 Poživajte se nekaj nekaj informacij, kako se zelo v kaosem. Poživajte mi, da? Rekordi v progras. Ok. Jesteveno smo izgledali nekaj, da je izgleda. Jesteveno smo izgledali Stabilitivne teore. Stabilitivne teore da je dinamikalne. Prvo je definitivno, da je Ljapunov, kaj je syntotik in vse stabilitivne. Prejdeš, da je vse bolj. Vse, kaj je vse bolj, je vse bolj, kaj je vse bolj. Zdaj smo izgledali o stabilitetu ljepunov. Zdaj smo izgledali... Zdaj sem kliknil na skrinju. Zdaj sem kliknil na to. Zdaj, čekaj. Zdaj smo izgledali o stabilitetu ljepunov. Zdaj smo izgledali o stabilitetu ljepunov. Zdaj smo izgledali o stabilitetu. Zdaj smo izgledali z capsule ljepunov nglošnjih vzgl. Zdaj smo izgledali, that nomi izgledali vzgl. .. stjepana and depending on the position, you can decide if fixed point is stable or unstable, and then again vectors, as I told you, define the direction of approaching or going far away pri tem vseh početnega. In neč izgleda, da ne se je zelo vseh, vseh nešlazeš zelo, da neč inšli zelo, boječ neč inšli zelo, boječ neč inšli zelo, boječ neč inšli zelo, in linišnja stabilitva potrebač je začnjena scenariša, da je linišnja stabilitva. In vse izgledajte, da, da je izgleda, da je z vsega početka, bomo vsega vsega vsega, limit cycle, limit cycle. Limit cycle. Zelo sem, da je inštak način, limit cycle, limit cycle kako je unika. Zelo je unika zelo obit. Čakaj, ne? Zelo je odrečen, odrečen obit. Zelo je, da je obit unika. Zelo je zelo, kaj je zelo, kaj je zelo, kaj je zelo, kaj je zelo, kaj je zelo, E samplepiano zelo, kaj je zelo obit, kaj je kasovljšit ko zelo je to, da je staviljnj, in stavilj stavilj. Počakaj, si bi te kanalcije kaj je zelo vzivil in skupaj na radošnjih medglas. Kaj je jaz tudi vziv, kako jaz potrebno pozdaj, jaz jaz zelo vziv. Zato, kaj je, da, kaj je to zvonila, kaj počut, na početku, počut, na početku, počut, no? Zato je, da, definitiv, tudi limitne vseks. Limitne vseks, so, da, prihledajte koncert, tako, da je invarjante, in vseks, in vzelo, invarjante, to, da, da, vseč. Zelo, da se pozite vživljage dinamikov, in dinamik vseč pomeni na to, da je zaplavnja. To je zelo, da je zaplavnja. Zelo, da se pozite vzve dva zaplavnja. Zelo, da je zaplavnja, tako da je zaplavnja, da je zaplavnja izelo, da je zaplavnja, da je zaplavnja, Prez. Stabobmanifold. Sopoš, da jste imeli stabobmanifold. Sadov. Sopoš, da jste imeli stabobmanifold. Stabobmanifold je stabobmanifold, kaj je vse počke, kaj, nekaj nekaj, nekaj vse počke, pada v vse počke. Innozaj, da počke bonovanje na počke, ma pribej, nekaj, nekaj, nekaj, nekaj, nekaj, nekaj, nekaj, nekaj, nekaj, nekaj, nekaj. Vse je inzaj nekaj, pa je inzaj, način. Vse je ta zelo vzelo vzelo vzelo vzelo vzelo vzelo vzelo vzelo vzelo na zelo vzelo vzelo, tako, da je dve stave, vzelo na dve stave, in dve stave, način, način, in zelo na dve stave. Pazca. Zelo je pricim. Vse. Vse. Daj, pa. Pazca. Vse. Daj, pa. Pazca. Vse. Pazca. Pazca. Pazca. Pazca. Zdaj tudi sem bilo ozirana, da je prijezena dojda, ali se je pozbila na toh zelo. Kaj je največe naši nočnih. Stavil, stavil manipul, je ozirana na skup. Zato je, da je tudi vso pomejn, da je prej ze, da je, da je, da je, da je, da je, da je, da je, da je, da je, da je, da je, da je, da je, da je, da je, da je, da je, da je, da je, da je, da je, da je, stavlje in stavlje, in tudi, kaj je meni, vseč je tukaj skeliton na dinamikalne vseče. Kaj je tukaj? Ne, ne, tukaj. Kaj je tukaj? Tukaj je to, vseč je, vseč je, vseč je, vseč je vseč, vseč je vseč, več, malo poslice, da je... to vseču ogladnju, ogladnju, tukaj, vseč je, vseč je, vseč je, vseč je, vseč je, vseč je, vseč je, vseč je, Menar tega, ki ni je ojevstrnit, je je, da moči se podobno podačiti... ki je na vse wedi mač, ki bo fakusnje, je fakusnje, ki so siti, siti 1, siti 2 in tako, siti 3, okej? Rešte pri, revene preferenske rešte, z vse stvari, z vse stvari, z vse stvari, z vse stvari, so na taj zvori, so stabilne in stabilne, je to koncept, ki je vse globale ideja, lahko si je, in u vse stabilne in stabilne, je to lokal ideja vsej spaz, vsej postav. In, kaj si vidi, kako jih biliš, če se zelo potenjeno in potenjeno, vse moji je globalizacija, zelo potenjeno in potenjeno neseljne kvaliti. Prežitosti, ki ještje, zelo je zelo sezerno, da se nešeljši poživiti pendulum, Tukaj je tudi unfair površen, ki je svoju samarbe dan. Tukaj so vsredi pa štev, ko je to vesredil. Tukaj je svoju senarja, očer antorite tukaj pravnej pendulum. Tukaj je tukaj машin, ki bilo vsega štev površen. Tukaj je tukaj in se kakvaj vsega štev površen. Tukaj je tukaj vse svoj menifor. Se, že, in there are some interesting concepts around the stable and the stable manifold. Is the fact that the set the point first of all at the core of the unstable dynamics of the chaotic dynamics, the complex dynamics of your system. What do I mean? It means that if I have to look some chaotic behavior, or strange behavior, Mi se da sem tukaj, da se ta dobro vzouto v glasbo. Zgleda, potenšelji, in potenšelji, bi se je neko inštačne dobro vzouto v glasbo, početno. In tako, je zelo nekaj vzouto, da je dobro vzouto vsak. If the system, if you don't know about it, If you want to search for chaos, it is better to stay near the saddle. All attractors lie on unstable manifolds. This is another property. If you want to find some strange attractor, maybe it is an effect of unstable manifold. Stable manifold also instead, so this is the property of unstable manifold, but the stable manifold also is as important as an important property because defined by the boundaries of the attracting set. In this case, for example, you can separate the different kind of motion. In this case, for example, you can separate these attracting points and you see if you take the other attracting point, they are divided by the stable manifold. The stable manifold is defined by its boundaries. This is another concept that you have to take note of that. Of course, there is also the possibility, and it is called homoclinic, that an unstable manifold goes to the other fixed point and becomes, so there is this possibility, unstable manifold becomes then an unstable manifold. This is called homoclinic intersection of the fixed point. Another interesting concept that you have to take in mind, to be in mind when you deal with chaotic dynamical systems. I finished the last part and I would like to start with the other one. Lecture 4, I would say. Yes, example, coded by behaviors. Share screen, I share my screen. Now, in this lecture, I briefly would like to discuss some examples to see how chaos is working. I will discuss three typical standard systems. Historical, I would say, historical systems. The logistic map, the Lorentz model, which is the logistic map, is a map introduced in the concept of mathematical ecology. The Lorentz model is a simplification of thermal convection in atmosphere physics. An analyze model is anamentonian system. It is introduced to model the motion of a star into a galaxy. I would like to discuss these three. I think we conclude these lectures with the logistic map. Logistic map comes from the population biology. The first model of population biology is due to Malthus, introduced this equation of evolution for a population, for the individual of a population. The problem of this equation, of course, is that the number of individuals in the population grows forever. And this is impossible, of course. It's unphysical, it's unbiological. So, the last, the last instead in 1838 introduced a modification of this model by introducing a term which stops the infinite growth of the population. And it's called, in fact, introduced the concept of carrying capacity. So, the environment cannot support, any real environment cannot support the growth, the infinite growth of a population. And, in fact, introduced the concept of logistic growth. It's, OK, this is a typical behavior of a population, a very simple behavior of a population, OK? So, the population grows until this term comes into play, comes into account, which stops. So, we renormalize the growth, OK? Very simple. I think you have seen this object many times. On the point of your chaos, this model, the continuous model is very unusual, unusual because, of course, trivial, because it's a one-dimensional continuous times system. And for, as I told you, for one-dimensional, according to the Poincare-Bendingson theorem, you can't have chaos in one or two dimension, OK? Only fixed point or limit cycles. But the interesting things, of course, if you take the discretized version of the logistic map. I wrote it. The discretized version is written here, so you have just t plus 1 is equal to r parameter x of t multiplied y minus x of t, OK? And this is the logistic map. It's a one-dimensional map. And it's interesting because, why it's interesting? Because for some parameter r, it develops a very complex behavior, so complex that Ulam, one of the ancient mathematicians, who also worked in chaos in ergodic theory, decided to use this map as a random number generator for the values r equal to 4. He proposed to use this map, the random behavior of this map pseudo-random number generator, OK? You can study, for example, I suggest to you to see this review by Robert May, Sir Robert May in 1976, and this is a very, very nice review on the logistic growth. I suggest you, I invite you to read this very nice review. OK, the logistic map has to fix a point, of course. I remember that the fix a point is obtained by, for a map is obtained by studying the equation, OK, x star. And you can study the stability by considering the derivative of the map, the modulus of the derivative around the fix a point. In this case you have 2 and r minus 2. And you can study the behavior of the map at varying, at varying, the parameter r is a control parameter. So study the behavior of the map, of logistic map, at varying control parameter, control parameter, parameter r, OK? Now, suppose that r is, now, let me write the map. So this is the unitary square, OK? 0, 1, 0, 1. You have, this is the bisect, this is the bisect in line, of course. And suppose that you start with r less than 1. If you start with r less than 1, you see that there is only one admittable fix a point that is 0, no? And you can see this by making, for example, either by studying the stability, of course. But also if you iterate the map starting from a given point and the iteration is a procedure, you take your initial point, you arrive at your function, no? And then you move to the bisect in line and then you iterate the procedure, you go again from this point on to the curve, you move toward the bisect in line, and so on. In fact, this procedure is very simple to see that goes toward to 0. So 0 is the only stable fix a point, OK? With r minus than 1. Now, increase r in the interval 1 and 3, OK? Now, what happens? The bisect in line intercepts your curve, OK? So another fix a point appears, OK? Another fix a point appears. And this point is stable. And this table you can see either by iterating the map. In fact, you arrive at r, but also making the stability analysis, no? OK, the fix a point, if r is in this interval, r is positive, so the 0 fix a point becomes unstable, while the other has a slope that is negative, sorry, it's negative, it's less than 1, so it's stable. In fact, if you iterate, you see that the iteration procedure gives you the behavior, so it brings you to the fix a point, OK? Something new happens, very interesting, if your parameter, your counter parameter is greater than 3, equal or greater than 3, OK? In fact, if you make the iteration, you see that something new happens, and you see that there is an oscillation behavior. It means, what does it mean, the oscillation behavior, that there is a limit cycle of period 2, OK? So there is a bifurcation, I would say, no, from a stable fix a point into a limit cycle of period 2. The fact that the map is very simple, can you explain analytically why this happens? In fact, thanks to the simplicity of the logistic map, you can extend the linear stability analysis to the period root orbit, and the period root orbit is the stable points of the second iterate of the logistic map. What does it mean? It means that, sorry, no, I would like to, OK. If this one, this is the second iteration, means that you have to replace this object, no, xt plus 1, is this object with the map. So you have to put, substitute this with, again, rx1 minus x, but also in this term you have to substitute rx1 minus x, OK? Of course, that depends on t, but it's boring for me. OK, I replace x with the first iteration, OK? It's clear, it's OK. This is a polynomial, second, of course, I have to study the fixed point, of course, of this equation, OK, to study the stability, and if I do this computation, and it is quite simple, it is algebra, it is simple algebra, you see that the fourth degree polynomial, that fourth degree polynomial has, admits four roots, this one has the old stable points, of course, no, they survive, of course, no, because if survives at an iterate, they survive also at the second iterate, because they are fixed point, of course, there appears two new fixed points that becomes stable, and you see here what happens, this is the second iteration of your map, so that polynomial basically, you see if you intersect the, sorry, if you intersect them in the bisecting line, you see that there are two stable fixed point, that are those, and the others are becomes, and this one is this one, becomes unstable, because the slope is larger than one, and also, sorry, but I have to, I would like to switch it off, sorry, and so, two new fixed point appears, what does it mean? You have these two new fixed point, and the motion way around between them, so you have, the motion is, you arrive at x3, for example, after some transient, x3 is mapped into x4, x4 is mapped to x3, x3 is mapped to x4, and so on, and this is a period two orbit, of course, and this is the, at the nice things, that this bifurcation, so the behavior from one fixed point to two fixed point occurs at the values of r3, so at r3, in fact, x3 coincide with x4 and coincide with x2, but x2, you don't see it, no? You see that at r equal to 3, you see that one fixed point becomes unstable, and two new fixed point occur, but these two fixed point and the behavior of your map oscillate between these two fixed point, way around. But this is not the end of the story, of course, because there is a lot of bifurcations, the bifurcation from r3 you have the bifurcation of period two. Period two remains stable until these values are two. From r2 to another point r3 you have a period four and then you have a sequence of bifurcations, so in the, for example, in this range of rk and rk plus one you have a period 2k, the change, of course, of stability is known by forkation, but I think we have already the cascade, this is called a cascade of bifurcation, cascade of bifurcation, so there is a sequence of bifurcation and here you see the table, this is the period of your orbit, this is the, sorry, this is the number of bifurcation, the index of the bifurcation, this is the period of your bifurcation, this is n equal to one, you have period two and at the point r2 where you have this value you have n equal to two, so you have period four, for example, then we have a period eight and so on, and this is the number of the bifurcation point, where the bifurcation occurs, the values of the parameter are, where this bifurcation occurs, this is delta, this is a parameter, the difference between two bifurcating points to next bifurcating point divided by the previous bifurcating point, the distance between two bifurcating points divided by the the distance of the previous bifurcating point, and this quantity is important because it seems to converge, in fact, you see this table, it seems to converge to some point, to some value, here I see there is a picture of the bifurcation point, for example, you have this one, this is the fixed point, this is double, you have the period doubling, of course, then you have another, you split in four, this is period four, and so on, and you go to the set of period doubling, the session of period dabbling. The important thing of period dabbling is that, of course, the sequence has a limit in one constant, you can prove it very rigorously. This sequence has a limit, Fagenbaum in 1975 highlighted some intriguing properties of the period dabbling, the universality of the period dabbling, and of course, he defined this quantity, okay, and this is considered the first Fagenbaum constant, the ratio converges to, the succession of bifurcation point converges to this point, but if you make this ratio, the ratio converges to this quantity, that is called the first Fagenbaum constant, but there are another universal feature of, there is another universal universal feature of the bifurcating, the period dabbling bifurcation. If you consider this quantity, for example, this quantity, this is the line one half, and you consider this segment and you construct from this one half, another segment, and you consider this segment, this segment and so on, so you consider this set of successive segments, you see that even in this case, you have a, sorry, even in this case, yes, sorry, you have a limit, so the ratio of these segments converges to this quantity, that is called second Fagenbaum constant. So, sorry, the value is one half, this value of one half is called the super stable orbit and this is important, but okay, the definition one half for this quantity is called super stable orbit, but you can find in every textbook, so it's a way to define this quantity. Alpha and delta are universal constant, no, characterizing the period doubling cascade for all quadratic unimodal maps and this is the proof of Fagenbaum, he uses a normalization group for doing that and he says that not only the logistic map, but any maps which locally has a maximum, suppose, no, suppose another map, but the condition that this map should have a single maximum, unimodal, it's called unimodal, this is bimodal, no, this is the bimodal, no, of course not, so all this class of map through the normalization group analysis Fagenbaum was able to prove that this scenario is universal and these numbers are universal, which is delta n, delta n is this delta n is the segment, so you take line one half, the super stable orbit, this is called the super stable orbit, you take this line and you consider the length of this segment, you can define the length of this segment at every bifurcation bifurcation point and so if you take the ratio of these segments, the next segment with the previous one, the sequence of these values the ratio converges to this universal universal quantity that is called second Fagenbaum constant it's ok professor? so if we do this procedure for every unimodal map we obtain those numbers yes exactly and it's a computation of renormalization group it's a universal characteristic of unimodal maps every unimodal maps have these numbers ok thank you again the universal is ok so the universal is larger than the ok but you can have also period doubling in ordinary differential equation system, so continuous time systems and probably because under the behavior of these systems there is a hidden structure that is the return map is similar to the if they basically if they motion if you take the return map or for example Poincaré map and you obtain unimodal map probably you fall in the universality class of Fagenbaum scenario and so in some cases it's possible to have these values for ordinary differential equation continuous time systems there is of course another universal scenario that you can obtain with maps with non quadratic maximum like this, this is a typical this is a casp also for this you can prove something similar to Fagenbaum scenarios but now the values of the parameter alpha and delta depends on z, so these maps are like this with a casp also for this you can prove something like Fagenbaum approach and you see there are two again the constant alpha delta, but this universal constant may depends only on z I think I finished this part ah, no this is the bifurcation plot of the of the map of the logistic map there is the the period doubling behavior you see this period doubling stops at the values of Fagenbaum at the Fagenbaum constant and you see that after that the maps behaves a complex behavior a chaotic behavior this chaotic behavior is interrupted by stability stability windows in which Fagenbaum scenario replikates and so on this is the typical object that is obtained to have this object you have to iterate your map discard a transient and after this transient you plot here all the state visited by your map ok and you change R and then discard the transient and you plot again the state visited by your map and you get this nice picture ok, this is called bifurcation diagram in internet on the web you can find a huge amount of picture very fancy and finally I will discuss the behavior of the map at the ulan point R equal to 4 where chaos appears and you see for example that the behavior of the map becomes a very unpredictable very strange at R equal to 4 the ulan point no stable periodic orbits exist no more trajectory looks random and there is no more periodicity ok, there is no more periodicity actually what happens actually there is an infinite set of unstable periodic orbits of infinite even of infinite period you have a lot of unstable periodic orbits so your system goes from one periodic orbit jump to the other R because this is unstable and you have a total periodic orbit of infinite of infinite period ok and what is interesting that if you look at this picture I don't see if you can appreciate it but this picture in this picture in this graph there are two trajectories starting from a very nearby initial condition in some trajectory you make a small error and you change the initial condition by a small amount and you iterate the map so you have two initial conditions that initially started very near after some while you see that until here you are unable to distinguish the trajectory but after some point you can see that they are ok and this is exactly the effect of sensitive dependence on initial condition one of the signature one on the landmark of chaos ok and Volpiani next week show you how to characterize these these characteristic they are right dependence on initial condition this is the typical the the typical signature of chaotic systems sooner or later to trajectory will diverge starting nearby very very close very very close sooner or later diverge like this but since the system is bounded this diverge can last forever but sooner or later you have to these curves must bend and this but if they come close they diverge again and so on and this is the origin of the complex behavior of chaotic system they continue stretching and folding stretching is this one they stretch they are stretched because the distance is increased but sooner or later you have to fold your trajectory because the system is finite and you come back and then you diverge again and so on and this is mix your trajectory in a very complex way in a very predictable way ok I think we can stop here and just relax for a while next model chaotic model will be the Lorentz model ok if you want you can have some relax are monotonical increasing functions consider as unimodal maps and if so will these also exhibit ok, the guy I think I found the answer but maybe you want to add something ok, the answer no, yes maybe the answer is that I make some the final function scenario means that your map your unimodal map at some point arrive to repeat ok, you have you have increased the behavior of bifurcation so consider that R change the height of your map but it remains a unimodal map no so you have you have to increase R there is a set of these values of these control parameter until you arrive at the values r infinity it is called where there is an infinity period an infinity period orbit ok I don't know if I reply moving R you are moving the the height of course of the map you are changing the map until to arrive at the fangenbaum fangenbaum point fangenbaum limit is that correct? he said yes then there is another question are there examples of universality classes where going beyond the linear renormalization group analysis show bifurcations there is not linear renormalization group analysis it's a renormalization group analysis because you have to ok, I have to enter in details but there is a you have a flow you have to construct in your renormalization group analysis what you do you make a change of your map and you try to fix the fixed point of your changing of your maps but this is not linear not linear processes you have to construct a class of maps depending on a parameter and you have to try to search a fixed point the fixed point develops universality properties ok but this is fixed point of the functions not of the state so you are trying to to change the functions in other way you have you try to make a transformation of your function in other way in such a way that you have a certain point unimodal unimodal shape and when you arrive to the fixed point you linearize around the fixed point and you see that the Faginbaum scenario appears and the other question it's ok, can I go on? ok the other system that I want to discuss it's a continuous time system and it's called ah no sorry we are we finished with this ok we have finished with this ok sorry I skip this ok the other system that sorry I have to rush because otherwise I can finish the lecture the other system that I would like to discuss is called introduced by the very famous meteorologist Lawrence in 1973 and Lawrence investigated this is called Reilly Binard convection in the atmosphere basically the physics of this problem is that suppose you have anode plate fluid and a code plate ok and this the hot plate can simplify for example the presence of land of ocean the heat your the air and this cold part this cold plate may represent for example the higher layers of the atmosphere are very cold and so you have for example and the question is how the fluid behaves in these two in between these two source of the hot source and the cold source basically what you have of course the upper plate is colder and the lower plate is colder than the lower plate so buoyancy the fluid here is hotter so is also with lower density and goes up to the the cold layer and you see this type of behavior but this tendency is contrasted by viscous and dissipative force anyway in any way conduction state occurs so you have for example the transport of heat through the fluid from the hot plate to the colder plate and this is the conduction state but if you increase the temperature between these two plate what happens the conduction the conduction state is no more stable and a new state of counter rotating vortices appears so what happens if the temperature is enough so the hot fluid goes up but the colder it reaches the colder state and becomes heavier and goes down and these behavior start to give rise to rolls or counter rotating vortices there is not conduction but convection behavior so the fluid starts moving making rolls if you increase just a little bit your temperature between hot and cold plate you have that this state is no more stable and something that is very unpredictable since the motion is very strange you see that the beaver or the fluid is no more so simple to model this behavior Lawrence make a very crude approximation of this phenomenology introduce basically he expanded the Rayleigh-Benard convection partial differential equation because of course is a problem of partial differential equation into three dimension into Fourier modes and then he retains in this expansion the only three first modes hoping that in this process the best part so the main part of the phenomenology could be preserved in fact Lawrence wrote and this is the equation that Lawrence wrote where xx is the intensity of convection y is the temperature difference in place and z instead is the deviation of the temperature from the linear profile because if you have a simple a simple conduction of course you have a profile a simple temperature profile from a simple temperature profile from the hot from the cold the hot is here and so this is the length of the plate the length of the distance between the plates so you have a simple temperature profile in conduction state and the deviation from this profile define the variable z sigma r and b are dimensionless positive constant sigma is considered a prandle number basically is the ratio of the fluid viscosity and the thermal diffusivity so thermal basically the two physical phenomena entering in this in this model and r is a renormalize temperature difference and is a control parameter basically so it's delta t and the parameter that r is related to delta t is your control parameter basically is proportional to delta t the difference between the of the temperature of the two plates and b instead is a geometrical factor explaining that the fact your geometry is the one that I show you this model you can use of course you can study the stability the linear stability analysis of the Lorentz model and you see first of all that this model is dissipative in fact the divergence of the field is negative because sigma b and are positive and so the phase space volumes are uniformly contracted so it means that you arrive, you start from a three dimensional space but you arrive in something that is no more three dimensional space basically is something that is related to two dimensional for example so your volume you are arriving on a tractor that is no more three dimensional and the fixed point are of course obtained by this equation and there are if you make the computation there are zero is a fixed point and there are these two stable fixed points are zero in the state of conduction so the first plot the conduction is the fact that that you have a transmissional heat from the hotter to the colder to the colder plate so all the conduction and instead of star plus minus plus minus are the two clockwise antikof rotation of the convective rolls if you remember this plot at certain point you see that there are rolls and there are two possibility of rolls two rotation two rotation state of course clockwise clockwise it's quite simple questions on that for the moment and then you can study of course the bifurcation as we did for the map the logistic map we can do the bifurcation behavior in this case it's not so simple but it's but there are some features for example for the of course r is the control parameter I use r as the control parameter r is my control parameter I fix the other and I study the behavior of your system by varying r at varying r we can have different scenarios of course for example from zero between zero and one you have that the conduction state the conduction state is stable ok and the other two fixed points are complex so you can see the two other the convection state you only see the conduction state ok so start becoming greater than one what you said that r zero loses the stability as one again values become zero larger than zero and this the other the other two fixed point becomes stable so conduction is conduction this is the conduction becomes unstable and is replaced by the convection so two rotating rolls appear several rotating rolls appears basically in this range of parameter the gain values the gain values part of the of these two start to becoming comps and conjugate but the steady the steady convection remains stable ok r r star plus minus remain stable at a certain point at a certain point for r larger than this critical value that is defined by this quantity even the steady convection state so these two fixed point becomes unstable ok as something new appears what appears ok it appears something that is very strange because you see that the attractor the motion is attracted by this very simple very simple no very complex behavior and this is called the Lorentz attractor ok ok, the characteristic ok, this is the values at which the the Lorentz attractor appears basically and of course is above the the critical value rc above rc rc is this value if you make the computation if you put if you put 10, 8 and so you obtain 8 third you obtain of course this value no if you if you stay beyond these values and this r is 28 you observe this behavior this attractor ok and what do we see, for example we can do two numerical experiments in this case you have to study the behavior of your system in numerical analysis numerical integration of the question of motion no, what you observe ok you observe that, for example if you start from two initial conditions that far apart they after some transient are attracted on the on this strange geometrical geometrical object the Lorentz attractor but if you start to initial condition they are very close you see that for example these two trajectory again start start nearby and after a while 15 they becomes different and the way and the way this happens if you plot the different in time between these two trajectory now you see that this error between these trajectory is small start increasing linearly and then saturate until these two trajectory rolls on to the attractor and this is again the sensitive dependence of initial conditions the signature of sensitive dependence of initial conditions ok some questions this is the fact that you remember that I told you that chaos generally appears if you look the stable and the stable manifold of fixed points of unstable fixed points and in fact if you construct the stable manifold of the fixed points of the unstable fixed points these are the two unstable fixed points you see that the attractor lives there ok the attractor if you want to search the attractor the attractor stays in the unstable the unstable manifold ok ok as we done for the standard map for the standard map for the logistic map you can construct the verification plot of the Lorentz attractor for example you have again the control parameter you fix these values of the other parameters of the model you find the numerical solution for example up to a time of 300 times steps discarding of core transient and you plot are according to z max for example if you decided that your your evolution has a behavior like this z if you take the z coordinate of your model this is time a trajectory of z should be something like this you take the maximum ok you plot all this maximum that you obtain in your trajectory after discarding of transient and you see that you have this typical plot where you have the this is the stable fixer point the bifurcation that I discussed at the beginning and then after that bifurcation you observe chaotic behavior you see even in this case there could be some stability windows so the system becomes stable again because becomes non non chaotic and this is the typical the typical bifurcation plot of Lorentz model ok just this is the last slide about the Lorentz model so we see that for example in this case you have that a separation ammunition condition grows exponentially in log-log plot you see that in log-log plot sorry that linear plot so maybe I did a bit this is log-log plot you see this is log linear plot sorry so this behavior is exponential because it's linear in log linear in log linear plot this is the separation of the trajectory is exponential ok and this means that if you start if you take the two trajectories at the beginning you have the propagation of the error becomes grows exponentially in time ok till to reach the saturation where the attractor no till this behavior stops when to trajectory arrives on the attraction question sorry ok and ok also Lorentz system show a radical evolution of trajectory associated with sensitive dependent of initial condition and the typical equation of the sensitive dependent of initial condition is this one an exponential divergence of nearby trajectories another evidence of the chaotic behavior of the Lorentz map can be obtained by considering the Lorentz rated map sorry the Lorentz system, the Lorentz model can be constructed by using the Lorentz the Lorentz rated map basically Lorentz rated map is this one you have for example considered z z is the behavior of the trajectory the component z of the trajectory you see that you obtain the maximum you take this maximum and if you plot this maximum the maximum at time t at time n against the maximum at time n plus 1 you obtain this typical typical map and if you see this map is something that ok is is a map which has unstable behavior ok and this map is if you for example you are able to fit these two branches of the map with some analytic function you have that you see that this map develops chaotic behavior ok and this is another signature of the fact that the Lorentz attractor is has a chaotic behavior ok the rated map this is called Lorentz rated map ok and I finish the in this lies there is also derivation derivation of the but I don't have time to show how Lorentz derived the equation is model from this partial differential equation using the Fourier's function and truncating the Fourier's function to a given order ok to the lowest order basically ok the other system that I would like to discuss is the last system that I would like to discuss is the Hannon-Heilz systems is another historical model and it is an Hamiltonian system and that to the conservative dynamic a simplec structure you remember that Hamiltonian systems are sympletic systems you remember by now you know what the meaning is you see that the question is how chaos manifests in Hamiltonian systems the model was motivated physically motivated by the fact that Hannon-Heilz wanted to study how a star was moving around a galaxy center with axiosymmetrical symmetry axiosymmetrical symmetry and the question they want to investigate through this model is if there is another integral of motion except the energy and the angular momentum the angular momentum to the fact that you are rotating in a galaxy ok the question is is there any integral of motion except energy because the system is conservative the angular momentum due to the fact that there is a symmetry of course so the question is because they want to understand if this possible integral no constant motion was able to restrict the motion of your trajectories so for example if you move on a surface of an energy surface you are constrained by your energy conservation of course your system moves on the hyper surface of energy conservation if there is an angular momentum there is another constraint of your motion in the intersection of the energy surface with the surface defined by the angular momentum conservation if there would be another integral of motion there are intersection of three surfaces ok so the question they wanted to answer this question the model is very simple so you have this two particles moving sorry a particle moving in a two dimensional so the system has two degrees of freedom this is the potential the potential is written like this ok and if you start to make the analysis of this model you have Fabio sorry may I ask a question so they constructed this model enforcing a third no they are asking if there is a hidden hidden symmetry so that a new constant of motion appeared ok ok actually for example the star explore a three dimensional constant energy surface in four dimensional space the physical train is this one and of course not only all this surface is explored but this part is explored because if you if you make if you have this one if you have the energy that is if you discard for example you have an equality of course so not only the surface, the energy surface is explored by your system of course there is a sub sub set of the surface explored by the star so the original question if no other isolated integral of motion exist the above region this one of non zero volume of course because there is a surface in a four dimensional space of course that by a single trajectory of the star so when you have this set and this set would be filled suppose that this set is a portion of your surface no if there is another integral of motion maybe you have again some limitation but if you don't have any integral of motion now the trajectory should wonder in your surface like this you could explore every part of this sub surface and so what you can do of course now you have four dimensional system the constraint of energy conservation reduce the dimension of your system to 3 but of course even in three dimension it is very difficult to see the motion to describe the motion what you do you make the punkary section and the punkary section that you want to construct is the punkary section where x is equal to zero and p of x is positive so the intersection comes from the positive p of x ok now I count only the intersection with positive p of x in this way a four dimensional phase space it reduce a two dimensional phase space of course this is the advantage of punkary section before analyzing the punkary section is better to say that anon high cementonian is the sum of course of harmonic part this is harmonic part plus something that is an harmonic part and this part plays the role of perturbation this part is an integrable system so you can write this in terms of action angle variables do you remember yesterday I discussed you that when you have integrable system this is equivalent to the motion on a torus ok because in the action variable representation your dynamics lives on a tori ok so this is the integral part and this is the perturbation ok and you can consider for example this as a deviation from your integrability ok so you have a perturbed basically a perturbed integrable system if you make some analysis of your potential you see that the potential has a very strange behavior no it's a I don't know how to can you see, can you imagine from the picture it's a well there are three escaping possibilities no so they are saddles you see that here there are the saddles ok, this is the fixed point the stable fixed point in fact if you make the analysis the fixed point analysis you see that zero is a fixed point but there is also this one and these two other ones scheme of the distribution of the fixed point and this is the stable fixed point and this is the saddles and the nice things that these saddles at the values e equal to one sixth you have that these saddles are connected by these lines that are triangles the defined triangles when u is equal to one sixth they are the equation of motion of course the Hamiltonian equation of motion that you can derive by deriving the Hamiltonian using the the rule of the Hamiltonian equation and now we have to study the Poincare section what you observe so this is the Poincare section the section is at x equal to zero at different values of a integrability different values of v means that you have deviation from the integrability and ok you have to the initial condition of Pox is given by this one by the constraint of energy conservation if you integrate your equation of motion you see that two different scenarios can be possible this trajectory this is a torus ok this torus intersects your section x equal to zero giving rise to this kind of plot and this means that the motion is this a signature of the motion what kind of motion is periodic periodic motion but of course if you change the initial conditions maybe you change I don't remember how did I obtain this plot but maybe I change the initial no, the energy the energy because no sorry, in fact sorry this is the integrable condition but if I increase the energy of course into a non integrable initial condition in fact you have the appearance of chaotic behaviors in fact the intersection with the Pankare section is very random and this is a signature of the fact that the values of the energy the values of energy I remember that is the deviation for the integrability basically the higher is the energy the higher is the deviation from integrability if you increase the energy you see that your Pankare section is this one you can do this for several values of energy for example three you can use three values of energy this for one divided by twelve for example and this equation basically is stable because you have closed orbits generally closed orbit in Pankare section are stable trajectories as I showed you in the plot before and so there are very clear but you see that here near the near this this the saddles you see that something happens in fact you see that these curves are thicker they same thicker if you zoom for example this point you see that some signature of chaos appears and in fact chaos appears where where there is saddles saddles in the Pankare these are saddles in the Pankare section not saddles in the original in the original model be careful in fact the intersect you should understand this is a map because the plot intersect because in the real space you can have crossing of the of the trajectories because because according to the existence and uniqueness theorem you can have crossing of trajectories otherwise you violated the deterministic nature of your system because I repeat this this is important important statement existence and uniqueness and existence theorem says that you can have crossing of trajectory because if you move here if you arrive at this point following the flow then what you have to do you can go this way or this way but this is a violation of the determinism there are two futures not one future but this is not true for maps no this is not true for maps in fact in Pankare maps you have crossing you have crossing in Pankare maps no ok now if you increase for example the energy a little bit you see that there are island so there are even periodic solution still survive but the sea increase so there are lots of randomly distributed points so the intersection with the plane start become randomly and you see so there are island plus the coexistence of stable trajectories plus chaotic trajectories if you again increase the energy so you move away from the integrability you see that there are lots of chaotic trajectories chaotic points and only few stable stable trajectories and this is why and this is how chaos in Hamiltonian models appears ok and the fact that you have this picture concludes answer to the question that there is no another integraval motion in fact your trajectory wandering on the allowed phase space and there conclude I think I concluded this part thank you thank you it has been a pleasure and I hope to be I was comprehensible