 Vsega se vsegačaj. Dobro, ki sem vsegačaj, s prvom vsegača ovoj nekaj kurs, ki je v kaosu, iz Fabio Cikoni in Angelo Vulpiani. Vulpiani, da, to je vsegača. Mislim, da je me vsegač, ja imam Fabio Cikoni, je v Rom, vsegač je z CRR, thank you. That is the Italian National Research Council. Okej, and generally I work on bio physics and polymer physics, especially in proteins. But I do statistical mechanics. V pravdu, da sem vseh tudi izgledačenja kajostiori, in da je to, kaj Mateo, prišli, da me dobro vseh lečenja v kolaboracijenju z Angelo Volpiani o kajostiori in dinamikovih sistemov, basicamente, in to je vseh lečenja. Ok. Why in a course of the theory of complexity we have to study chaos. But there are many, many, many reasons. And especially we would like to convince you that the deterministic chaos is something that you experience in everyday life from weather forecasting, turbulence and whatever. So lots of phenomena are governed by instabilities. And these instabilities manifest as unpredictable behavior. So confusing behavior, confusing scheme, and you have the difficulty to extract from this behavior a conceptual scheme to frame this phenomena. Ok. In this lecture I would like to explain what is deterministic chaos, the mathematical definition, but even the popular definition of deterministic chaos. And especially under which condition it manifests, in which systems you really observe chaos. And the other things, why it influence our life. And in particular I would say it has changed our idea of science in the sense that the main goal of science is the prediction. Interpretation of reality and prediction, especially prediction. And chaos in some sense gives some restriction to our possibility of predict the future. And so it's a revisitation of this concept. And finally I would like to frame the chaos in the huge theory that is called complexity theory. Which part of the complexity theory is concerned with chaos. So it's place into the theory of complexity. Oh, the first part, so my part of the course should be organized. It should be because maybe we have some variation, but I have four topics basically. The first one is the first encounter with chaos. I would like to show you some example, some basic definition of chaos, and some interesting aspect of dynamical systems. The second part, the second part. So the other topic is a bit harder because it's mathematical, it's more mathematical. And concern the language of dynamical system. What kind of tools we have to use, what kind of definitions, and what kind of application we have to do with chaos in reality. And finally, I will give you some example of chaotic behavior, studied in details. So we have some kind of lens and we have to look at, looking at these systems in very details. Finally, and this is probably the most important part of my lectures, is the statistical approach to chaos. It's basically, imagine for the moment that chaos is some disorder evolution. You don't understand anything about evolution. What you can do is say, okay, I try to probabilistic approach. I use the tools of the probability. And in fact, this is the connection also to the statistical mechanics. In statistical mechanics, the probabilistic approach is needed because you have a lot of degrees of freedom and you can follow each particle, but you have an ensemble of particles. Avogadro number of particles. So it's useful to look at each particle, its trajectory of particles. It's impossible. And you have a coarse grain description, a statistical description of approach. In chaos, it's something similar, but the fact that you don't have a very large number of degrees of freedom. You can have also small number of degrees of freedom. But the motion is so disorder, so unpredictable, that the statistical approach is one of the possibility to understand something. Okay. Some question about this program? It's boring? Probably, yes. Okay. Here I put some reference about these lectures. The first is a book that we wrote in collaboration with Massimo Cincini, Angelo Wuppiani. Angelo Wuppiani will be the next lectures. And another book that is very... Maybe it's the first book on chaos, a systematic book on chaos. It's a very good book. It's a bit old, but very comprehensive. And another one is Tabor. Tabor is a Russian mathematician. And it's very simple, very readable, even sometimes not very precise, but it's a very good starting point for beginners. And finally, this is a very, very, very nice, nice. Hard to read, but very interesting review by Ekman Eruel, two of the father of chaos, and review of modern physics of the 85. And if you have, if you want, you can also have a run. You can browse the pages of Scholarpedia, because there are lots of figures, lots of interesting definitions. People writing on Scholarpedia spend a lot of time writing the subject, the page. So it's a good source of information about chaos. And now, okay, let me introduce the concept of complexity and the position of chaos in this general framework. Okay, complexity, what is complexity? General, the definition of complexity, the popular definition of complexity come from the very, what is the eraser? Okay. The very, very interesting and beautiful article of paper written by the Nobel Prize, Phil W. Anderson. And he introduced the concept to explain complexity. The concept of the wording, let's say, more is different. Is it correct? Okay. What does it mean? It means that suppose you have a system made of several units, points, particles, animals, agents, and whatever, insects, fishes, and whatever. And suppose you want to, as you know, you isolate one of these guys and look the evolution of these guys. And you say, you may think that the evolution of these guys is very simple, so you understand very well the behavior of the single unit. But suppose that these units interact in some way, start interacting. So there is a possibility, the collective behavior of this system, so the full system cannot be predicted by the behavior of the single units. Something new happens. For example, suppose in a liquid you have a phase transition, you understand very well the behavior of a particle of fluid. But at a certain point, a critical point, you have the phase transition. The fluid organizes in a certain way. It becomes, it changes state. Okay. And this is the concept more different. In this light, I summarize this concept. So the behavior of a complex system cannot be predicted looking generally, it's not a general state, but in general it's looking the behavior of isolated unit components forming the larger system. I repeat, you isolate your units. The unit is simple, but if you put them together, you can have some emerging behavior that is not contained in the evolution of the single object. More is different. Okay. For example here, okay, yeah, I have some example. Suppose you have one ant, but the social life of a colony of ant is very different. You follow ant. Ant is moving in following a trace or in following some, the path toward the food, toward the food. But if you have more than ant, you have a trail of ants toward the source of food. The single behavior is different. Now you have an organized behavior, a trail. The same happens, for example, for other insect, the cicadas or crickets. The single cricket can sing. But if you have lots of singing, of lots of cicadas singing together, they start to synchronize. And this is a phenomena very, very, very observed in nature. Very surprising also. And this is the collective behavior. Simple units, collective behavior, unexpected. The same for you neuron. For example, you have the brain. Single neuron is a cell that can fire if there is enough potential differential potential. So there is a threshold of electric potential, the neuron fire. So make a discharge. If you put them together, you form the brain. The brain is a complex object because it develops a loss of function in our organism and generally in organism of living beings. And so the passage from neuron, from neuron simple behavior to complex behavior, of course, is another manifestation of, I would say, unexpected the collective behavior. Of course, pedestrian, for example, again you have synchronization of pedestrian. Single pedestrian moving generally in straight line or generally escape object. If I can go there, I can change the direction. But my motion is very simple. Generally I go straight line. But if I am surrounded by lots of pedestrians, maybe my behavior is very different. Maybe we organize on lane. It's perfect to go following the other ahead. Or, for example, in some case, you can have panics. Pedestions that in some concerts, for example, if something happens very, very dangerous, they can behave in panic. And also bird fish and mammals, the single birds fly straight line. But if you have, for example, a colony of birds flock, you see that the flocks in the sky move forming very beautiful, very beautiful forms, very beautiful figures. And I remember you that one of the study of this kind of behavior was the motivation of the Nobel Prize to George Ovalese. The interpretation of the shape of these swarms. OK. So the managerial quality behavior is the subject of complex theory. One part. This is only a part of the story. Because there is a part of the story that concerns chaos, is that complex systems, complex behaviors can emerge also in situations that are very, very less simple than many agents in interaction. A complex system is something that to be harder or predicted, controlled in general. So if you have a phenomenon that is complex, you have worked very hard to make a control and understand this phenomenon. But it happens also in very simple systems. And this is the situation that happens especially in chaotic systems. You can have a simple system with, for example, 3 degrees of freedom, 3 degrees of freedom, x, y, and z. And the evolution of these objects is very complex, very unpredictable. Could be very unpredictable. And I try to explain why this happens. OK, here I give you some definition. Prediction means that you have one present, you will have one future. And this is deterministic process. One present, one future. Uncertainty instead means that one present gives rise to multiple possible futures. For example, in stogastic system you have a stogastic process, it is missing, you have a probabilistic process. Each outcome has a probability to appear. And what is important, and this is a statement that I would like to put on your mind, to be in mind, chaos theory tell us that determinism is not equivalent to predictability. This statement is very clear in your mind. The reason why? The reason is basically, I try to explain this concept with this sketch, with this plot. Suppose you have a set of initial conditions, so you are present. And there is not so much resolution. You don't have much resolution, experimental resolution, for example. So you have an error. For you, these states there is a resonance. For you, these states could be equivalent. But each one, since your system is determinist, each one moves in this way. So f e devolution is particular, and we see how this is particular, probably this set of configuration. You can assume just a single configuration because you don't have enough resolution, spreads all over the space, all over the space, making the prediction inconsistent. Because you say, I start from this box, I arrive here, but I arrive here, and so on. And this is the core of chaotic systems. The impossibility of make prediction. So, let me start from this. And let me explain why these not are equivalent. So I make the figure. This is not equivalent because determinism concerns the evolution law. The evolution law. That is one to one correspondence could be if determinist is one to one correspondence from past to future. But predictability is something more because of course the system the system so the evolution law plus our information of initial states. So our initial condition, our present. Ok. It's clear. This is the at the core of deterministic chaos. The determinist is not equivalent to predictability. Because determinism concerns only the evolution law. This is one to one application, one to one transformation of the system from the past to the future. One past, one single future. But if I would like to make prediction not only I need a deterministic system. But also I need a very, very interesting system in our initial state. And these two things are not the same. Ok. Any comments? This is basically a philosophy, but it's a starting point of any lecture on deterministic chaos. Determinism is not prediction. I would like to make for example some historical notes this concept and already in the past somehow was clear what's prediction? Prediction means that suppose you start from a point here initial state of your system and the system evolves according to your evolution law. Ok. It arrives at this point. But suppose you take another position because you are not sure that your experimental gives you the right answer about the state. You have another trajectory. Suppose you have fixed some kind of tolerance delta that it's enough for making your prediction. You say if these two points are within delta, in a distance within delta I can do a prediction. I fixed tolerance. I decided to start with a resolution epsilon. If this tolerance basically is not very dependent on this resolution initial resolution I can say that I can I will able to predict our future within an error delta a resolution delta. But suppose that happens something like this that your evolution is quite unstable and you go here this is the initial point the reference point and this is another one. Of course the distance between these two points is much larger than delta and you are not able to make prediction. Is that clear? So the final prediction is that you want that the final error should be something that is bounded by your instrumental precision. If this not happens then there is no prediction. Ok Ok, so in historical note about this this concept because I think it is very important. The first who had in very clear in mind the concept of determinism was Laplace, Pierre Simon de Laplace a French mathematician, physicist and and in fact he said that determinism is a prediction based on scientific causality and it's called the concept of determinism Laplacian determinism of determinism Laplacian in French. In fact he wrote and I try to read we ought to regard the present state of the universe has the effect of its anterior state present future and as the cause of the one which is to follow so he says the past is the cause of the present and the present is the cause of the next one and now he says given unintelligence god which could be comprehend all the forces by which nature is animated and the respective situation of the beings the conditions, the initial conditions then this intelligence sufficiently vast to submit this data to analysis it would embrace in the same form the motion of the greatest bodies of the universe and those of the lightest atoms and this is the program of determinism it's the concept in a very clear way Laplac says what is determinism one state one present one future and for Laplac all the nature was a deterministic system but he says for this intelligence nothing will be uncertain and the future and the past will present at this size in fact one of the promoter of the fact that the study of planet motion wasn't nothing but solving differential equation from Newtonian dynamics so something that is very predictable like the solar system actually the solar system isn't stable and it will we will see that isn't stable so the prediction is possible but is not very long time but for the moment we are safe because it's stable it will stay stable for a longer time okay the other the other step toward the the chaos theory was done by Poincaré Harry Poincaré very brilliant mathematician and scientist too and was an into chaos in fact was an intuition by Poincaré and he studied the three body problem for example three body problem means there is the sun and two planets or two stars and one planet okay because the two bodies solving exactly solving is not chaotic while the three body problem could be chaotic studying this kind of problem so the three body problem Poincaré understand that very complicated orbits were possible very complicated behavior of the system could be possible and effects he wrote a very small chaos a very small cause which eludes us the term is a considerable effect that we cannot fail to see and so we say that this effect is due to chance or something about the law probably we have to resort to probability but this is because ignorance but he made another step that is important but if we knew exactly the laws of natures and the state of the universe at the initial moment we could accurately predict the state of the same universe at the subsequent moment but that is but even if the nature law no longer had any secret for us we have a complete knowledge of the laws we could still only know the state approximately if this enable us to predict the succeeding state of the same approximation this phenomena has been completely predicted and this answer this is the punkary answer to your question but it is not always so a small difference in the initial condition may generate very large differences in the final phenomena a small error in the former will lead to an enormous error in the latter prediction then become impossible and we have a random phenomena something that is appears like a random phenomena chaos is written in this part ok we have finished we can go and relax I am joking of course so this is the word this is the birth of chaos theory punkary understood everything about chaos a century after Laplace punkary indicated that randomness in determinist becomes compatible because of the long term unpredictability ok another sum I will continue on this historical path and early mathematical work on chaos was done by Birkov, a mathematician very very very clever mathematician littlehood in the 40s smelj in the 60s and finally the triad I would say the triad of Komogorov, Sinai and Arnold Sinai and Arnold was the students of Komogorovs the problem about chaos that other scientific communities for the moment didn't care about chaos and decided that chaos was a curiosity a mathematical curiosity or something that was referred only to mathematics however the reason probably is because mathematical papers were difficult to read of course in other cultural concepts for example for me it's very difficult to read a mathematical paper but also because the hypothesis the answer yes Matteo yes Laplace in fact Laplace indeed but it didn't make the step forward exactly Laplace and Poincaré was a different attitude toward the determinism but also Laplace understood the fact that the uncertainty was important in the prediction Matteo ok Matteo then ok the theorem hypothesis were considered too unphysical for example no Pendulum, Hamiltonian with something specific properties it's a bit difficult to understand the meaning of the mathematical hypothesis but the situation really changed when with the advent of computers digital computers in fact Lawrence and we say the father with Poincaré the father of chaos basically the mother theory of chaos in 1963 integrated on a computer a three dimensional system and he found very strange behaviors he found this behavior this is called after that the Lawrence attractor which is a very complicated structure and this is the effect of chaos with a system with few degrees of freedom so the complexity is not a matter of many bodies of course but could be also intrinsic in few elements in few degrees of freedom find a real coin the strange attractor to describe this kind of attracting set object where the motion was set in and the strange attractor is the locus where the motion of the deterministic dissipative system converge and they have strange geometrical properties and Mandelbrot in the same period introduced the fractal geometry that is the basic tool the mathematical frame to interpret framework to interpret the characteristic and to characterize the geometry the geometrical properties of this object I finished the part the historical part the philosophical part I will say and I start to introduce a math about dynamical systems ok, we say and I repeat that the deterministic system is considered forward deterministic if two identical copies of the systems that are in the same state evolve in the same manner at the same time evolve in the same manner follow the same trajectory so the theory means that there is only one way in which the system can evolve forward is a one-to-one correspondence between past and future and this is given this is basically can be justified by a theorem that is called the uniqueness existence and uniqueness theorem existence uniqueness is correct uniqueness theorem solution of ordinary differential equations ordinary differential equations what this theorem says only the statement of course, not the demonstration because I am not a mathematician so and I would say something suppose x0 your initial condition is an unsingular point it means is not a fixed point ok do you remember in the previous lecture by Joshua the fact that Lotka Volterra has fixed point suppose that in this region there is no fixed x0 is not a fixed point belonging to a certain subset u of the phase space if f of x this is your differential equation in the dimension satisfies the Lipschitz conditions Lipschitz means that is bounded so there is some kind of disbound so you have a point x and a point y this difference can be bounded by some k greater than 0 and less than infinity so k greater than 0 and less than infinity such that so if this happens now you can find in u an interval t, t0 such that there exist a unique solution depending on your initial conditions of equation a ok unique means that uniqueness means that you can cross trajectories of dynamical system can't intersect each other there is no self intersection because otherwise you can go this way or this way and you are violated determinism or two trajectories trajectories a and trajectory b can't cross so a single trajectory can't intersect but also different trajectory because in this point you violate the uniqueness and existence of theorem is it clear and you violate determinism basically determinism is something that is related to mathematical properties what I have to say that this problem doesn't hold for critical points doesn't hold for critical points and it's local what does it mean it's local that in u so there is only a subset of u where this solution can be fined it works only for a subset if you change you have another another solution but you are not the theorem does not grant that this can be matched is local the theorem in this form is only local everything is clear this is a maybe you find this theorem in math courses math lectures in the second here the university or the first year I don't know I can skip this recap because ok you remember the important thing is the distinction between determinism and predictability this is the light motive for our lecture bear in mind clearly this determinism doesn't imply predictability because of the possible ignorance of initial conditions and let me start with very interesting also historical historical notes I would like this is the fact that you and you have deterministic system that is periodic for example deterministic year is periodic you can measure time so the importance of big determinism deterministic means that you are able to construct clocks and in the past it was very important was a very technological challenging to build clocks because you would like to have a precise measure of time especially when you make navigation sailing in fact the fact that you have the new world discovery of America of the new world increases the transoceanic transoceanic navigation so and if you are on the on the open sea you would like to understand your position in terms of latitude and longitude longitude is the position with respect to the Greenwich maybe I mistaken by the Greenwich Greenwich Greenwich, I don't remember the Greenwich meridium and the other is the position with respect to the equator and this was done by making during your floating during your sailing measuring from the port for the initial point your starting point starting point harbor starting harbor time at intervals equally spaced intervals and the position of the polaris or sun in this way the angle you need to know the angle with the sextant the instrument for which you look at the sun and you have the azimut of the angle the altitude, the height of the sun in navigation this was very important and with this technique and some maps you were able to understand your position on the ocean because on the ocean you don't have any reference point of course and so the Dutch scientist Oygens decided to exploit the socronicity of the pendulum oscillation to realize the first pendulum clock with the hope to obtain what is called marine marine chronometer actually this is probably the sketch of the first pendulum the first pendulum devised by by Oygens and okay but the fact that on a sailing ship the floating of the pendulum changing the isochronicity makes this object not very useful for navigation in fact after that Oygens invented devised another chronometer to measure the time with accuracy with very accuracy in the navigation okay but now for example pendulum remains anyway an instrument to measure time on the land this is another historical historical and that's this first object that we encounter the pendulum I don't want to spend too much words on pendulum because it's the mathematical pendulum in something that is a mass attached to the wire in an extensible wire and this is the pivot point and this makes oscillations small amplitude oscillation areisochronos and this is the basic principle of a clock so if you have an oscillation behavior that is very regular you can measure time in skip this is the equation of the pendulum you saw in the mechanical courses in every mechanical courses theta theta is the angle with respect to the vertical so theta is the angle with respect to the vertical and the theta and this is the theta dot is the velocity so you have two ingredients for discuss two quantities for define the state of the pendulum and of course this is the predictable predictable object simple predictable so predictable that I use it for constructing clocks ok another feature that is important that I would like to become familiar with this notion is the fact that the energy conservation and the if you plot the potential of your pendulum you see that the motion is defined from the values of the energy that you give at the beginning for example the kinetic energy and the position at which you start you leave your pendulum and if you are here you say that you say that you have rotation so the pendulum has enough energy to make a complete rotation around the people but if you give this kind of energy the pendulum makes only oscillation around the equilibrium position and these are called liberation the the curve of the energy level separating these two kind of motion is called separatrix is called the separatrix this is blue curve the blue orbit separate liberation from rotation the separatrix corresponds to the fact that the pendulum is performing a complete rotation from the top in a time in infinite time and return to this point with zero velocity start with zero velocity from the unstable point it makes a rotation in infinite time and ends at top with zero velocity ok it's the motion of vertical vertical this concept of separatrix is another word that you have to write on your notes because it's one of the basic concept of chaos theory separatrix separatrix it separates exactly different kind of motion but this concept will be in the following very important if I had time this is a ok another things happen when if you introduce the friction on your pendulum if you introduce the friction of the air and and you see that the equation changes in this way so the phase portrait which is called the phase portrait is something that is the plot that I show you before you have theta and theta dot for the simple pendulum for the simple pendulum you have something like this the equilibrium state the cycles these are cycles not limit cycles the difference will be clear the cycles do the fact that the pendulum is making oscillation oscillation those are the motion of the rotation and this is called the phase portrait this is the unstable point the unstable point and this is the separatrix ok the separatrix this is going like this for example going like this clockwise or anti-coquice if you put some friction what happens that this point becomes stable so after a while if you wait enough time your trajectory sooner or later will fall into the stable point the attracting point in fact the phase portrait changes in this way the separatrix is no more no more existence is spiraling into the fixed point but the unstable point remains basically ok and this is another ingredient where you try to increase the complexity of the behavior the last step toward the formation of chaos to the development of chaos is to make this object forced so the next ingredient is that you add a periodic forcing to the to the pendulum the forcing in fact you have periodically driven dampened pendulum dampened because you have also already fiction it's a pendulum of course but you add a cosine sinusoidal forcing of your system and in this case you see that chaos appears in fact this is the trajectory that you obtain from a general initial condition this is time and this is your omega and this is the angular velocity and the angle you see that one is fluctuating around zero omega but the angle makes some kind of very unpredictable excursions if you fold this trajectory into minus pi just recall in the fact that you are doing the face portrait you remember face portrait of the pendulum you try to re-reconvert the trajectory into a scheme like this you see that you observe a very complex picture and this complex picture is of course already replicated if you instead of doing this so you fold all trajectories in this way on this square you make another representation of your motion that is called stroboscopic representation stroboscopic representation ok which means I have my face face theta omega and theta this is the time the time coordinate I have a forcing every period every period t t plus 2t 3t and so on I consider a plane here I start with a trajectory this trajectory is wandering intersect this plane in this point will intersect the next plane in this point and the other one for my ability of drawing it will intersect our the next if I put them together so I collect them together in just one plane I observe this kind of behavior and this is for example also unpredictable you don't see some regularity you are not able to make a prediction which is a starting point at time t what is the point at time t and so on sorry and another thing you say is the face diagram where you can find chaos this is the forcing the external forcing of your pendulum and this is the amplitude of your forcing you have some kind of diagram where you see here if you take the couple of values here you have chaotic behavior here you have a stable behavior so stable also known as stable behavior and in fact the system admits also chaotic behavior you start with some oscillation unpredictable oscillation but after that of this distance in the system start in a period 2 or period 4 behavior and the curves in this case you see the motion start to collapse onto this very simple attractor even the previous behavior remains complex this picture that is very beautiful and this movie is very beautiful and is basically the stroboscopic view but instead of considering a single plane you consider a set of planes that are shifted by so basically let me you have the plane this is the stroboscopic view that I showed you before but if you consider this plane another plane here and another plane here of course every t of a period in interval of a period and you consider a set of these planes so a set of stroboscopic view and you put them all together what you observe this behavior and this picture is important because it has planes how chaos works you see here there is a mixing you are in in a rotator and the system chaotic system makes these two these two is mixing basically your face space so you have to consider you have a bunch of points so the chaos basically distort this cloud but also stretch stretches stretches this cloud but also it falls so it falls onto itself and you see this continuous mechanism that is represented by this pendulum this is a simple the force of pendulum this operation is continuous and this is why it's difficult to make and this is why the motion appears very complicated very complex questions comments yes I shifted I have this is the stroboscopic view stroboscopic view is defined by only one one period this is the stroboscopic stroboscopic one stroboscopic two stroboscopic three so on but suppose that I decided to take the stroboscopic plane at time t you will find after a period you will find this period so put this and this together take another stroboscopic surface put this this together no leto stroboscopic sections put them together and you see the evolution the evolution of chaos is that clear? what so basically if you come back if you come back if you come back let me see to come back this is only this is the stroboscopic view suppose if I change the plane I have another stroboscopic view if I change the plane I change the interval I have another if I superpose all these pictures I obtain this movie it's a superposition of stroboscopic but this is important to see because I would like to to show you this picture because I can read the question if you want yes ok Matteo thank you but because the stroboscopic view is some kind of privileged description because you have a forcing that is acting exactly at a given period every period if you take the other maybe if you take another random random sequence of stroboscopic sections maybe you are not sure that if the randomness that you see depends on the fact that you have randomized your stroboscopic your stroboscopic view I don't know if I answered yes thank you sorry no no I'm not the period is of the forcing sorry you are right you are right this is important the period is given by this forcing the period is t that is 2 pi divided by omega ok now I have ok I would like to make some recap about the lesson of the dampened so the dampened pendulum and what I would do this way because you are privileged I would like to make some privilege even to ok the lesson from the driven dampened pendulum is even simple that there is the ballistic pendulum models when periodically forced may give rise to an irregular aperiodic motion aperiodic motion you have transition from typical of pendulum to an aperiodic motion aperiodic means it never replicates you see that there is unpredictable behavior as I show you from the picture the errors are amplified t to spoil our ability to learn yes actually stroboscopic view is something that is related to the Poincare section since since you have a system with many with for example you can have a system with many degrees of freedom let's say 4 for example if you want to have a visual idea of your system you need to reduce the dimension 4 to 3 and the way reduction, this projection should be done in a way that the effect of chaos so the effect of chaos is not altered by the fact that you are projecting so if you make a projection suppose you have a regular motion but if you make a projection you should have something like this maybe the motion is very simple because for example in 3-dimension it's very simple, it's a cube like this but if you project this into the you make a simple projection into the 2-dimensional place or 3-dimensional curve you have something that could be chaotic something that this could be you have the impression that the motion could be chaotic but if you want this and you should be careful in choosing the planes that are you using for visualizing the motion Poincaré section is a thing that allows you to do this but also stroboscopic view is another ideal way to represent chaotic motion so in stroboscopic view you don't have some spurious effect and even in Poincaré section you don't have spurious effect if you make another projection or you take another point maybe you have some arbitrary behavior that you could interpret as chaos but it's not chaos for example in this sense in Eulerian because the question who is the question is in Eulerian hydrodynamics you fix a point and you see how the field behaves around this point but if you change the point you have another information you don't lose information if you take a random selection of points instead if you make a random selection of section or random selection of representation of your chaotic systems you can have a spurious effect you can interpret chaos instead you will have a regular behavior that you are looking in a wrong way basically I answer and finally I would like to talk to you again so ok even the deterministic symposition can be chaotic the other information chaos needs nonlinearity because if you make the perturbation the periodic perturbation of harmonic oscillator you see oscillations you don't see chaos so the ingredient of chaos is the nonlinearity but there is another ingredient and it is called lower critical dimension for chaos that is something similar to lower critical dimension for phase transition you need three variables three dimensions but you say ok, what are you saying you have the velocity and the angular and the angular position two dimension but you don't forget that you have time time is another variable that you have to consider in fact any system which depends on time so it is called nonautonomous and we will see the definition if you have a system that is nonautonomous like this equal f xt you can make it autonomous because the evolution load depends on time depends on time it is nonautonomous nonauton if you make this transformation x dot equal to f x tau where tau dot equal to 1 so it is not more autonomous and you have another coordinate that is time and this is why you have chaos for the driven pendulum because time plays the role of another coordinate the fact that you need three coordinates to have chaos is a consequence of an important theorem that is called Poincare-Bendingson theorem some one of you knows the Poincare-Bendingson theorem I don't give you the demonstration anyway it means that let be this theorem says that let be a closed set of your of r2 on the plane so you are on the plane consider be a subset of the plane and let f1 to the component or two dimensional dynamical systems so under some hypothesis that this b is an invariant set so the system remains is forced to remains in this set because the field from the boundaries forces your system to coming back to coming back so you can't escape the only possibility that you have the fact that you don't have intersection you can't have intersection is that you can limit cycles or fixed points the only states, the only dynamics that you have in two dimension in a bounded set is fixed points or limit cycles Lotka-Worterra in fact you remember the example of Lotka-Worterra of Joshua you have only two possibilities fixed point, a limit cycle ok finally, and this is the fact of the unpredictability at work and this is the let's say a joke but it's very simple to explain the meaning of unpredictability suppose that you are two friends Bob and Alice and the starting point is there are no behavior of pendulum one can say you are not able to predict the behavior of the pendulum because you are not able to integrate the equation analytically if you would have some function some solutions of the model you can make the prediction that at each point at each time but this is not the case in fact this is a wrong idea of unpredictability because suppose that you have Bob and Alice two friends and Bob by email the status of the pendulum they are playing they are making homework together and Bob say numerically this pendulum and these are the initial conditions send an email to Alice but doing this he makes a little mistake on the two less less significant digits ok so the error they differ something very simple very small, very tiny is 10 to the minus 5 but Alice doing her homework integrates on the computer this is the trajectory of Bob after a while the trajectory of Alice follows the trajectory of Bob so she says ok I'm happy but at a certain point there is some discrepancy and the discrepancy becomes larger and larger during time but it can fluctuate the discrepancy and this is the fact that the chaotic system generally amplify the initial error and this is why you can do prediction because the initial error can be amplified too much more than your resolution your resolution ok and this due to the fact that in a sense that I will explain better chaotic systems are generally unstable unstable in a very so in the face space you can find very instabilities so it's unstable it's not only one point one single point it's unstable but there are a lot of instabilities due to the fact that the evolution is intrinsically unstable it's not local but it's somehow general ok comments so you are saying this is the concept of if I understood your question is is there possibility if I fix if I fix some resolution delta final resolution is there any possibility to understand my initial error the initial error that I have to put on my initial condition so that these two points could be predictable with the resolution yes, in principle yes and problem of chaotic system that if you want delta very acceptable so not so small this epsilon is very, very small exponentially small because basically the error and this is will be the subject of Volpeani's lecture probably the error so if you want resolution to grow its exponential so if you want you fix this of course you have to solve this exponential equation to get epsilon is if you take the logarithm of delta epsilon is epsilon e to the minus lambda t delta so if you want delta that is order one of course and your prediction should should should should live for a time t you need that epsilon is exponentially small with respect to delta and in general this is not possible in the atmosphere for example you have that the prediction is generally a reliable prediction is is a weak because this exponential is called the epsilon of the exponent and you will see of the atmosphere is something that is of the order one of the over week so and this is why because the atmosphere is unstable and the prediction of a week is reliable other time other weeks are very are very bad like predictions this is an interesting questions because it's about predictability of weather forecasting for example I finished this one we have time yes a lot of time for talk to you again lecture two and now we start with mathematics ok we start with mathematics ok mathematical speaking what is a dynamical system dynamical system if you read the book of mathematics refer to dynamical systems and chaotic evolutions if you read the book of mathematics you say that you find this definition a dynamical system is the triplet omega t of t and mu what are these guys ok, one we have already encountered in fact I called you encountered with chaos is the phase space omega is the collection of of for example rd of point rd that are accessible to system evolution ok very simple so is just phase space for the pendulum is the the angle and the angle of velocity here is the sample and for mechanical systems the coordinates and the velocities or in Hamiltonian for formalism the momentum and the generalized coordinates for ecosystem for example the number of species of the number of chemical reaction the number of components of chemical species in economics could be the prices the values of assets of assets and whatever of your portfolio and so on so there are the coordinates, the numbers that you need to describe the state of your system generally for mathematical reason that they don't control very much omega is supposed to be a differential manifold so something that is very smooth this is the requirement because maybe over you have you want to do some derivative you need some properties but for our purposes this is not important any case if you read the mathematical book generally say that this is a differential manifold ok here there is the scheme of this the state of the system and omega is the collection of the state ok very simple ok and this is done ok omega is the phase space or the phase of the state of representation the second one the second one is of course your evolution law ok is a transformation of what of course of the phase space into itself t of t is an application of omega onto itself could be even onto a subset of omega but anyway this is not important for the moment and this is the rule which predicts your future evolution you have future states by the knowledge of the present state so a deterministic law we are restricted to deterministic law of course if there is a unique consequence the law is deterministic if there are many possibilities many future possibilities you are considering a stochastic evolution but this is not the subject of our lectures there are two possibilities of course for the for the evolution one discrete time so suppose that your points that jump from one time to another time to another time so discrete instant state instant time but you have of course also a continuous time that is in mathematical language is called a flow because you have a flow like the motion of a fluid on the phase space and this transformation is nothing but the solution of your differential equation and is called a flow ok very simple only definitions the other maybe I already the other classification of the evolution law is autonomous and non-autonomous autonomous means that it depends on explicitly on time the evolution depends on explicitly on time for example this for the the flows and this for the maps of course as I told you before you are able to make a non-autonomous dynamical system you can make it autonomous introducing in d plus one dimension introducing the time as a further coordinate for example if you consider the pendulum the example of the pendulum those are the equations that you have for the pendulum this is the angular velocity this is the derivative of the angular velocity that is changed by the forcing and this is the coordinate that plays the role of time ok this is the recap of the classification discrete time maps autonomous non-autonomous continuous time dynamical system autonomous and non-autonomous but there is another classification of the dynamical systems conservative and non-conservative what does it mean conservative means that it preserves the volume in the phase space non-conservatives means that you have the formation as function of contraction of the volume in the phase space and mathematically why in which way I mathematically express this condition suppose you have this A is a subset of your of your system of your phase space in to the flow for example and of course it is displaced but also deformed because consider that you have your evolution you have your set it could be dramatically deformed by your evolution this point go to this point this point goes to this point again you can have another this point this point goes to this point so it's dramatically it's dramatically changed the question is it's only deformed but also the volume changes and it is the way to reply to this question is this consider so the evolution of the set at two different time is generally close this is infinitesim if you look at the surface for example you look at the surface this portion of the surface you see that this is the the position of the point on the surface and this is the field so this point is displaced till to this point if you want to compute the volume of the area of this parallel people this volume you have to make of course you project on the normal and you say that the variation of this of this volume is given by the s multiplied by the normal the field so your transformation of the points multiplied by dt and this is the volume of this geometrical shape ok it's clear I'm projecting to have a perfect cylinder I use the volume of the cylinder you have the s this is the normal but you have to project normal then I have a perfect cylinder and I compute the volume of this cylinder ok it's correct and this is the volume of the cylinder this one if I integrate over the surface all over the surface because this is a portion of the surface I integrate all over the surface this this expression so what's that is that have you ever seen this expression this the integral over the surface of the scalar product of the field with respect to the normal is that the flux the flux in the electric field basically is the construction that you do for the electric field this is the flux but you remember that there is a theorem called Gaussian theorem that allow you to transform the flux so the surface integral unless you change this quantity with the divergence ok and this is done because now you know how the volume change on the divergence of the field of the mechanical field ok questions comments ok there is another derivation of the fact that the divergence of the field is the quantity that you have to look if you want to understand if the flow is a span or not preserved the volume of the phase space and this given by the evolution of the densities basically the demonstration is this the computation suppose you have suppose you have a system dynamical system and suppose you have a set of initial condition in in a set ok and suppose that in the number of this ensemble these initial conditions ensemble of initial conditions is fixed is fixed because of course also to the fact that trajectory cannot intersect the number of point should be distinct from the solution they can collapse so the number of point starting point is fixed you will find them forever ok due to the fact that there is uniqueness and existence so this is your set of initial condition and this is the density of this the number divided by the volume if you want to make the variation derivative but you can apply the continuity equation if you put this inside the so you use the derivative equation basically and the derivative equation is this if you expand the divergence of these two object you see that this divergence can be written as the field for the gradient of the density for the divergence of the field but consider this object what is this object is there derivative time derivative of of a function if you have a function which depends on which depends on rho if you have a function I put rho of t rho t if you want to make the derivative with respect of x what you have the derivative of this is d rho respect of t plus sum e derivative of rho with respect of these coordinates x, y multiplied by the derivative of this of the derivative of the coordinates so if you use this you see that at the end of the derivation stop yes at the end you see that you end with this equation d rho depends on rho and here again the divergence what happens if the divergence is positive rho becomes becomes is growing sorry, if the divergence is negative rho is growing what's the meaning of rho is growing the number of points is is fixed, you can change the number of points if the density is growing you have 5 points but your density is growing but here you still have 5 points so the number of points is fixed the density is increased the volume shrinked ok and this is the other and this is the other definition if you want to make the exercise you see that for the pendulum you make the divergence of the pendulum this is the equation of motion if you make the divergence of the pendulum you see that the system is conservative is zero so you don't have a variation of the phase space of the volume of the phase space if you take instead the derivative the dissipative pendulum you make the derivative and you say that gamma the dissipation is the rate at which the phase space volume shrinks and finally for the Lorentz model you see that for example the divergence of the field shrinks again because sigma 1 and b is negative I can stop here ok just to finish the same concept applies to the maps because the maps have transformation on the phase space and if you know if you want to know how volume change in the transformation you have to compute for example infinitesimal volume you have to compute the Jacobian the Jacobian of the transformation ok the Jacobian of the transformation if this is the map then is the derivative the matrix formed by the derivatives of f and if the determinant this is the way how volume changes change and if you have the determinant of the Jacobian is 1 the volume is preserved is minus 1 the map is contracting in here you have again you can do some exercise this is the map you make the derivative the Jacobian in this case is only the derivative you take the modulus you don't know if this map is contracting it depends on the values of x because if it's contracting should be minus less than 1 it depends on x but in this case for example this is only expanding because the derivative is 2A A is positive and the modulus instead refold your motion expanding and folding is again the effect of chaos in fact this map generates chaos stretching and folding and finally this you can say that depending on the values A and beta you can have so the determinant of Jacobian is this and you can have contracting or expanding depending on alpha and beta if alpha and beta is equal to 1 for example cosine and this is a sign you have 1 and the map is conserving ok I think that you are sufficiently bored and next time I will explain you the the concept of measure ok, thank you so we are a bit late but if there are questions from the chat no, just appreciation ok, thanks a lot