 Ok, voljamo v tem nekaj, ta ta temorji je dvesi. Pozerno bi stvarno, kaj prednatil jeznanja na več starom na logisti na spet'kih mapu vznečen za prejznazena tokologija. Uz deathsi imam razvoj, ali tudi izdenema, kaj je jeznanja ali nekaj ziloč nekajno. Vz catchesi izdenema jeznanja. začaj. This is the smart change of variable. And if you take this and put here, and you perform just some exercise with trigonometric calculus, you obtain this equation. You obtain that this, it must be plus minus 2 plus k. So you have some ambiguity how to choice the sign k, k must be an integer. And if you require the continuity, you understand that the proper choice is the following. So this means that with this change of variable, you start from, OK, this is x t, x t plus one, we have parabola, and with this change of variable, you have a tenth map. Now, this of course is much simpler, much simpler, and you can compute the probability, the invariant density of this. We compute yesterday. So the invariant density of this is the invariant density y is one, OK. Now you want to compute the invariant density of x. How? You have just reminded how you compute the probability distribution when you change the variable. It's a well-known trick. So if you write that if x is a function of y, so you have that the probability rho x invariant, x is equal to rho invariant. So you know this. So you have that the invariant. This is just, OK, this one, modulus, where now you have to write x as a function. So you do the exercise. This, you know, this is a constant. You insert the derivative and you obtain the result. OK, this is the trick. This is a trick with a method, if you like. Of course, the question is always possible to use this trick. The answer is no, of course, because you have to be smart enough to change from a generic shape to a situation which is under the control, OK. This is also useful for another purpose. We discussed it. So, now, yesterday we just to give you some intuition what happened in a simple exercise for you. I give you the result and you can just consider this what we call Bernoulli shift, which is in a compact form twice, modulus one. So, it's something like that, OK. So, you can do the trivial exercise, you realize that the invariant distribution is constant and you can do this exercise. I assume that initial time rho is something like that. This is one. Something like that, OK. It's something like alpha, one minus x, x minus one alpha. Alpha must be smaller than two, OK. And if you write the Peron-Frobenius operator in this case with this initial condition, you arrive to this result, which, of course, ten for t go to infinity, ten to the invariant distribution. Not only ten to the invariant distribution, but also in an exponential way. OK. So, this is just a case where the system is gothic and there is also a relaxation to the evolution of the probability distribution is also attractive. Now, yesterday somebody asked me about the uniqueness of the invariant distribution, and I start to discuss the fact that actually in a dynamical system you have an infinite invariant, an infinite invariant distribution, which is just, you can obtain, simply looking at the unstable fixed point, unstable periodic orbit. So, this is a mathematical point of view and also from a physical point of view, this is a very unpleasant situation because you wish to have a unique invariant distribution. So, you realize that from a mathematical point of view this is not possible. For example, consider this case. So, there is a trick. No, this is a trick. This is a method. This method has been, this idea has been introduced by Ruel and Sinai, but apparently Sinai wrote somewhere that the original idea was due to Kolmogorov. OK. Apart from this idea, the idea is the following. So, let me discuss in one dimension. So, you have a situation like this. OK. And so, you have some invariant measure, invariant measure, and you have an invariant measure, which is smooth. For example, in this case, you have an invariant measure, which is just equal to one. OK. And then, you have an infinite invariant measure, singular, as I discussed yesterday. For example, just if you have, you take f and you take this, you have something like that. So, all this, you take delta, all this point, you have an infinite, infinite singular invariant measure, but which is disturbing. Why are these disturbing? Because this, of course, is unstable. I was going, unstable, if you start from something which is different from this singular measure, you escape. OK. And so, you expect that this is the good, the good invariant measure, in which sense you want to formalize this idea. The formalization due to the, the formalization is the following. And here, some noise. Some noisy term. Let me call eta. Eta is not a Brownian motion. It's not a white noise. It's a noise. Something which is independent in times, bounded something. And then, now, with the introduction, maybe you can put the term epsilon just to, to stress that is a small thing, small. Then, when you introduce this stuff, the problem from a mathematical point of view is trivial. In, in the sense that for sure now the system has a unique invariant measure. Why? Because this now is a Markov process and you can invoke some very general things and you have that introducing some epsilon. You have that you have a certain invariant measure which maybe depend of epsilon, which depend on epsilon. Then, this is unique. There is only one. Then you say, what, what, now you say, why you introduce this? OK, from a mathematical point of view, because in such a way you solve the problem of uniqueness of the invariant. From a physical point of view you expect, OK, come on. There is always some noise, the interaction with the rest of the units in the computer that has random, random of approximations. But there is some source of noise. And then they promise that you have this and then you take the limit epsilon go to zero on this stuff. And this is, so, if you have a certain invariant measure, let me call tilde, OK. This is what Ruhel and Sinai call natural measure. So, what do you expect to observe in reality? So, in reality when you perform an experiment, you don't have to solve something. So, you don't have to solve infinite problem. So, this is a way to remove this, say, this infinite invariant singular measure are removed when you introduce some noise, OK. Now, you can wonder how I can write the analogous of Peron from Benius for this system. You repeat exactly the same argument yesterday. Exactly the same. There is no difference. Just some decoration. So, the formula are a bit horrible, but not so horrible. So, you have something like that. I write just in a formal way. So, Ruh at time t plus one is equal to the integral of... Let me assume that... No, let me call this... Let me put epsilon here. And this is, as a certain probability distribution, let me call p of eta. So, you have an equation like this. You can write explicitly using the... If you write explicitly using the property of... You have something like that. It's a bit horrible, but so the life is tough. Where these are the random pre-images, the point such that f over x k is equal to x minus eta. You have this formula, which is a bit disturbing, but so that's it. However, this is the idea why it is possible to remove with this argument, with this physical argument, the no smooth probability. OK. This is the question where it's possible to have... To have the presence on... Now, consider something more complicated situation. Up to now, it's got just a one-dimensional map. One-dimensional map is nice for the mathematician, but so we are physicists of course, it's a model which is really just exercise. Maybe from a mathematical point of view, we are not so simple, but for us it's not particularly interesting. So you can wonder what happened, in this argument, when I consider high-dimensional situation. For example, two-dimensional map or the Lorentz model. Last week Fabio Shekoni introduced you to the Lorentz model in an unknown map. So you can wonder, I can repeat all this game for the unknown map. So, yes, I can write, and the evolution for the unknown map is simple. So I discussed yesterday, you are just instead to write F prime, you have the Jacobian, but there is a serious problem. The same for the Lorentz model. So if you have an equation like this, the Lorentz model d is equal to 3. So you know that there is this distinction between conservative and dissipative system. If you write the equation for this stuff, you write the equation, the analogues of the real equation, you have this. You study in your course on elementary calculus on mechanics. You have this equation. In the case of you will, this is outside because the divergence is zero. This is general, completely general. Why not? Now what is the, you write this equation, and apparently you are happy. No. You have to worry a lot about this equation. Why? Because there are two cases. The divergence of F, divergence means sum of Fn. So if the divergence of F is equal to zero, these are conservative systems. We are happy. The volume of the phase space is constant. If this is smaller than zero, this dissipative system. So you have a contraction of the phase space. The fact that you have a contraction of phase space from the point of view of this problem, of this problem is horrible. Why? Let me just convince you with a very simple argument. It's not necessary to. I do this argument in continuous or discrete time. It's actually the same. So I started with a certain x0. And this evolves up to xt is equal to stx0. And I consider a small volume, let me call delta v0 around this point. So something like that. And this evolves. So you have a certain delta vt. So how is delta vt? So let me consider the case where this is dissipative. Is this dissipative? This is delta v0. Something negative. Something which is constant here. It decreases in time. With c positive. So this evolves. And this volume is distorted. Now, I started with rho, rho x times 0. Rho x times 0 delta v0 will be equal to rho of st of xx at time t delta vt. Just a conservation. If this is small enough. Just a conservation of the probability. Now, you invert the formula. You invert the formula and you write the rho x at time t is equal to rho x minus t at time 0 And here you have delta v0 over delta vt. S minus tx. Minus tx. Thanks. You have this. You know this. This is a smooth function. You can start with a very smooth function. You start with a smooth function. And this is the trouble. The trouble is here. Is here, because this is something which is exponentially growing exponentially in time. Because this decreases. So when t goes to infinity, when this increases, this diverges to infinity. So you start with something smooth, but this increases exponentially in time. So if this increases exponentially in time, then approach to something which is smooth. It's impossible. This is just a consequence of the phase-space contraction. You have this in the Lorentz model. You have this in the random map. In all the cases where you have the contraction, you have this trouble. So this means that you can write a nice equation, but you start smooth and then this must diverge. This must not explode, but diverge. So this means that the invariant for distribution cannot be something smooth. It must be something different. This is just a simple observation that from this is clear that you have a problem. You have a serious problem. Then you wonder in which sense this can be singular. It can be singular in a trivial way or can be singular in a trivial way. Let me explain. For example, imagine that you have a pendulum with friction. So you have a pendulum with friction. So you have q and p. You have a pendulum with friction. You have something like that. You have a spiral. So this means that if you start with this set of initial conditions, this evolves in time. But you have a contraction. You have a contraction. So you have delta v t equal delta v 0. This goes to 0. But this is not a serious problem because at the end, at the very end, time goes to 0, q and p go to 0. So in this case, the invariant for distribution is just delta. At the end, you have the invariant for distribution of this stuff is just delta. Delta of p, delta of q. And that's it. It's a bit disturbing, but it's not so surprising. So another situation when you have the wonder pole, did you study the wonder pole question? In the case of the wonder pole, okay, I don't try the wonder pole. What you have? You have something like that. You have a limit cycle, which is stable, is attractive. So you start from here. This evolves. And then you collapse. You collapse to relax to the cycle. So in this case, you have that this is smooth if you look along the curvilinea coordinate. It's not smooth if you look globally. So you have that in some sense, this is singular respect to the plane, but it's not singular respect to the line. So this is singular, but it's somehow in a bit stupid way. In the sense that at the end you have a reduction of the dimension. Instead of to live in 2D, this is leaving 1D. And this is another case of singular, which is singular in the sense that this diverges. If you look in term of the Lebesgue measure in 2D, this is divergent. But if you look in term of the Lebesgue measure along the curvilinea coordinate, this is not singular. So these are examples of attractor, attractor, which are smooth attractor. Which are smooth attractor where simply we have reduction of the dimension. And when you study this in terms of this variable, the situation is completely stupid, because these are attractor, which are smooth, which are smooth. But unfortunately, unfortunately I don't know, it depends on fortunately, otherwise I get trouble with my chair. At least. This is not the unit situation. It's possible to have some attractor, we are not trivial. These are trivial attractor, one point, one line, maybe some smooth 2D surface embedded in 3D. But it's possible to have something which is not trivial. Why? This is an empirical evidence. The mathematical proof that this is true is another story. But I know, I guess, as we proved only some very special case that really exists mathematically, but this is another story. So consider the Lorentz system. So you discuss with Fabio Shikoni in Lorentz system. So what is the essence of Lorentz system? The essence of Lorentz system is that for certain value of the control parameter, you have that the system is bounded, is bounded, remains the Lorentz, bounded motion is bounded something asymptotically in time. After certain transient is bounded in a certain region, the motion is aperiodic and you have volume contraction. You have delta vt is equal delta v0 exponential something. You can compute this c. It depends on the parameter. We see it's smaller than 0. Now, you can wonder, so since you can wonder what is the attractor? The attractor roughly speaking is the place where the system will tend after a certain transient. In the case of the pendulum, just the point, in the case of the pole is this line, roughly a circle, but it's not important. Now, you can wonder how is the attractor in this case? So the first, for sure it's not a point. You wonder how is the attractor? You know, in Cancellino, Vero, Vistokoso, we don't see it on my mind. Natural, the attractor. You can start. Point, no. Of course, you see that otherwise you see that the system collapse to a point. Point, for sure, no. And limit cycle? Limit cycle, no. No, because if you approach to a limit cycle, the motion must be periodic. So, since you see that it's a periodic, this, for sure, is no. These, you have to exclude, because otherwise you have a periodic. Now I say, pointed with dimension 0. This is dimension 1. You can try with a smooth 2D, 2D, what is the name? Validity. Is possible to have a smooth 2D variety? Manifold. Sorry, thanks. Manifold. Is possible to have 2D manifold? The answer is no. Why is no? Is no, because there is an important theorem proved many, many years ago. Do you know that if you follow some of the course in differential equation there is a theorem, so you know? Just for the people. Everybody know the Poincaré-Bendisson theorem? Poincaré-Bendisson theorem, one of the main result is that on a 2D manifold is possible only to have periodic or a periodic motion. Or fixed point. So, in 2D smooth you cannot have in 2D manifold you cannot have a periodic motion. So, also this is impossible. Also this is impossible as consequence of bending of Poincaré. Also this, no. So, cannot be a point, cannot be a line, cannot be a 2D manifold. The next step is a set in 3D. A set in 3D is not possible. It is not possible because we have the phase construction. So, also this is impossible. Also this is possible because we have the phase construction. So, you have that. This is not a point, this is not a line, this is not a manifold, this is not a region of of of rt. Must be something different. Must be something, which must be intermediate between this and this. OK. So, just from this remark you realize that there is a problem. OK. The problem is must be something different from this stuff. And the answer where, so the idea, the congeture I guess has been there is no rigorous proof is that this is a set. What is now is called fractal fractal object. I guess you are familiar with at least with some vague idea of fractal object. And so, you have to one problem is to characterize the structure of the attractor. This is called strange attractor. The name has been invented by well many years ago. Strange, why strange? Because of strange it's not it's not common object must be something different. And yes, the idea maybe is fractal, maybe. Just look, try if it seems that they are fractal. OK. So, you know the problem in this you have to understand the structure of the attractor. This is one problem. OK. So, you have to understand the spatial complexity this complex system school. So, this is an aspect of the complexity. This is the spatial complexity. This is the part of the business. Another part is the temporal complexity. Because now OK. This is one point. Angelo, can I ask you if you can recall this Poincaré theorem? I mean if there is an intuition of why you cannot have the intuition is the intuition is the following. In so Poincaré. OK. In today only periodic or quasi periodic or fixed point. Why? Because, OK. You can have fixed point. This is obvious. You can have periodic motion. If you try something different you realize that it is not possible in today. You have this point. You have no unicity. But this is possible in 3D. In 3D you have something like that. So, you have the if you are forced to remain 2D this is impossible. But this is possible in 3D in the sense that this is the projection. So, in order to have chaos you need at least three dimension. In the continuous time. In discrete time, no. This is enough. Because in discrete time you have no the problem of the continuity. This is the this is the end-waving argument to understand if you try to do something different you see that it is impossible. OK. This is not the proof of course. This is not the proof. But this is the essence of the theorem. If you try to do something you realize that you have enough space enough dimension to make it more complicated. In 3D, in 3D there is no problem. You can have something very strange without any self-intersection. In 2D you cannot avoid the intersection. So, it is something like in the case of random walk in random walk when the dimension is larger than 4 you can the self avoid the random walk and the usual random walk is the same. Why? In the small dimension in dimension 2 the problem is different. It is something like that. Just try a connection with different aspect of statistical mechanics. So, now we have arrived to a problem characterization of the strange attractor. Another problem is the characterization of OK temporal of instability let me call it. I put instability in inverted comma instability usually have instability of a fixed point of instability of a certain solution. Now, here the idea of instability is more general and this is very rich argument. OK, let me start from this. Let me start from this. And about this point there is from a technical point of view there are some very strong connection also with some part of statistical mechanics disorder system I mentioned just briefly so what is the problem here? With the Fabio you know that if you look at the Lorentz system if you start with a certain delta 0 here your delta t beled like this at certain time you have a saturation so if you decrease you have something like that. And so roughly in this part this is something like delta 0 exponential. But you see that this is very rough observation and you want to have a precise definition of what of this instability and this rate of expansion. So intuitively you say let me take long linear stuff and this slope is this number. But this is very crude because this is true but only if delta 0 is small otherwise this is not true and this is true but t must be not too large to formalize this. And in addition this can depend on the initial condition of the different slope. So you have a lot of this of this problem and you want to formalize this idea. Ok. The formalization of this idea is absolutely not trivial. Is absolutely not trivial. Let me discuss the unique almost not trivial but relatively simple situation is the one-dimensional map. Then the one-dimensional map the formalization is simple because but you are just to invoke the ergodicity. In more than one dimension the situation is not trivial is very complicated and you have to use a very sophisticated theorem due to a soviet scientist Valerio Zeledeck 60 years ago. Yeah. And this is what is called the theory of Lyapun exponents which is a very rich branch of the mathematical physics which is very useful even in statistical mechanics for example in disorder system in low-dimensional disorder system you can use the do you know the method of transfer matrix for spin system? Ok. If you use the method of transfer matrix automatically you have this kind of problem and what in statistical in the disorder statistical mechanics is the property of self averaging the system of self averaging is nothing but a physical term to indicate ergodicity. There are perfect correspondence between these two stuff. So, Lyapunov I guess this is the time to stop. Ok, and then I stop here and in the next in the next hour I start to discuss the problem of the Lyapun exponent I started with 1D map in this case everything is very simple. It is very simple because we do and then the case with the dimension even today is not simple at all. Even today the problem is not simple both from a mathematical point of view both from the computational point of view you need some effective method and I had to say that we Italian gave a very important contribution to this stuff Ok, so let me just mention the name of the people who gave important contribution around 1960 Valerio Zeledeck he was a Soviet mathematician and around if I remember correctly there are three Italian and one Polish guy Giancarlo Benetini from Padova Antonio Giurgibbi Luigi Galgani and Strelchin I don't remember the name of Strelchin Jean Marie, if I remember correctly I am not sure I am not sure of the spelling So this gentleman produced the constitutional theorem that there exists the solution of the problem from a mathematical point of view the problems are well posed and there is the solution then these are physicists Ok, we are very happy that somebody proved the theorem but now we have to do the computation and so you need a method you need an algorithm and all the algorithms used even now are based on this idea of this gentleman Ok, so they produce a very smart algorithm which is actually simple to use and translate for us who are theoretical physicists and the idea of the cell deck and this result has a strong connection Ok, as I said with statistical mechanics of random sheets Ok, I stop here and I have a question I don't prefer the question after the Eh, this should be understood But from what I understand the only