 Zdaj, načo vsega rekača od vsega lečenja, vsega najbolje vsega tudi vsega pridljamo ponkare, ponka re, in stroboscopična mapu, da vsegače dinamikovati sistem, in da vsegače poslednju kaos, In je to vse reprezentacije, ki njič ne se vse nekaj načinjali. Zato je spetno, tako je kaotic, da se povsem, da je povsem kaotic. Zato je povsem informacij, da je zelo lahko. Se svojte lahko vzaš z načinje, z witchjama, vsi je stajno počakvala. E點, ki je počakvala. Z travelần jazam that, našli sem genel vanen imam, da sem res vsem, našljenje squares, in tudi eno bo, da sem je to bo, neveda, ki so bo, neveda, našljenje squares, in je zelo, da se je vzelo, in da se je vzelo, da se je vzelo, še je ta stručenja, na kvama, na daj na rečenji, in je to dobro, zelo, da se je način, da se je način, da se je začniti, da se je začniti, in je to, da se je začniti, način, način, transformacije. In transformacije kononičnih je zelo vsega tehnika, kaj je zelo tehnika generacije. Zelo vsega nekaj? Zelo vsega nekaj? Zelo vsega nekaj? Zelo vsega nekaj? Zelo vsega nekaj? Zelo vsega nekaj? Zelo vsega nekaj? Zelo vsega nekaj? Zelo da govorila do stran delom zelo to je vzelo, da je vzelo. Nisem zelo, da je to močnjega, ker je nezelo, nezelo bi ne bo zelo, ali ni se neglade. Ne začnevoj, da je zelo, ali je zelo. ... če še, da bi онежne izračiti izračati... ... rečne poživna, točne pa, ovo, da je pa... ... neko pač, ovo je, kot kde služeno. ¨ Here we go. ¨ ¨ These... this... this... ¨ This... this... this... the program is very simple in ... ... one dimension, mm? Because in one dimension for example If you have a... sestem, kaj presev te energije, danes lahko se počutite v tem, kako je potenšel. In, kako se počutite, kako je potenšel, in kako se počutite energije,こと, da je tukaj nače. Vse si speno se, da je ta temoč del, je ta temoč del, ki ti se, tu je tukaj nače, nije skorod naredo, tudi svega ideja si pripoče, v zelo tako svoj, jasno, je to inoč dob griza, pojad tukaj ne bom diaz, tukaj ne ima pripočen obiet, pojad nače, tukaj te ljubi ki bi, ki tej nače, tej nače, There you see, this is the liberation and rotation of this pendulum here at this level, this critical level, is called the separator, which separates the two kind of motions. It is very simple for a system in one dimension. What kind of motion is this? What kind of system is this? This is the same phase portrayed of this one. Znamo se, če je to? Znamo se? To je potenčnja, potenčnja. Partične vrste, vrste, vrste. To je... Vrste. To je postavno, partične vrste, vrste. Vrste. Vrste. Vrste. Vrste. Vrste. Znamo se. Što je? Quarter. Što je? Što je? kako je infinitiva, zelo to je objavljena objavljena. Vsih objavljena, kako je in energija, vsega vsega, kako je vsega, vsega, kako je vsega, kako je kratična vsega, kako je vsega, kako je separatrič, kako je vsega, kako je vsega, kako je vsega, navodilo. More difficult in the situation to understanding the situation, the quality of the situation in more than one dimension. Two-dimension, three-dimension it is already affordable, but he higher dimension of course there is very difficult to be understood. For example in two-dimension what you can do is where is vector field and To je dela všechno-vettorku, je zelo vsezvanele, što se se odliče vsezna, bo je odliče vsezna. Kajวess, da boče, da se potrebe s pismo vseznjane, to je za učin, nekaj je však in izgledaj se in vsezna, da srezne vsezna. Zazvalo se, da so tako deločne, na vsega vsega, da se površila, načo je ta nekaj dinament, in vsega se ga tudi odrečiti, načo je, da bi je trajaktori, da se začne bi, da je to vsega trajaktori, bo trajaktori, da nekaj je trajaktori, načo je trajaktori, da nekaj je trajaktori, neskrič, ko je odstavljena. Však je, da je vsemoj počak, je to občatno, da je vsemoj način pa da je odstavljena z neko odstavljena, da je boj delarizena. In da je vsemoj način, da je začne odstavljenje. Prejdeš, da jošša izgleda noženja na konceptu nulklijenja? Nulklijenja je to, da je začala, da je začala, da je začala, da je začala, da je začala, da je začala, da je začala, da je začala. If you aren't able to make this analysis because your space is a higher dimension, you can, of course, you have to consider the fixed point that are equilibrium state of your of your of your motion, the fixed point satisfies this equation for flows and this equation for maps. It means that you remain in a fixed point, of vorever in that state. z vsej pridljaj, da jste priči. Ale pa, da jste vse vsej pridljaj, da vsej pridljajo, da nekaj možda je začal tega, začal tega, ki je zelo. Po izgledu, ne? Kaj je možda? Jeste poču, ki so priče vsej pridlj. Jeste poču, ki se priče vsej pridlj. Kaj je priča? Kaj je priča, ki je priča? To, ki izgovorite iz vseh domne, zelo jaz vse je nekaj nekaj nekaj nekaj, zelo jaz vse je na prezat, na radočni, matematikni, Aleksander Mikhailovich Ngliapunov. Osprej, izgledaj začetne stabilitve. Ako zafrešte stabilitve, nekaj ne nešte najbolje. Przelaš je tudi bolo stvari. Tudi je tudi stvari, to je bilo. Radjuz Delta. To je tudi stvari. X star. Tudi je stvari, delta. Zato, da se in tudi stvari izgleda, da se pravno vseče, da je stabil nekaj. Vseč, in to je druga sljeda. Vseče, da je. Vseče, da je stabil nekaj, Zdaj. Premaš, kako potražite, kako zdi expertsom prišla počke, čistmenje, representično, nešto, stranje, posledno, primizučno. OK, zato moram pokazati drugo lahko izpersim t파, kak الo so vignetite starring na vse epsilon, tako, da, kaj ta vse trajžaktor je vse zelo vzela v delta epsilon, tako, da moja moja moja moja moja moja moja vse epsilon. Zelo sem vzelo in tudi je srečen. Navrša. I to je tukaj. Zelo si, da echo, da si sem vzelo, da sem vzelo v termine delta, zelo se je, da se demeti. Moje objevak je, da sem vzelo, da si, da mi pomega, The fixed point, less than delta, will remain on the ball epsilon forever. The other notion that Ljaboljov introduced was the notion of asymptotic stability. Thus, you can define a delta, so a ball delta, such that if you start within delta, Je tudi nekaj razlogi, zato jao je, da je razlogi, zato je izstavno. Zelo je, da je izstavno. Zelo je zelo naredito. Zelo je asymtodist, zelo je. Zelo je stavno. Zelo je asymtodist. Zelo si, nekaj da so stabili, nekaj da so stabili, vse smo vsebežjene in stabilite. To je tebezmatematijne definicije, več hodnje definicije stabilitivi, ali zelo tudi bolj, da imajo našličenje, in da se našlič je operativne klj累erije in pa jaz sem način na stavo. Pasko. Ok. And the criterion is, for example, very simple for one dimension. No? The one dimensional system is very simple. In fact, this is your equation of mode. Your equation of mode. So, yes. You plot f of x and if you decide, if you look at the sign of f of x, you can always, also you can soon decide if your fixed point is stable or unstable. Of course, this f of x crossed the x-axis in the fixed points. In this case, f of x is positive, so the field goes this way toward this fixed point. Here is negative and the field is negative, so f of x is negative. So, I go this way and this point is stable, of course. In this sign, instead you have the field is negative because f of x is negative. So, I point this way. Here is positive, so it points this way and this fixed point is unstable and so on. But there is another criterion. Yes? Yes, please. Is it possible to visualize the notion of asymptotic stability with this kind of graph? Of this one, which kind of the asymptotic stability of this in one dimension, you mean? I mean, asymptotic stability, maybe you have, because you have point and something you should go, probably you should have something that's going, I don't know, I'm guessing, OK, I think on that better, but maybe something that is going like this. And I don't know. For the moment, I don't have any example, but I will think about that. Thank you for your, OK, I will think about that. Yes, so another, so, but the, OK, of course, this is of qualitative analysis, but the real analysis, the real mathematical analysis is done by considering the derivative, of course. In fact, the slope, you see, the slope says that this, if the slope is negative, the fixed point, the slope of f of x, if the slope is negative, the stable point is stable, so, if the slope is positive, of course, this is, the fixed point is unstable, OK. Now, what happens in two dimension? And I, in two dimension, the situation is a bit, is a little bit complicated. In fact, you have to study the linear stability analysis, what you do. Basically, you linearize your dynamics around the fixed point. And how can you do this one? You have your equation of motion. You have your equation of motion xi equal to fi x1 xn. You make a variation, so a linearization with respect to, linearization means that you are considering xy with respect to the, and this is called delta xy, no? You are considering this quantity. And you want to write the equation for this quantity. And the equation for this quantity is quite simple, because you have, make the derivative, so delta xy is this dot, sorry. Delta xy dot is something by, nothing but j, the f with respect to xj evaluated at the fixed point, multiplied by the variation xj, OK. This is the, OK. Basically, I do the differential, the differential of my equation of motion. OK. In terms, in matrix form, you can write this delta of x is equal to L of x, where L, of course, is the stability matrix, is the matrix with entries, with these entries, OK. So, L ij evaluated at the fixed point, is nothing but the derivative of f ij with respect to f ij with respect to j at x star, OK. This is the definition. And this is called stability matrix. Stability matrix, OK. Once you have the stability matrix, you know what you have to do probably, you saw in the courses. And, OK, it means that you have to diagonalize this matrix. So, you have to find again values and again vectors. That's a diagonalization. Once you have the stability matrix, your task now is very simple. The task is compute eigenvalues and eigenvectors. I'm supposing that you are familiar with eigenvalues and eigenvectors. Every one of you. Good. The eigenvalues could be, of course, real or complex conjugate, because the matrix is real. The problem is real, no? So, you can have either real eigenvector or at most complex, but complex conjugated. OK. If you want to just a little bit explanation of why this is how the diagonalization works, is that what you do generally in this case, you suppose that you have your matrix L. OK. Suppose that this matrix L has a set of independent dimension. You have a set of independent eigenvectors. OK. That means that they form a basis. No? They form a basis. In the independent eigenvectors. OK. Then you can, what can you do? You can, for example, expand your equation, your linear equation, your expand the solution of your equation that you are looking for. In this basis, no? In, for example, you have some k, ck, uk. You expand this one. And on the other side, of course, you have the same expansion. But since the matrix crosses all the scalars, you can write, sorry, this is c dot, because this is c dot. You are looking at this constant. This is the equation. This is expanded into the eigenvectors. On the other side, you have ck, l, uk. But luk, by definition, each uk is an eigenvector. So it satisfies this equation. And so you end in the simple equation k uk, lambda k, lambda k, ck. Now, since these guys are independent, you are allowed to identify the coefficient of this combination. And your equation for each ck is lambda k ck. Very simple. This is a linear equation. The solution is ck of t is equal to the exponential lambda t, lambda kt, ck with zero. Where this constant defines how you start nearby your fixed point. In fact, so the solution is this. And the constants are set by initial condition, how close you start from the fixed point. Uk determines the direction of approaching or of escaping from your fixed point. And luk determines the type of motion around your fixed point. Very simple. Maybe. OK, now you have the classification, very simple classification, two dimension. You consider the classifications. Very simple. If you plot your eigenvectors, eigenvalues, sorry, in the plane, in the complex plane. Because as I told you, they could be real or positive, real or complex conjugate. So this is the image of lambda, real part of lambda. And this is the imaginary part of lambda. Now suppose that lambda 1 and lambda 2 are real. Real means that they are on this axis. If both are negative, lambda 1 and lambda 2, what do you expect? You expect that your motion converge toward the fixed point. This is the eigenvector, for example, the direction of... And your motion is going... Sorry, let me... This is the vectors. The arrows are otherwise... I'm confusing you. So you have both going in this direction, in this situation. And this is called a node. But the node, if the eigenvalues stay here, so no more negative, but it becomes positive, the node becomes unstable. You run away along the direction defined by your eigenvectors. The other possibility is that lambda 1 and lambda 2 are real, but they lie in the opposite part of the imaginary axis. So this way, then what happens? The scenario is you have, of course, the direction defined by your eigenvectors. This one means you are unstable, you run away from the fixed point, but this one says that you are stable, so you run into the fixed point. And this is called saddle. The motion here happens... And this is saddle, because if you consider the surface, you fall from the saddle. So there is one part, your part you arrive this way, but the other side you go away from the saddle. It's exactly like a saddle. So the right picture is... Those are stable, and those directions are stable. In fact, it represents the saddle of a horse. Finally, if you have a complex conjugate, it means that you can have this situation, complex conjugate. The real part of the eigenvalues is negative, so you lie this part of the eigenvalue. So this is mu plus i omega, and this is mu minus i omega. If the real part of the eigenvalues is negative, so you lie this part of the axis. What you have? A spiral, because... Yes, a spiral. In fact, it means that the... Since mu is negative, you exponentially fall into the... You are attracted by the fixed point exponentially, and of course, there are also the rotation to the face. So you have spiraling around the fixed point. This is a spiral, or consider focus, it's generally called focus, but spiral is enough to visualize, to remember this. Of course, the spiral is stable. If you are spiraling this way, but if mu, of course, lies... If the eigenvalues lies in this part, mu becomes... mu is positive, so you spiral away from your fixed point. If finally the elliptic... If you are in this situation, where you stay here, so negative, the real part is negative, what you have is an elliptic point, so you stay around the fixed point, so you can circle the direction, for example, in values. You make a rotation around your fixed point, this is called elliptic point. Of course, the role of eigenvectors, as I told you, gives you the stable and unstable direction of the motion near the fixed point. There is another situation that sometimes in the book you don't see discussed, and when you have degeneracy, degeneracy means that some eigenvalues, in this case, since we are working in two-dimension, two eigenvalues are identical, lambda1 is identical to lambda2. Just a brief mathematical reminder, the situation when the matrix has degeneracy in eigenvalues is called defective matrix, and there are two notions that you have to be reminded. The algebraic multiplicity, so consider the characteristic polynomial, this is a polynomial. If your polynomial can be written in this way, plus another polynomial of degree n minus m lambda, m, so the generacy of the matrix, this is a root over the m, this quantity is called algebraic multiplicity, multiplicity, but to this algebraic multiplicity corresponds to geometrical multiplicity, and the geometrical multiplicity is the dimension, so the geometrical algebraic geometrical multiplicity, multiplicity is the dimension of the eigenspace, yes, the eigenspace spand by the eigenvector is differentiated to these eigenvalues, and there are two cases, of course, either the multiplicity are identical, so m, so generally you have this property that the geometrical multiplicity could be less than the algebraic multiplicity, the algebraic multiplicity. And what does it mean in terms of, so if you, for example, ok, let's discuss in this way, it's very clear. Consider the case of that, the dimension, the geometrical multiplicity is equal to the algebraic multiplicity. Then you can find, this means that you can find two independent vectors, because the dimension is two. You are still able to find two independent eigenvalues, but the solution, in this case, is written very simple in this way. Of course c1 and c2 are the initial conditions, and if lambda is positive, you have the star, the rate of approaching the fixed point is the same for both directions, because it's rude by this law, it's symmetric, but the situation could be even worse. The worst case is when the dimension is one, you are able to find only one eigenvector, and instead the multiplicity is two. And then you have only one eigenvector. What can you do in this case? In this case, you have one fundamental solution to construct your general solution, because the problem is linear. So the task, when you have a linear problem, you have to find the fundamental solution, combine them, and obtain the general solution. In this case, you have one fundamental solution that is simple e lambda t u1. The only eigenvector you can have at disposal. The other solution, you have to find the second fundamental solution. And what you do? You guess that your solution can be written in this way, e lambda t, a vector a to be determined, multiply by t, a vector b to be determined to. How do you determine this? You take this, and you substitute in the equation of motion. The equation of motion is this one, and I would like to leave you as an exercise to show that these two vectors, constant vector, satisfy this uA, b is nothing but u1. So the old eigenvectors, coincide with the old eigenvectors. The second one, instead b, sorry, a, instead a, satisfy the equation L minus lambda, u, y, u, sorry, a is equal to u1. This is called a generalized eigenvector problem. In this case, you can do the analysis, and you have, of course, a different behavior, because consider that now your solution, of course, the general solution in a combination, a linear combination of the two, and I wrote here this here, and the approaching to the fixed point if lambda is negative is this y, and of course, there is a deformation to the fact that you have also a linear part. You have not only the exponential behavior, but also a combination of t and the exponential behavior. And this changes the local topology of your flow around the fixed point. This is a scheme that I give you for, since you, I will lend you the presentations, you can find the summary of this. In the dimension, the situation is somehow similar. It's a bit tricky, but somehow similar, because, of course, you have to compute again the secular equation. The secular equation means that you have to diagonalize your stability matrix. And you have a situation like this. So, if this is the number, this is the set of eigenvectors, this is the set of eigenvalues, so you have a situation where you plot in the imaginary plane, your eigenvector. Of course, this condition corresponds to someone says this condition corresponds to all eigenvectors lies in this part of the plane. Stability, exactly. But there is the possibility that someone of these eigenvalues crosses the imaginary axis. And in this case, you have in stability a saddle in the dimension, a saddle in the dimension. But also there is a possibility that all eigenvectors or eigenvalues lies on the imaginary axis. In this case, you have a center or a marginal point. And this is a definition that I never understood very well, but in when the dimension is even, this center is called a elliptic point. As a elliptic point. But not a change. You have to, for example, to understand the stability, you have to compute your eigenvalues, put these eigenvalues on this imaginary plot, imaginary axis, imaginary plane, and complex plane, complex plane, and you have to see the position of the axis, to the imaginary axis. Finish, stop. Very simple. The same happens with maps, of course, the difference is that instead of considering the whole imaginary plane, you have to consider the unitary circle. In fact, you can see that basically if you solve the same equation, you have that the stability and stability depends on the sides of your eigenvalues with respect to the unitary circle in the complex plane. For example, here you have stable behaviors, stable points, but if you have these objects, you have a stable behavior. Inside the plane, inside the circle, you have stable behavior. Outside, you have stable behaviors. OK? But the classification is basically the same. The only difference that now if you have maps, you have to remember that the unitary circle is the criterion through which you can decide the unitary circle. OK? The unitary circle. I don't know that. You know that in this course there is a lot of time spent through symplatic maps. Because symplatic maps symplatic geometry, symplatic representation because of course symplatic means Hamiltonian. It's a synonym of Hamiltonian. Sorry? In the chart. Where? Yes? Of course. The maps, we use the circle. The circle in sense, yes, you have to put the eigenvalues with respect to you compare the eigenvalue with respect to the unitary circle. Unitary circle in complex plane. Yes, this is important. OK, again, for the symplatic map, you have to compute the stability. In this case, the stability satisfies again the properties of symplatic maps. The stability matrix is a symplatic matrix. That is something like nemonically you should remember that there is some strange no, some strange map some analogy with the orthogonal matrices. OK? What happens before symplatic maps? For symplatic maps I would like to make this statement. No? OK, the symplatic map you have a symplatic stability matrix. Simplaticity. Simplaticity. No, why? Is written. Simplaticity to use a constraint also on stability. And it is easy to prove that if lambda is an eigenvalue of the stability matrix even 1 over lambda is an eigenvalue 2. So, if lambda is an eigenvalue of L also 1 over lambda is an eigenvalue of L. OK? This is a general property of stability matrix of symplatic maps. How can we prove this one? Of course in dimension 2 is quite obvious because you know that the determinant of a symplatic matrix is 1. And the determinant generally is the product of the two eigenvalues. So, if you have lambda 1, lambda 2 is nothing but 1 over lambda 1. And it is simple in two dimension. In dimension of course the situation can be done in the proof can be done is very simple because you use the properties of the matrix the symplatic matrix you write you write a j a transpose j you can write this one multiply by this you have a j is equal to j a transpose minus 1 first you make first these algebraic passages you are multiplying by the inverse the inverse of that OK? And then you use properties of the transpose the matrices you know that if a matrix has eigenvalues a lambda transpose matrices has eigenvalues so if m is a matrix with eigenvalues lambda it's transpose which kind of eigenvalues has with respect to lambda exactly. And the proof is here in fact if you use the polynomial the fundamental polynomial here you make a transpose of that but the determinant of a matrix this matrix and the transpose is the same so at and a cannot have a different eigenvalues so they share the same set of eigenvalues OK? This is very simple now you can say that if one of your symplektive matrices admit eigenvalues eigenvectors e eigenvalues lambda with eigenvalues this one then you can write this multiplied by the inverse you can have one over lambda is equal to a transpose one to the inverse is e one over lambda e I didn't do anything just multiply by the inverse from here to here but then I use this identity in fact if now I multiply for j both members I can use sorry, I am doing a mess with my OK, if I multiply by j OK I can have and use this identity I can write this one in this way a j e is equal one over lambda j e OK what does it mean it means that if e is an eigenvector even j e is an eigenvector 2 but with eigenvalues one over lambda OK some question about this very simple proof is clear in the algebraic what is the implication of that now in geometry in geometry I think there is a lots of geometric geometry about symplectic manifolds the dimension should be only even so symplectic geometry works in even dimensions 2 in fact Hamiltonian dynamics lives in even in even dimension OK that's good question, thank you because baby I forgot to stress this point OK what's the implication of that so the question no what's the implication of that about the stability the implication is sorry I can't go the implication of that of the fact that you have a symplectic matrix is if the stability of a symplectic matrix is an if if you have a symplectic stability matrix that has a consequence that in 2n dimension of course even dimension there are 2n eigenvalues and eigenvalues are paired in this way if you have the eigenvalues here one here you also have after n eigenvalues you have one over lambda i OK and this is called pairing pairing means if you plot your eigenvalues into the complex plane and you consider the unitary circle because we are considering maps of course they are organized in a very strange way in a very symmetric way distributed in a very symmetric way in what is called what I mean what does it mean it means that suppose that you have eigenvalues lambda but you also complex of course a complex the red one but you have also the inverse also this inverse is eigenvalues and this inverse stay here stay here it is very simple to prove plot for example you have this here these eigenvalues stay here but if this is complex conjugate see this is complex you have also the complex conjugate and the complex conjugate of this stay here but there is of course another inverse of that conjugate eigenvalues should live here so this stability this stability of of simplicity maps is quite complicated for example because if you run soon you can run outside the unit circle for example if instead the eigenvalues is real then you have here inside the circle and one over over lambda is like this if the eigenvalues as rho equal to 1 you have just only the complex conjugate complex conjugate behavior of course you have to remember that for simplicity maps the stability the stability distribution as eigenvalues satisfy these strange rules and consider also these nice properties that these two eigenvalues lie exactly are symmetric with respect to unit circle you see also these two one are symmetric but be careful they are not the couple due to the pairing here and here the pairing is here her here but you need some this is a curiosity you saw that the distribution eigenvalues is somehow symmetric with respect to the circle in this way the property pairing is a general properties of Hamiltonian systems in fact Angelo Volpiani show you that the pairing properties if you compute the Lyapunov exponents Lyapunov exponents are indicator of chaos and they generalize the concept of instability from fixed point to a whole trajectory but this is the subject of Angelo Volpiani's lectures ok I will stop here just just with you I will stop how much ten minutes ok questions, comments any curiosity no linear stability next point is no linear stability in a pause