of years in the future and thousands of years in the past in order to track historical records.
The accuracies depend on how far away you are in time.
If you're within a few decades, you get unbelievable accuracies in centimetres or metres,
depending on exactly what you're measuring and even a thousands of years away.
You get good accuracy. It's an amazingly mature subject, which for my work, in my opinion,
is really the great example of very high end calculations conducted by the human race, basically.
Very exciting stuff. Now.
What else that I want to mention about that. Oh, I believe the actual body integrators they use have all sorts of add captivity built into them.
They have variable orders, variable step sizes. It's it's a wonderfully mature technology.
Have any of you connected with this stuff to any of your work in such areas?
Uh huh. So what department are you in? Uh huh.
Yeah. Okay. And by the way, I think a lot of the bodies are essentially point masses.
That is good enough for many purposes. The Earth isn't the point mass, but further off planets often are.
Relativistic effects are important. Most of the bodies have that included in them.
So that's a great thing to be aware. So let's say a word about chaos.
And this is a huge subject.
Let me in particular include in my heading the notion of leaping off exponents.
So as I say, chaos has always been there but never really got focussed on until computers came along.
Really in the seventies I guess is when it hit the headlines.
The central question. As people would regard it nowadays.
Four possibly chaotic systems would be this.
If the initial data are perturbed.
And it even makes sense mathematically to talk about infinitesimal perturbations.
Of course, in practice, no perturbation will be infinitesimal.
But mathematically, you can imagine an infinitesimal perturbation.
And you can ask how fast? Does it grow with time?
And the advantage of infinitesimal perturbations is that you can let time go as long as you want, and they're still infinitesimal.
The big distinction is between systems where perturbations grow algebraically and perturbations where systems where they grow exponentially.
So algebraic growth means of time to some power.
Exponential growth. Means O of E to some number like lambda T, where lambda is bigger than zero,
or the real part of lambda is bigger than zero, depending exactly what you're measuring.
Now a chaotic system needs to have this second property. That's not all it needs.
For example. Here's an ode to EU Prime equals U.
The solution is U of T equals either the t.
That solution grows exponentially. And moreover, if you perturb the initial condition, that perturbation will grow exponentially.
But of course, that's not interesting. The interesting cases are where you get exponential growth with a global bounded ness of some kind.
So this is the situation where chaos turns up.
You have exponential growth of perturbations, but still you have in some sense some kind of approximation to global bounded ness of orbits.
Now, I'm not trying to give a precise definition.
And in fact. Although chaos does have precise definitions,
this is not an area where people have settled very happily on one particular definition that everybody uses.
It's a bit complicated, but you know it when you see it, as they say.
Probably most of you know about STROGATZ, his book.
So here's one of the editions of it.
Steve STROGATZ from Cornell was an early person writing a text book, and it's just a fantastic book that everybody loves.
So, Steve STROGATZ, if you want to know about chaos, that's the place to start.
I'm curious. Raise your hand if you were aware of STROGATZ this book.
Yeah, that's impressive. Raise your hand if you've looked at. Have his videos online about chaos.
Okay. He's also a person in the news.
He writes columns for the New York Times and for other magazines and so on.
So he's one of the well known applied mathematicians on the planet.
Just to say a word about the history. So I mentioned John Kerry, who was very important in the pre-history, if you like.
And that's going back to the nine, you know, 1900 ish.
He was concerned with the Three-body problem and the solar system and so on,
and he realised that effectively chaotic things could happen, though he didn't coin that term.
Some other key names were in the 1970s.
There was Roussel and Tompkins and there was Feigenbaum was a key person, so-called universality.
And then there was Bob May, who became Lord May, who showed chaos in biological systems.
And there was Jim York from Maryland who invented the term.
But then the really famous original person was Ed Lorenz.
So. Ed Lawrence at MIT.
And was it 1963 wrote the great paper that sort of put chaos on the map, even though the name wasn't there yet.
He died about ten years ago. He lived to be 90 years old.
I checked this morning on the Web, you get 453 million hits on the word chaos.
They're not all mathematical chaos, but many of them are. And if you get a glance at the others, I've given you the first page of his paper.
This is really a great paper. It's so readable.
If you look at it, you just feel inspired for how much he saw using the primitive computers of those days.
It's it's really great.
He was a a quiet kind of he was kind of an on the spectrum, brilliant genius who didn't say much but knew he was on to something very real here.
So Ed Lorenz. Let me give some examples.
As I say, it's pretty hard not to find chaos if you look at a big dynamical system.
So one field in which people look for chaos is what they call billiards.
Billiards means bouncing particles. In the simplest case, you have a billiard table and you say,
What about the orbits of a ball bouncing around that table if the billiards are on a rectangular table?
That's not chaotic. The perturbations grow only algebraically.
On the other hand, if you have billiards on a curved table, well, it's not always chaotic, but it's depending on precisely the shape it usually is.
And there's a whole business of quantum chaos in which people study chaotic and non chaotic orbits of particles moving around in point particles,
which relates to eigen functions of the in the plus operator in quantum mechanics.
It's a highly developed field. So billiards is one example.
Here's another example if you have any linear equation.
No matter how many variables, it's not chaotic.
It can very easily have exponential growth of perturbations, but that would be in the context of actual solutions going off to infinity.
You don't have the bounded mass needed for things to be really complicated.
Another thing that is never chaotic. Is any one variable or two variable system.
First order autonomous ODI.
So such a problem cannot be chaotic.
An example would be the Vanderpoel equation that we've looked at. And the reason is.
If you have, let's say, two variables, your trajectories are trapped in a plane.
They're completely described by a phase plane. So.
Suppose you're in a plane. There are theorems about this.
Of course. It's impossible to get tangled up in a plane because you can't cross a trajectory.
An autonomous system is completely determined by its value at a point.
So you could have trajectories that look like this, but notice how constrained they are.
They can't really. Since they can't cross, they can't really get tangled.
So there's the hand-waving argument that two dimensional systems cannot be chaotic.
But as soon as you have three dimensions, then chaos is very common.
Now. What about the weather on planet Earth? Which was.
Lawrence's motivation. Well, surely that is chaotic.
It's a bit hard to say because, of course, that's not a well-defined problem.
It's an infinitely complicated system. So, properly speaking, chaos applies or doesn't apply to a precise set of equations,
but many reasonable sets of equations that model Earth are indeed chaotic.
A famous one is turbulent flow in a pipe.
Now it's obvious that, loosely speaking, that's chaotic.
It's very hard to predict crazy things happen.
But it's an example of how challenging it can sometimes be to connect mathematics and physics that the rigorous study of this is a bit controversial.
And there are those who have argued that. What looks like chaos in a turbulent flow.
Is actually, in theory, always transient. So it will look chaotic for a time like our planets that eventually ionise,
but then eventually it has to become a smooth flow again eventually can mean on a doubly exponential timescale.
So it's this is spectacularly academic, that discussion.
Obviously in practice it's chaotic, but it's not completely clear that in an infinite circular pipe modelled by the navier-stokes equations,
technically speaking, one has chaos. Another example.
Now. I guess that's all I have on my list. But let's say a little bit about planets.
About the solar system. So the solar system has a lot of particles in it.
So you would think, given that even three particles, it would seem, can be chaotic.
Obviously, the solar system is going to be chaotic. The reason it's not so simple is that the sun is so big and heavy.
So there are a lot of epsilon in the solar system compared with the big sun in the middle.
So there's actually some controversy here, too.
But I think most people now believe it is chaotic.
So it would seem that with timescales on the order of tens of millions of years or more.
Now, what does it mean to be chaotic?
Basically, the sort of question that arises is might the solar system eject one of the planets out to infinity in principle?
And I think many people now think that perhaps yes, the answer in principle is yes, though it won't happen in a hurry.
Let me mention some key names here. Just two pairs of names in the 1990s.
Sussman and Widom at MIT. Didn't major calculations which sort of put this problem back on the map after Poincare.
And then more recently in France, the big names, a big name really is Lashkar.
And another key person is Gastineau. Okay.
I guess that's all I'll say. Well, I'll say one more thing.
Chaos is a phenomenon, but also it's something of interest in engineering and people.
It's such a huge thing. People have done everything with it.
For example, if a system is nearly random, then it should be that if you want to achieve a particular outcome,
some set of initial conditions in principle will do that.
Like I said, might there be initial conditions that at times 200 made those planets collide?
And so this would be the general subject that people might call control of chaos,
where you're using the exponential divergence to achieve interesting things.
It's been used in cryptography, for example. You can interplay between the deterministic and the seemingly random aspects to encode messages.
So it was a huge subject. Now I want to say something a little more precise.
So let's talk about one of the favourite examples, which is the Lorenz equations, and I'll call that the next example.
These equations are often the first place that people see chaos.
But for my money, planets orbiting around each other are really the most obvious example, as in what we just looked at.
But let's call this section 4.10.
So these equations go back to 1963.
Now you know that to get chaos, you need at least three variables and you need to be non-linear.
And sure enough, there are three variables and it's non-linear.
So in some sense this is as simple as it can get and still be chaotic.
And let me write down the most famous version of these equations.
X prime equals ten Y minus ten x.
That's a linear equation for how X evolves.
Why Prime equals 28 x.
Minus Y, minus x, z.
So that's a nonlinear equation for how Y evolves. And Z prime is x.
Times Y minus eight thirds. Time Z.
So that's another non-linear equation.
Notice the nonlinear nonlinear sureties are as simple as you could ask for just quadratic products of two terms.
The precise numbers can be changed. It doesn't have to be exactly 28 and 8/3 and ten.
But these particular choices have become so standard that 99% of discussions of the Lorenz equations use these numbers.
So if you start analysing this, what you find is that one solution?
Is X equals Y equals z equals zero.
Plug that in. And you see, of course, the right hand sides are all zero.
Therefore, the equation is satisfied.
So this would be called a fixed point of this dynamical system.
A fixed point is a point which doesn't vary with time steady state.
It's an unstable fixed point, but it is a fixed point. There are two more fixed points.
And they are at x equals y equals plus or minus six square root of two.
And Z equals 27. So if you analyse those, plug these into the equation, you'll find that again, the derivatives are all zeros.
So it's a fixed point. It's unstable, meaning that small perturbations will grow away from those fixed points.
So when something's unstable, in a physical sense, you would expect not to observe it in the real world.
Now what happens is that when you compute most solutions to these equations, they move around in a seemingly random fashion.
Lorenz claimed that in 1963, many people thought he was wrong.
They argued that it was errors in the computer. You know, this is 1963 computers.
Just imagine how slow and moreover, the floating point arithmetic in those days was not as established as now.
Every machine was different. People didn't understand it so well.
It was hard for him to persuade people that it was true. But he was very careful and he did persuade people.
It took 36 years. For a proof.
So there's a famous guy called Warwick Tucker, who I guess is the head of the math department at the University of Uppsala.
And he wrote a paper called The Lorenz Attractor Exists.
So this is the gulf between pure and applied mathematics.
With chaotic systems, it can be very hard to prove things. But he really did prove rigorously, though, using computer assistance of a certain sort.
He proved that there is a strange attractor, as they call it, for the Lorenz equation.
And this was in a famous French journal in 1999.
What's the. Layup on off exponent.
I didn't define it, did I. But the layup on off exponent which is in my heading.
Is Lambda. So it's, it's the lambda when we get E to the lambda t behaviour and more precisely it's the maximum such,
so it's the maximum exponential rate at which perturbations can grow infinitesimal perturbations.
These things are hard to calculate. I think for the Lorenz system it's thought to be about 0.9057.
So about one. So perturbations grow on average at a rate of about E to the T.
The strange attractor, which I haven't tried to define and I won't, is a mathematically precise sense of what the crazy orbits look like.
It's a set in three dimensions.
The reason they're three dimensions is that we have X and Y and Z.
And so the strange attractor is a three dimensional object.
But one of the wonderful things.
About chaos is that these are geometric objects, although they sit in three dimensions, may have a lower dimensionality in a precise sense.
And I believe that the dimension of the strange attractor is known to quite a few digits.
It's something like 2.0.
6 to 7 1600.
I'm going to plot it in a second. So what that means is that the chaotic orbits of the Lorenz equations.
After an initial period, essentially lie on a object which is nearly two dimensional, but slightly more so.
It's a little thicker than a two dimensional manifold.
So let's play with that now. Okay.
There are lots of ways to play with the Lorenz equations. Everybody's got a method.
So let's start with Matt Labs method. If I type Lorenz in MATLAB, I there's a built in demo.
So let's see what that does. This has been there for ages.
I'll press start and you see it just.
Uses Odie something or other. I don't know which one to compute a trajectory.
And you can see the this is the strange attractor emerging as you have this chaotic motion.
You see it sort of looks two dimensional. It's got these two butterfly wings, if you like, but they're not quite flat.
If you rotated around,
you'd see there's a little bit of thickness there and there is a precise way that mathematicians measure these non integer dimensions.
Any comments or questions. Stop.
Let's do another demo. Let's try and check.
Fine. I'll use a cheap, gooey. So the graphical user interface to the O.D initial value code.
So there we are, one of the built in demos is.
The Lorenz equations. This is a much slower way to solve the problem.
Telephone is never the fastest, so it's a coupled initial value problem, a system.
And I'll click on Lorenz Equations. And then if I type solve, you can see it's only going up to time 15.
So this is a very short trajectory. And then in the end it's plotting the three components U, V, W, X, Y, and Z.
And you can see this familiar random looking behaviour.
One of the nice things about a system like this is that you can play around and change things very easily.
So what happens if you change some of those coefficient ten or 28 or 8/3?
I'll just make one change that I happen to know does something interesting.
I'll change 28 to 22. So there you can see that for a while it looks sort of similar, but it's obviously then settling down to a periodic orbit.
So that's not a chaotic system, evidently. Let me also draw your attention to the book Exploring Odes, which I mentioned.
So if you look in there, of course, there's a chapter on Chaos, chapter 13, I think.
But then if you go to this appendix, I like so much appendix B with the more examples.
A bunch of those examples are chaotic systems, so I think it starts with 64.
So in fact, when we were writing the book. I asked myself whether.
Please. Can you tell if this is working? Yeah. I asked myself whether you might have a scalar, chaotic system.
Of course. In principle, it must be possible, but it has to be at least a third order because you need three variables y y prime and wide.
So sure enough, if you look on the web for scalar chaotic systems, you find one.
And this is a simplified version of that.
Notice is a third derivative here and we've cooked it up so that you're getting chaotic behaviour in this scalar ODI.
If you look at the other examples, I won't linger.
But so there you have a forced nonlinear pendulum which appears in some books.
Here you have the Roessler equations. This particular orbit is periodic, but you can see a kind of a period doubling phenomenon.
With some parameter it would have been simply periodic. Now you can see the two different amplitudes reflect a doubling of the period.
This is the universal so-called root chaos that Feigenbaum made famous.
Here's a three body problem, just as we've been looking at.
This one is not interesting. The initial conditions don't do anything remarkable.
This one, on the other hand, is interesting, chaotic.
Here's a double pendulum. That's a nice example that you often see in people's offices because you can build one physically.
Here's the so-called hand on health equations.
And I guess that's it for the chaos. Now.
I thought I had an example that showed the growth of perturbations.
Was that the very first one? Did I fail to? Where am I?
Here. Let me remind myself if I do em again.
No, that's not the one I want. Uh, I don't think it is.
Well. I'll run that while I figure out what I'm looking for.
No, I guess it's just in the next one I'm going to show you. I thought I had one already.
Okay. So let me finish up by discussing another famous example of chaos in the billiard department.
So if you think of billiards, the physics is so simple.
The particle is always going at a fixed speed. So how many variables are there?
Well, at any time it has a position, and that's two variables.
And it has a direction which is a third variable. So billiards has just the right count to be interesting from a chaos point of view.
And the example I want to mention is called the senior billiard.
So this is our last example. Seni is a famous mathematician and physicist billiard.
And I'm also going to mention the so-called Siam 100 degree challenge.
Which I think came up earlier when we talked about optimisation in two dimensions.
So if you look at the handout, you'll see that the.
The reverse side of Lorenz's paper is this thing that I published back in 2002 where I asked people around the world to solve ten problems,
each to ten digits of accuracy. So these are the ten problems.
Nine of the problems are not chaotic. And what that means in practice is that you can get ten digits of accuracy on an ordinary 16 digit computer.
However, one of the problems is chaotic, and that's number two.
So if you look at problem number two.
You see, the way it's described there is that you have an infinite set of circles in the plane.
And you have a particle moving around and a particular.
Orbit is specified doing things like this. Well, that's a chaotic system.
It's a billiard problem in which the boundary is not just straight lines, but curves.
And it can be proved that that's chaotic. In order to prove that, you don't really need to think of the infinite domain like this.
What Seni did was realise that the simplest version of this would be.
That kind of domain. Because if a particle.
Keeps going. Well, that's equivalent essentially to being reflected.
So it turns out. The compliment of an infinite set of circles is equivalent to the interior of this simply connected so-called seni billiard.
So that is his billiard. And he proved that the motion is chaotic.
And I want to show you that in action. So this is a code that I actually haven't printed up, though.
I'll put it on the web called M 30 Circles.
So suppose I say am 30 circles. And.
It asks how much of a perturbation I want. So I'll start with a perturbation of ten to the minus.
Ten and you'll see as we run it what that means.
I'll tell you before we run it,
we're going to track the 100 digit challenge problem with the specified initial condition and then also with a perturbed initial condition.
And we're going to compare the two orbits. So.
You can see that's actually two particles bouncing around.
And then finally, after a while, they become distinguishable.
The red one has gone off. Who knows where? And the black one is still there.
So. Let's press. No, I'm going to run it once more before pressing.
Enter. Okay. Let's try a different perturbation.
I'll try ten to the -12 now.
10 to -8. We keep losing the particle.
I want to find one that where we keep the red particle. So I try ten to the minus nine.
Must be possible a. It keeps.
Okay. That's a little better.
Okay, good. I like that one more. So you can see that for a certain time.
Although the trajectories are diverging exponentially, they're so close that you don't see that in the in the picture.
But then eventually they have diverged so much that.
They're no longer correlated in any way. Now what happens at that point is that.
The exponential divergence ceases. Everything is now algebraic.
Nothing is growing exponentially in this picture. Apart from those initial perturbations.
And that's when I press return the second plot that comes up.
So this plot shows the distance between the red and the black dots as a function of time.
And we started with perturbation ten to the minus nine. And you see a pretty systematic exponential growth.
The slope of that curve is the the up and off exponent. But then once it gets to scale one, no further exponential growth can possibly happen.
So you see, mathematically, if we started with an infinitesimal perturbation, this behaviour in principle would go forever.
Let's do it one more time with a very small perturbation. Suppose I say.
And 30. I'll say one E -15 and see what happens.
Okay. So there you can see the exponential growth is just barely beginning to level off at the end.
I think now we still have three more minutes, so I want to go back and play some more with the Lorenz equations unless somebody has a question.
Yes. Is there a formal way of. Assessing.
Like when faced with the. To assess whether it is stable or not, whether we sort of just sort of just playing with it.
I think the essential answer is, no, there isn't. You just play with it.
And of course, play is a loaded word, but you do very serious computing.
But no. And this is why, for example, the solar system is controversial.
It's very hard to assess these things and small changes in parameters can change the answer.
And there's a lot known about that. This famous Feigenbaum period doubling.
Yeah, there's an infinite cascade as you've change a parameter and the different regimes get exponentially closer and closer together.
Although, you know the general structure of such things,
there's no way of knowing a priori whether a particular parameter choice will be a chaotic one or not, I think.
Other questions. I guess I thought it would be fun.
Yes. Another question. The economics.
But it depended on the size of your tax evasion. In principle, no.
So the rigorous definition,
you have to maximise overall infinitesimal perturbations because some of them are maybe going to be in lucky directions that don't excite the growth.
But loosely speaking, it doesn't depend.
I wanted to try, I guess, changing the parameters a couple more times.
So we'll go back to our Lorenz demo.
This was coupled initial value problems. So we know that with ten, 28 and 8/3, we get chaos.
Now, if anyone has a particular choice, I'm happy to listen.
But failing that, let's just try a few things at random. What if we change 28 to 38?
That looks chaotic. Of course, we don't know. What if we changed 22 to 24?
I think I did try this one. Now.
Is that chaotic or periodic? Well, let's run it on a longer interval.
Let's take an interval of 30. It's it seems maybe to be settling down to something.
Let's try an interval of 60. Oops.
Maybe it's not. So that probably is chaotic. But of course, this is no proof.
Let's go back to 28 and change one of the other numbers. So what if we change 10 to 9?
That looks chaotic. What if we changed the five? That's not only periodic but very boring.
It's obviously settled down. So of course there will be some critical number in between at which the behaviour changes.
Six seems to be high enough. What about 5.5?
You can see how addictive this sort of thing is. What about okay, we're going to nail it, right?
5.25. We're doing by six in here now.
5.1, two, five. Ah.
5.1. Oh, I got the wrong way.
Sorry. 5.15. Now 5.175.
5.2. So you see, it's absolutely fascinating and infinitesimal changes can change in theory or whole dynamics.
Okay. Boundary value problems on Thursday. Remember to turn in your assignments, please.