 The question was why can, I read it before, why can you have chaos in dimension lower than three for discrete maps like logistic or random maps. Ok, exati. The question is the condition to have chaos in larger than or three or larger than three dimension is a condition that you have for flows, continuous dynamical systems. For maps, these conditions is not applies. In fact, as we will see in the future, in the next lecture, we can have chaos also in one dimensional map. The condition larger than three dimension to have chaos is a requirement that holds only for continuous, only for differential equations, so continuous time dynamical systems. Is that ok? It's a good, it's a good question, good remark because, ok, thank you. Sorry, I removed my mask, so if you don't mind. Ok, good morning. Let me start with a small recap of our lecture of yesterday. So, first of all, with some discussion, we discover that somehow determinism is not equivalent to predictability. Because determinism is only properties referring to the, can you hear me? It's a property referred to the, only to the dynamical, to the dynamical law, to the evolution law, why predictability involves also the dynamical law, but also our information on the initial conditions. And if there are cases where our information of initial condition is very rough and the dynamics isn't unstable, you can have some unpredictability in the future. Ok, this is the first, I would like to, this is important distinction that I would like to, you be remind. The other lesson that you have that even still, even simple model, like the pendulum, when forced with a forcing that depends on time, shows chaos. Is able to have a behavior that appear unpredictable, that appear random. Ok, in fact I show you the evolution, you remember the picture when the states were mixed and were stretched and folding continuously. And the other thing is that the condition necessary to have chaos is of course no linearity, because linear system are always predictable. And the other condition is that you have, you work in three dimensions, at least larger than three dimensions, exception. And this is important, this is good remark by the chart, but the guy in the chart that does not apply to maps, to dynamical systems, which have discrete time. Ok, ok, and can I erase the blackboard? Ok, and also yesterday we started to introduce mathematical notion of dynamical systems, and the notion is given by these three objects. Omega, t and mu, where of course omega is the phase space, the set of states are available to the system. T is your evolution law, and generally I put this t, because I mean that is something that depends on time. Ok, this is a transformation, t is a transformation of phase space into itself, or maybe even into subspace of omega, of omega, and mu is called the invariant measure. If I don't remember correctly, we give also the classification of dynamical systems, and the classification is discrete dynamical systems. So t could be discrete if t belongs to z or to natural, ok. That could be a flow if t is defined by some differential equation, and these both can be autonomous or non-autonomous. If the evolution law, this is the map, let me write x t plus 1 equal to f at the state x, are non-autonomous, if this evolution depends explicitly on time, and the same for this. And the last classification that we saw was conservative and dissipative. Conservative means that the volume during the evolution, consider that during the evolution, the volume or set in the phase space is displaced and deformed. So, ok, is the volume of this set, and it's evolved, are the same. I would say that, let's call phi of t of a are the same. I say, so if v is equal to, see, sorry, this a, if the volume of a is equal to the volume. If the volume of t of a, I say that the evolution is conservative, while if the volume of the evolution of the evolved set is less than v of a, I would say that the dynamics is contracting. Ok, dissipative. And this is the recap of the lesson, the yesterday lesson. I can go. Ok, sorry. Let me start another notion, let me introduce another notion, which is the equivalence between flows and vector fields. This is a mathematical subject, but sometimes in the books, especially in mathematical books, this equivalence is done. So, if you have a differential equation, it defines a vector field, because if you take this, of course, this quantity, this is a vector field in the phase space. But it's also true, the vice versa, if you have a field, you can construct the flow, the line flow of the field exactly when you do it in electrostatic, for example. And so, there is a equivalent, and the flow lines are defined, of course, by the equation v of x. Ok, where s is not time, for example, could be if this is the electric field that you can construct the flow of the field. And s, generally, is not time, for example, is a coordinate, a length coordinate. Ok, so there is, this statement is there is equivalence between vector fields and ordinary differential equations. Ordinary differential equations are equivalent to vector fields on phase space. And generally, we call, when we discuss the evolution of the dynamical system, continuous time dynamical system, we generally use the word flow. Just to, somehow, mean that this is equivalent. So, the dynamics is nothing but the flow of points, which follow the vectors of the dynamics. Ok, flow, when I, this is generally say flow, when I, in this lecture when I say flow, I mean that is the solution of the differential equation associated to the vector field f. Ok, clear? Question? Just a definition, nothing. The other two things that I would like to define is, for example, is the notion of trajectory. What is trajectory? Ok, you see the definition is here. The definition is given in this, in this slide. Ok, the trajectory is the successive position occupied at two time t by your point, which is displaced by the dynamics. Ok, very simple trajectory is the elements of omega, such that displaced from the flow in a given time interval of time. Zero, zero, generally zero, and t. Ok, we can use, we can use s, because if you want to specify, and generally is given by your evolution, of course, the low, evolution low of your coordinates. Ok, for example, for three dimension you have parametric equation. This is a param, in mathematical speaking, is a parametric equations of your trajectory. Ok, but there is another notion that is important. The general is confused, the general is confused with the trajectory, and it's called the orbit. The orbit instead is the geometrical locus. Ok, so the question is geometrical locus. Ok, let's say a curve in a dimensional space defined by this property. So, where t is now is not anymore important, because t is only, you don't see t anymore. Infact, what you do is, if you have your representation in with time, of the trajectory with time, you can eliminate, you can remove t, and you have the equation of the orbit. So, and time now is no more important. Infact, I used this notation minus infinity to infinity. For example, the orbit of a planet around the sun are elliptic curves, and of course the orbit is spanned in time, but as a whole set of motion could be considered a geometrical object. It's only a geometrical object. It's an ellipsis. But general, these two concepts are confused, so maybe with some abuse of language, we speak of orbits and trajectory at the same time, and I confuse you much more. Ok, this is, we have already discussed, this is the fact that the maps also can be conservative and for this, you have to compute the Jacobian of the transformation, because a map is only a transformation on the phase space, depending on time, but it's a transformation on the phase space. You compute the Jacobian, that is this matrix, the Jacobian matrix of the derivative of f. It means you construct this quantity, for example. Suppose you have three uf3, x1, f1, x1, x2, x3, x2, x1, x2, and x3, x3, f3. You start to make the table, df1 with respect to the x1, df1 with respect to x2, df1 with respect to x3, and so on. So this is df2 with respect to x1, df2 with respect to x2, and so on. And this is a 343 matrix, and this is called the Jacobian. If you take the determinant of this matrix, it's called Jacobian, you discover that the determinant of Jacobian, you discover that if this is less than 1, the map, this transformation contracts the volume on the phase space, if this determinant is equal to 1, the map preserves the volumes. And it's called conservative. Now, if you remember our triplet, the only thing that you have to discuss now is the invariant measure. This guy, that is called invariant measure, what's the meaning of invariant measure? Someone can suggest, someone wants answer, any guess? Okay, invariant measure, it's a mathematical definition, of course. I say that mu is invariant with respect to the dynamics T, so the evolution T, if mu of T minus T A is equal to mu A. Okay, this is the definition. What does it mean? It means that T of A, T, it means here, you see, better than. T of A, it's the set of the preimages of A. Okay, if apply, it means that if x, x belongs to T minus A is here, x belongs to the vector, of course. Then, after the dynamics, it folds into A. Okay, this is the definition. It's a preimage of, and this is T of x. Okay, questions? Solid definition, no? It's a mathematical definition of preimage. So, this means, and the fact that the measure satisfies this question, this property, we can say that this is invariant under the dynamics. Okay, consider that if omega with the measure is considered a measurable space. And if this measure is normalized to one, this measurable space becomes a probabilistic space. Okay, so if you read the mathematical books on chaos, on deterministic chaos, generally you say that the dynamical system is nothing but a transformation on probabilistic space. Okay. And why we use a measure to describe deterministic systems? Because they are not probabilistic systems in strict sense, but somehow it's describing the evolution of a system in terms of measures, on the evolution of measures is something that is more simple. And so, it's convenient, I would say. Okay, and now let me make a very brief excursion on the concept of measure, because I don't know if all of you are familiar with the concept of measure. Have you ever seen the notion of measure before? You see? You, yes? Yes. Yes, no, I'm not formal for the moment. You know. Okay, but do you want to have an idea what is measure? Or my idea of measure? Okay. Okay, the statement is in other words it's important to understand how initial distribution evolved toward the invariant's measure. Okay, this is one of the problems of dynamical systems. The invariant measure is something that is very important in dynamical systems. Okay. What is measure now? Of course measure is a generalization of a very simple and intuitive concept of length, volumes, weight, charges also. And for example, you say that this length is 10 centimeters. Okay. But as usually in mathematics, this intuitive notion should be formalized very precisely. Okay, and this is what we are trying to do now for the moment. Okay. Measure generally so is a procedure, so a function on sets, which gives to a set some values that is the weight of the set. In some sense it says how big is this set. But big, not only in dimension could be big also because it contains a lot of points. So it's heavy. Consider two bags of sand. One is heavier because there is more sand and the other is lighter. And we have to compare these two bags. And this is what measure does in practice, for example. In fact, you assign to each set a number on positive reels. And in which way? Okay, in which way? I go there for democracy. Okay. I would like to give the idea of measure for more general sets. More general sets means that I can use for define these sets the intervals. The notion of interval for some set is usual, is useful, is useless. For example. Because there could be while set. Here I have two examples, I put two examples of sets. For this is the counter set. And the counter set is generated in this way. Is a fractal set in the end is generated like this. You have zero, take the segment zero and one. First generation, generation zero. Remove divided this interval in three pieces, equally spaced pieces. And then in the next generation, you remove the central piece, the central part. But you can do, you iterate the procedure. Divided these in three pieces, equal pieces. And remove the central pieces. The central piece, okay? You go here. Like this. Okay. You iterate the procedure. And you can ask what kind of set I arrived to. This set is fractal in some sense that Angelo Volpiani maybe will explain in which sense is fractal. Because it's a strange dimension. This set is full in the sense that it has the same order, the same order of the real segment. While I removing pieces and pieces and pieces continuously. We can show, and you can find it in every, also in internet, you can find a simple demonstration, madriha, that don't have time, simple proof. That this object has the same, the final set has the same power of the real. Could be put in correspondence with the reals. It's full. What is strange is full, notwithstanding that I removed a lot of things inside of it. Okay. Questions? The power of the reals is the fact that you can't make reals in correspondence to the natural set. They are uncountable. So rational, for example, the rational are countable because you can put them in correspondence with the natural. Real cannot be put in correspondence with nature. They are many, many, many than natural point. And this is called the power of continuum. This is the power of discrete and this is the power of continuum. Yes, exactly. This is exactly the cardinality. This is the power, it is considered the cardinality. And this is called Aleph zero. Aleph, I don't know Aleph. Aleph is from the Jewish alphabet. This is Aleph one, Aleph zero, and this is Aleph one. Aleph, I don't know. Maybe Aleph is, okay, sorry. Okay. There is a question from the chat. Yes. So Matteo is asking, I didn't get why we use the pre-image of t in order to define the invariant measure. Is it possible to use the image instead? Yes, because in this case, this is generally do if you don't have invertible maps. If the map is invertible, you can say, you can say this is equivalent. But if the map is not invertible, you have to work with the pre-images. So invertible, not invertible. Okay? Thank you. You're welcome. Okay. So fine, I just want to go faster because it's not very important in the future, but it's a concept that you have to consider if you want to read some paper on the dynamical system. So in other words, defining the measure of interval is quite simple, because the measure of interval is the difference between the two extremes. Okay? But if you have a complex set, how do you define the measure? Consider you have, for example, the irrational or the counter set. You can't use the segments because there are no more segments in these. I remove a lot of segments. Basically, these are isolated points. And the segment can define the measure of isolated points. Okay. And in this perspective, the help comes from the definition of abstract measure theory of a set omega. Okay, we want to define, given a set omega, we want to define the measure of all subsets in omega. And this is called the power of omega. So I would like, my idea is, if I have omega, omega, a set omega, I construct, if I have a set omega, okay, I construct this strange set, family of set, that is the set of all the parts of omega. This is omega, it's a set of all the parts. So all subsets, all subsets of omega. And I call this the power of omega, power of omega. I would like to define a measure on this subset. Of course, this cannot be possible, in general, because the power of omega contains a lot of wild, wild sets that couldn't be measurable. How do you construct, for example, the power of omega? If omega contains only two elements, the power of omega contains omega itself, the empty set. And also all the subsets that you can form from these two elements, okay? And this is the power, in this case, it's countable, of course. It's countable. Okay. Yes? Those p and omega is not the same, you see, because omega, the question is, yes, because they belong, they are subset of A. Yes, because they are subset of omega. Ah, sorry. Yes, in this case, omega, yes, these are redundant. Yes, of course, you are right, sorry. It's a redundancy, yes. You are right, thank you. And of course, once you have this set, okay, given, so we would like to give a measure on this, on the power sets, no, on the power of omega. But it's impossible because we have a lot of wild sets that are surely, I cannot even define, for example. We would be satisfied if you could assign a measure to a good collection of the sets in p of omega. Huh? I don't want to measure everything in p of omega. I would like to have the measure of some subset. This subset is called the algebra, sigma algebra. And define the collection of measurable set. Okay. The algebra belongs, of course, to the power of omega. And the algebra, the sigma algebra, has to, to have, to has, has to have three properties, for example. No, for example, three properties, basically. First, omega and the empty set belongs to the algebra. A, if A belongs to the algebra, also the complement of A belongs to the algebra. Also, the complement of A belongs to the algebra. The complement of A, if I, I don't know if you are familiar, complement of A. If this is omega, and this is A, the complement of A is all that still stays outside the A. Okay. And this is the complement of A. That is generally defined in this way. Okay. So, if A belongs to the algebra, also A, the complement of A belongs to the algebra. And if you have a collection, a countable collection, countable means that you can consider one, two, three, so countable, no? They are in correspondence with the natural number. So, you have a countable collection of set of the algebra. Even the countable union belongs to the algebra. And this is called the closure under countable union. Once you define, when you are able to define this class of subsets, okay, then you say that you can define, you are allowed, you are allowed to define the measure. And the measure. So, it's an application of, it's an application, a function of the algebra toward the real, the positive reals, onto the positive real with the property that the measure of the empty set is zero, of course. And if you take a collection of elements of the algebra that are disjoint, so the intersection is zero, is the empty set, if I is different from J, so here there is the plot, the scheme of this, and this is called countable additivity, then you can define the measure. So, in the conclusion, and I stop here for the measure, the notion of measure based on countable infinite collection of sets, the algebra, is the concept of Lebesgue measure, because due to the fact that the mathematician who introduced this theory of measure, that is a generalization of the theory of Peano Jordan, that is the standard way to introduce measure in general analysis courses. This is an extension, basically with Peano Jordan you can measure lots of sets, but with the back measure you have much more sets that become measurable. Finish, stop. At the moment I have defined, I hope for you, I have an idea, I give you an idea of what these three objects mean. The question that I think you may ask is, why do we care about the measure in dynamical system? We can look at the trajectories that we are satisfied, for example, instability of trajectory, fixed point, we can see, we can understand the dynamics by looking, for example, the trajectories. The question is, of course, reasonable. I can answer to you in three ways for three reasons, for example. I would say that, first of all, in varial measure means that the evolution preserves the probability. Since that, if your system evolves, you don't need to recalculate the probability. You don't need to change your probability or recalculate them. In varial measure is always the same. The second thing that, and this concerns basically these lectures, these set of lectures, chaotic evolution amplifies errors. In varial measure, these set of lectures, chaotic evolution amplifies errors. In some sense, a single trajectory is not so important, not so determinant, not so typical of your system. You don't learn too much looking at single trajectory. You prefer to see a set of trajectory and generally starting from an initial condition, a box of initial condition. We are interested, not in the single evolution, that is not representative, because it is not representative. If you say a lot of points are moving around and you say, what is this? I don't know. But maybe your information is much more important if you consider instead of a single evolution a set of initial condition around your initial condition, a box of initial condition. And finally, more importantly, in varial measure is important for ergodic theory. Ergodic theory stands at the fundament of statistical mechanics. And ergodic theory says that you say that a system is ergodic if you can replace the time averages of observable during the dynamics with the average on the phase space. In formulas, it means that this is the time averages of observable phi. If you are not able to compute this, you can replace the averages over the measure, the invadiant measure with respect to the dynamics. And this is the foundation of statistical mechanics. The ergodic hypothesis. Let me give you some example just through. Consider, for example, a simple model like this. Very simple. X dot is equal to one. Y dot is equal to omega. And you take this model of one. This is called torus, the rotation of torus, because if you consider the trajectory, they are straight line, of course. Those are straight line. If you integrate, this is very simple, they are straight line. But consider that you are identified this part with this part and this side with the other side. So basically you have a leaf and you are doing this and you are folding again this. And this is a torus. Torus is a mathematical object, defines like this. And maybe you use the torus when you were child and you want to swim and you can swim, of course. Yes, OK. Now apply the definition that I gave you before. So the omega is the set belonging to the torus. So there are points in this square. And the transformation basically is very simple because you have to integrate. So the translation on the torus is basically a translation. OK. So sorry, a rotation. OK, it's translation, but since you're coming back, it's a rotation. And the measure, of course, is the Lebesgue measure. That is the Lebesgue measure. Basically it's the measure of very simple. It's the measure that you use when you make integration, the x and the y. So you divided your set with this simple box, infinitesima box, and it's called the Lebesgue measure. OK. The torus, as we will see in the following, the rotation of torus since a very, very unnatural, very artificial system. However, we will see, how we will see, that actually the rotation of tori represents the integrable system. So when you are able to integrate an integrable system what you find is that the motion is nothing but the motion of the dimensional tori. OK. So that's why this transformation is important in mechanics. OK. And finally, I would like to show you that this is the notion of omega t and mu for the Hamiltonian flow. Do you know Hamiltonian dynamics? OK. Everyone? OK, perfect. Good. OK, Hamiltonian equation. Hamiltonian equation are defined by, you give an Hamiltonian that in principle could depend on time and you define the motion of the system defined by the Hamiltonian with the Hamiltonian equation. In this case, omega, of course, is the set of the momentum, generalized momentum and generalized coordinates, maybe in R2n. T of e t is the flow, the solution given by the Hamiltonian flow, the Hamiltonian equation, OK. And the invariant measure is, again, the Lebesgue measure, OK, this measure. And this is why this is the invariant measure. In fact, if you recall the lecture of yesterday, we discovered that the system can preserve the volume of the phase space and this is an infinite volume of the phase space, an infinitesimal volume of the phase space if the divergence of the field defined by the equation. If the divergence is 0, in fact, if you take the divergence of this field defined by the equation of motion, you see that this is identically 0. It means that if you have a volume in the phase space, this volume is deformed, again deformed, again deformed, but its extension, so its measure, remain the same, remain the same. And so we say that this is, so this is the invariant measure. But if you have, sorry, but if you have conservation of energy, the situation becomes more difficult, for example, because you know that the conservation of energy implied that the motion occurs on the manifold defined by the surface of energy. So the manifold is defined as the set of pq belonging to omega, such that satisfy this condition. OK, this is the manifold. The manifold means that, for example, the motion occurs manifold considered that there is a surface space. So the motion occur on this surface. OK, h equal to a, and the motion leaves there. OK, in this case, the invariant measure is defined on the surface, and you can, and you see, you can discover that the invariant measure for a system living on the energy, for a Hamiltonian system living on the energy surface is something like this, where the sigma is the element, infinitesimal element of the surface. OK, this one, this sigma, and this is the measure that is invariant with respect to the Hamiltonian flow on the manifold, OK, on the constant energy manifold. Here there is a very simple demonstration of that. Do you want to, do you want that I show you why, because this is there? Do you want me or not? Shall I make, OK, the demonstration is very simple. And the proof is very simple. Suppose that you have the surface A, you are interested in, and you make displacement to another surface, energy surface, constant energy surface, that is called A plus delta. Well, delta is supposed to be small, supposed to be small. So, you have, this is the equation for this, for this surface, and this is the equation for this surface, where this surface is obtained by displacing along the normal, the points, OK. And we would like to know which kind of displacement I need to apply in order to pass from this surface to the other surface. If you expand this in, you make an expansion of this because epsilon is not so much, is not so great, so because delta is small, you make an expansion, and then you see that now this cancels with this because of this condition, OK. Can you read? And then you see that epsilon is not in both, but delta E divided by N, the normal to the surface, scalar H. But if you remember the geometry, some notion of differential geometry, you know that the gradient to a surface is already normal to the surface. So this quantity is not in both the length of the gradient, OK. And now you see that delta, OK, delta, the variation of the volume, OK, is given by this quantity, OK. Question, yes. Dependent on the answer, maybe I have a question or not. We are saying that if there is an invariant measure, if the measure varies and it is not invariant, we will have dissipative dynamics, right. In principle, yes, in principle, but not so general, OK, but in general if you have dissipative dynamics, your system is shrinking and so it is going, so your measure is changing in this sense, OK, then. But if the Hamiltonian system, of course, depends on time, even the definition of invariant measure is not a good, OK, it, maybe it preserves also, even the Hamiltonian depends on time it preserves. OK, I just, OK, let me think tomorrow I answer, OK. And in any case, dissipative systems does not conserve the invariant measure, even if, even if, even if, you can have the invariant measure on the attractor, for example. So, even if you start with a system, with a volume like this and you arrive on to the attractor, for example, the dynamic brings you on the attractor, OK. Once you have, once you stay on the attractor, the dynamics become, can become invariant on the attractor, OK. For the Hamiltonian systems, the invariant measure is all, it is defined on the whole space, but for the dissipative systems, the invariant measure is only defined on the attractors, OK. So, I can skip. Lotka Volterra you have already seen. And I give you just classification of dynamical systems. Maybe it is so premature, but it is too anticipated, but I would like to give you an idea of what kind of classification one use when deal with dynamical systems. This is a class, it is a general classification anyway, and you have five levels of hierarchy, which depends on the complexity of the dynamics, for example, no. And so the degree of randomness basically. The weakest requirement is ergodicity and it concerns kaotic and non-kaotic systems. So this is the weakest condition that you have, definition of systems. OK, generally ergodic means that lots of systems are ergodics, ergodic for example. Mixing means that you can have, for this system, you can have relaxation to equilibrium. So chaos, in this case, is sufficiently conditioned, but it is not necessary to have the relaxation to equilibrium. Komagorov systems are generally the definition of kaotic systems because they show the sensitive dependence to initial conditions. And also see anosov are considered globally unstable. So every point of this system is unstable. You have lots of unstable trajectories. Every trajectory is unstable. And finally, you can have systems that are so complex. So the evolution is so complex that can be considered equivalent to a coin tossing. They are deterministic, but are so complex. So the evolution is so ergodic that quite resemble, no, actually resemble Bernoulli system that is a coin tossing. And so this is the classification that you can find in some books. Okay. And finally, I would like to show you the global, the goal of dynamical system theory. The global dynamical system theory is you have to understand the effect of the evolution on a state and at a longer time to have the possibility to make some kind of prediction even if transient behavior is important. In this goal there are two possible path, two possible strategies. One is called the topological differential geometrical approach. It studies the trajectory, the properties of the trajectory. Fixed points, periodic orbits, attractors, bifocations and instability in general. The other instead is called ergodic. It concerns basically the behavior you try to understand the behavior of trajectory not looking at single trajectory but looking at the measure. Okay. And here the scheme you have dynamical system one approaches the topological one and the other is the ergodic. Okay. Do you want to stop some relax or okay. Ten minutes, five minutes, ten minutes, okay. Ten minutes is enough, okay. So we'll be back at noon 15. Can we start? Are you ready? Okay. So now we after we have defined the general definition of dynamical system we start how to study this kind of systems in practice. And the first things that one can do for example visualization of the dynamics. Since the dynamics is very complex very complicated you would like to understand the behavior of the trajectory for example if it's chaotic, not chaotic how it behaves and so on. And the visualization of course is in dimension the greater the tree is of course impossible but however to have a qualitative idea of the behavior we call resource to the code pankarek section in the first lesson I already introduced the pankarek section but I call it the stroboscopic view but they are the same basically. Okay. Pankarek section means that you can define a plane for example and or a d minus one surface consider the intersection of your motion with the surface but only the intersection that crosses the plane from a given side so if you go this way you can consider for example the intersection in the plane that goes only in this direction you don't count for example you go here and you go here and so on you can count for example the intersection in the other way and this is called the pankarek map pankarek in fact introduced this map in 1881 and he called this return map because effect is the return map onto the surface anyway it's a good it gives a good idea for example of your motion and of the behavior of your system and the pankarek map of the flow so it's defined by the intersection so it's a map because one intersection you have an next intersection intersection exactly is a map of course and you define generally this map in this way but it's generally constructed numerically because you don't have a technique to construct the pankarek map anyway it's important for example because a periodic behavior on a map is a set of isolated points for example a periodic motion appears on a pankarek map a pankarek section as a set of isolated points for example because the motion comes here and then moves comes here so if replicates so you have for example you have only four points if the period is four ok but if the motion is quasi periodic quasi periodic so you have on the pankarek map just align ok not points but a full line if the motion is quasi periodic quasi period periodic and quasi periodic motion are generally stable motion are not chaotic so the signature or non chaotic behavior on the pankarek map is points isolated points or a single a single curve ok the advantage of the pankarek map is also invertible in fact if you reverse if you integrate your trajectory is backward in time you have of course the inverse of the pankarek map and ok so the map is invertible and what else and it's a natural way for passing from continuous time dynamical system to discrete so but in a way in an important way because if the system is conservative the map is conservative so it preserves for example lots of information about the the dynamics ok so if the for example is the flow is ergodic the map is ergodic if you have the map signals this if you have conservation the map is conservative it preserves areas ok while the flow preserves volumes preserves the areas on the surface ok the problem is the caveat is that this in choosing choosing a good surface section so is a very difficult very difficult subject and in general there is no strategy no general strategy for doing that ok another discretization that coming from the the the flow is the map of the flow what are you doing in this case this is for example I don't know if you are familiar with integration scheme but this is you learn integration scheme basically what do you do you divide your interval of time in steps ok you divide your interval of time in steps and you want to discretize your dynamics and the way to discretize your dynamics is very simple because if this is your if your but you have to do you integrate in each in this interval so in school h is the amplitude of each interval you say that the integration of this on this interval is something like this x x at time plus h is equal x at time t plus of course the integral t t plus h of d tau f of x and tau ok I integrated in this interval ok and this is the this is called the flow map and the flow map because it's the map of the flow we are discretizing the flow basically ok and the flow map has also is is another interesting map that we see in the following ok and it could be used for example to have an idea also of your of your the behavior of your of your system I give you just an example of the Poincare map and the system I want to to study is called elastic elastic pendulum again we we don't move from away away from the pendulum of course in these in these lectures ok and the pendulum elastic pendulum is something like this you have a mass connected to the pivo to the pivo by a spring so consider in the original pendulum this quantity was inextensible but now the the length of the pendulum is extensible and ok you can write your Hamiltonian where you have the coordinate for example if you put a coordinate system in this way this is the mass x-way for example this is x and this is y ok you can write the Hamiltonian and because you have considered the energy due to the fact that there is the gravity so your mass is subject to the gravity and you can write the and this is the the energy the potential energy of the gravity and the other potential energy comes from the fact that you have a spring you have a spring ok if I ask you a question how many degrees of freedom is this system pi to 2 how many variables I need to describe the phase space one I mean 2 degrees of freedom x and y but I need also the momentum so my phase space lives in 4 dimensions ok 2 degrees of freedom 4 dimension of the phase space ok and for example if you ok since the energy is conserved since the energy is conserved you have the advantage that you can use only 3 you can remove one of the of these since this is constants you live the motion lives on the energy sorry this is h lives on the surface defined by the conservation of energy you can for example you can consider this one because you can invert this expression and you remove for example pi from your description so actually you have only 3 3 variables for describing the state of your system because the system lives on a surface defined by this equation but you can reduce again the phase space if you take the surface of Pankaré a Pankaré surface for example a nice Pankaré surface is the one that you obtain when you put in this ameltonian y equal to 0 and if you compute the intersection a different initial condition so you make the integration of your motion for several initial condition what you observe is the Pankaré section and what is very interesting at this Pankaré section it tells you that the system has a very very complex behavior in fact you can have stable behavior in these regions they are called stable islands but also in this way in this part of the initial condition you have stable island so the motion is not chaotic but if you move away from this part you can have for example chaotic sea so this system can develop chaos for example another very simple system in which chaotic behavior can appear depends on initial condition just to stroboscobic map as I told you is another way to make a Pankaré section the difference that is you instead of considering a single plane you consider planes that are defined by your clock so by the fact that your system is periodic so your dynamical system is periodic so your field depends on time but in a periodic way so this period define this kind of planes for example this is q and this is this is x y so this is the phase space but this is the time so at each time at each period you can find the intersection with the planes with the planes at a given period and you project all these planes onto the initial plane and you see also in this case the behavior of your system you have an idea the system is three dimensional but you have for example in this case on a two dimensional description if your system is chaotic is non chaotic if there are stable points and whatever again I give you the example numerical example of that this is the system this is a double well consider that this the system is a a particle moving in a double well and also is forced by this this external forcing ok this is the potential energy of the system you can do the you see the evolution with certain initial condition of course is chaotic so the motion goes in strange way from one well to the other well basically jumps it remains it is the well then jumps and so on going back and forth between the these two wells and if you make the stroboscopic the stroboscopic representation you see that of course the motion is chaotic because you see that the section behaves in this way what you can see this this simple model is of course mixing in fact you see that there is some kind of mix mixer I say like in the kitchen no and and this is the typical behavior of chaotic of chaotic chaotic evolutions this mixing is due to the fact that you have simultaneous so this mixing mixing is also the definition of the system ok do you remember in the classification system I define the system of mixing in fact the chaotic is chaotic system in general mixing and there is the fact that this mixing is the simultaneous action of steering due to the instabilities of the dynamics but also folding because of course the system lives in a finite environment so it cannot be stretched steering sorry stretching is the right right right word stretching stretching ok stretching is due to to the instability but since your system lives in a finite environment so it's not infinite generally physical system lives in infinite environment you don't expect the escape to infinity so you have the stretching but soon the stretching is accompanied by the folding discontinues the stretching unfolding stretching and folding produced the effect of chaos that somehow resemble some mixing effect ok ok now to go to Hamiltonian dynamics and I would like to make some consideration about Hamiltonian dynamics and it's importance in chaotic system because also Hamiltonian dynamics can be chaotic ok generally Hamiltonian dynamics you can find Hamiltonian systems in physics everywhere ok for example in mechanics in celestial mechanics physics many body systems physics of suit whatever is characterized by degrees of freedom so you mean that this phase space is two dimensional so the phase space is the is defined by this set of vectors the states where you have the momentum and the generalized coordinate and these are called canonical variables ok canonical because canonical because they are they are good ok the evolution of the system of the canonical variable is defined are defined the evolution is defined by the Hamiltonian equations and if you look at the structure the Hamiltonian equation you soon observe that they are not symmetric they are not symmetric because there is a sign that is strange no but ok comes from the definition and this is the p and q are not equivalent no because q are the coordinates and p are the momentum they are independent but they are not equivalent and are the same things so they are not the same variables generally when I write a dynamical system I write something like this f of x without specifying without specifying x and y no generally say x because they define the phase space in this case I have to specify that the phase space is there are two distinct coordinates in the phase space so there is some high an asymmetry which is which is evident no ok but there is a representation that is important in dynamical system where this symmetry is lost and it's called I go there and it's called a Simplity formulation it's a Simplity formulation it's a compact formulation of dynamical systems and it's also it's helpful it's very helpful because it's a light some geometrical properties of amiltonian systems basically someone usually say that the amiltonian dynamics is nothing but a geometric differential differential geometry on Simplitic manifolds so actually the amiltonian dynamics can be interpreted in terms of geometry differential geometry in fact in many books you can find this this analogy that is very helpful because later the properties of the geometrical the geometrical differential geometry onto dynamical dynamical systems there is a mapping ok the way to write this representation is in this way you introduce you introduce x the vector x containing both momentum and coordinates and x is defined like this for i going from one to one it contains the coordinates for the rest of the vector it contains the momentum and the way you define the equation of motion here amiltonian equation of motion is this one where j is a matrix that now contains the asymmetry of the sign in fact is defined like this this is the o is the null matrix of order n i is the identity matrix of order n and this is of course the general matrix f the field the amiltonian field in this way is written like this matrix with the gradient now can use the usual representation of dynamical systems ok the fact that this the symmetry so the fact that this the amiltonian can be written in this way so it's a symplactic it enforce a lot of properties a lot of constrain on the amiltonian dynamics and one constrain is evident when you try to make the change of variables if you want to make the change of variables from one amiltonian system to another amiltonian system you have to be very careful so the symplactic structure restricts the choice of eligible transformation not any transformation is admittable ok and how can we see that we see that in this simple way so in mathematical terms you have your transformation from X to capital X to the canonical variables to other canonical variables this is the starting amiltonian systems and this is the final amiltonian system upon the transformation now the requirement is that you want that this transformation preserve the structure of the equation of amiltonian equations so in the coordinate X you have this equation but also in the coordinate X as capital X you should have the same equations the same form of the equation in the capital in the amiltonian the new amiltonian ok the transformation which preserves the Hamiltonian equation are called canonical transformation and now this is the property of this transformation very simple in fact you make the computation you have the starting your starting system you apply this transformation and you discover that the derivative in time transform in this way with respect to the capital coordinate where am matrix of the transformation if you make the same thing with respect to the the gradient so this is the derivative time but you have also to transform the gradient of course and the gradient transforms according to the transformation in this way where am again is the Jacobian matrix of X if you put them together you want that this object form again amiltonian equation you arrive to this equation ok and immediately you see that the Jacobian matrix should satisfy this properties where j I recall I remind you that j is the matrix that we have defined at the beginning is to beginning and is is this matrix where j is in matrix so any canonical transformation or simplicity transformation generally are confused are synonym any simplicity transformation must satisfy this properties this properties means automatically that the modulus of the determinant is one in fact is very simple because if you apply the properties of the determinant and this is good because you know that amilton dynamics should preserve the volumes of the phase space ok so the the automatically the fact that the transformation the Jacobian is a matrix in general is satisfied this equation of course the determinant of the matrix is one and this is of course natural because as I told you amiltonian dynamics should preserve sorry ok and ok we will see that it preserve the volume of the phase space and of course we can show that for example if you integrate amilton equation for example the integration of amilton equation is a canonical transformation ok in fact if you try to to make the the computation you see that it satisfies it is a simple transformation ok this is basically the integration can be considered a change of variable from t to t plus tau your integration step ok and it is canonical and in fact it is called the simplicity flow map this implies again that it is canonical implies that you will theorem the conservation of the of the phase space the volume of the phase space now the fact that the system is implicated has also a practical this is important I think it's also a practical implications because if you try to integrate on your computer an amilton anamiltonian dynamics for example you construct the map of the flow you try to construct the map of the flow you construct the map each time step but this integration scheme should be a simpletic integrator and in fact the difference that you can see is this if you make a naive let's say naive discretization of your Hamilton equation for example you have this iteration scheme at time t and this is the state at time t at time t plus h is the state at time t plus these two terms ok and if you compute the determinant of this transformation and you make the determinant of the flow map no what you observe that of course there is a a conservation of the volume up to a term of h square ok so you have one because the amiltonian dynamics preserve the the back measure so preserve the volumes but you have a term of order one so the precision is up to order one if you change a little bit the scheme and instead of doing this you consider the momentum this momentum not at time t but at time t plus h so this is basically what is called in numeric analysis the basic scheme you improve much more you are making a small error of course but the advantage is that the determinant of of zero is one so the determinant of this transformation is lattice one sorry it's not zero but it's one and this is an error so this scheme is considered truly simplectic simplest simplectic scheme that can use on a computer to simulate the dynamical the amiltonian dynamical systems ok of course if you want to go to higher order you have to work very lot very hard but it's impossible in fact there is a class of integrations of integrators that are called integrators that are very useful if you want to consider hamilton dynamics ok how much time I have including the the interval ok I try to just use slides ok just this is this is only an information not important but if you read the book and you can find the fact that the determinant of the Jacobian of simplectim matrix is one of course is a consequence of the preservation of the is another code is another indication of the fact that the volume of the transformation of the of the amilton dynamics is always preserved ok in fact this is the way when you change the variable during integration ok if this is determinant one so the volume are the same but the hamilton dynamics preserve a lot of many invariant not only the volume the volume is very very but there is a lot of cascade of integrals and then Poincaré integrals or Poincaré cartane integrals and basically if you can prove you can prove for example that amilton dynamics preserve also this form also this quantity and this is in differential is a differential form but you can do also in integral form ok where CDT is a closed form is a closed curve is a moving according to hamilton dynamics another invariant for example is the Poincaré cartane differential form I show you what is this and this is another differential form which is conserved by the amilton dynamics this is in the differential form this is an integral form ok and pictorially these two things are something like that and I can stop here with it pictorially and you can find this picture in books of amiltonian dynamics pictorially means that if you have your face space p and q and this is time the integral of Parcaré cartane is if you take a curve of initial conditions and you make this curve in initial condition flowing according to hamiltonian flow you obtain another curve of course every curve that envelope that encompass the flow this is called the tube flow preserve this integral that is called integral of Parcaré cartane if you consider this line instead of at a different time at a given time so these states have initial condition simultaneous initial condition you have that no more this term contribute no? because you have a time so you don't have this term because dt is of course because the states are simultaneously and you end up to the same integral it's called integral of Parcaré and it's called where c is the integral of simultaneous initial condition simultaneous state canonical transformation is able to preserve both integral in the books you can find some sections devoted to do the fact that there are integral, invariant integrals that are called invariant of Parcaré or integral of Parcaré cartane and ok I can stop here so if there are questions or comments I would like to answer and also I would like to know if some feedback about the audience if the level is sufficient or high to low I'm going to fast, slow some honestly frankly it's boring boring course too fast you mean what your educational history you are from the bachelor bachelor so you have never seen these things but you can you can follow but for example this part is completely it's just an information just to give you an idea because my idea is if the students want to read they should have at least some information some vague information of what the meaning just looking at the content if you say Parcaré cartane what's the Parcaré cartane I have to read, I have to skip probably and this gives you an idea what is that so you can adjust the reading of your book according to the knowledge I'm trying to give you some information about how to read the book of dynamical systems and even mathematical books because there are lots of mathematical books interesting mathematical books but they are a bit hard to read and I would like to that you can orient on the great landscape of dynamical systems that is very huge because it is an old an old topic so in the next time I will discuss if you survive if you will survive we discuss canonical transformation how to construct canonical transformation