 Okay, thank you. It's an honor to speak at this conference honoring Poincare. Poincare is certainly one of my mathematical heroes. I suppose that's true for anyone in dynamical systems and and many other fields as well as we've been hearing about in the previous talks. So here's a famous quote of Poincare's that I always found as one of my favorite parts of his monumental treatise on celestial mechanics where he's trying to describe his first encounter with chaos. This is my my English translation. So if one tries to imagine the figure formed by these two curves with an infinite number of intersections, each corresponding to a doubly asymptotic solution. So I'll explain more about that later. These intersections form a kind of a trellis, a fabric, a network of infinitely tight mesh. Each of the two curves must not cross itself, but it must fold on itself in a very complicated way to intersect all of the meshes of the fabric infinitely many times. One will be struck by the complexity of this picture, which will not even attempt to draw it. So nowadays one might refer to this as a homoclinic tangle or homoclinic chaos. And I'd like to describe a couple examples of homoclinic chaos in the three-body problem. Simple examples. I'll focus on the first three there, Poincare's particular kind of chaos that he discovered, and then some more recent work. Chaos near infinity and the Sitnikov problem and chaos near triple collision in the Sitnikov problem. And then I'll just mention a few other kinds towards the end. And there's many repetitions have worked on this, including I worked somewhat on triple collision. So I put myself at the end, but there's a list of some, probably not all. So, one of the things I'm going to attempt to do is, I'm going to attempt to draw some of these pictures using a computer. And I just put these guys up because, well, I'm a Macintosh fan. I'm sure if Poincare were around today, he would have one of these computers. And he would probably use Mathematica to do his computations with. So these are two steves of the 20th century that one wonders what Poincare would have done with a personal computer. But actually it turns out that the computer sometimes gives you the wrong answer. It shows you things that maybe aren't true. And it turns out that Poincare really did have a good imagination. And he might have been fooled had he seen some of the pictures that I'll show you. Okay, so here's Poincare's three-body problem, I'll call it. It's the planar, circular, restricted three-body problem. So we have two large masses. The green mass is heavier than the blue mass. And those are the primary masses. And they're just going around on circles and constant speed. So they're just obeying the two-body problem. And then the other mass, the black mass is the third body. And that's supposed to have mass zero. Well, it's influenced by the other two, but it doesn't influence them. So it's moving on some orbit which is being affected by the other two. So one thing you can do, which helps to simplify things, is to immediately put it into rotating coordinates. So this is actually the same orbit as in the previous slide. But now we're in a rotating coordinate system. So the two primary bodies, which used to be going on circular orbits, are now fixed on the x-axis. And you can see that the black particle is really moving on a pretty simple curve. It's an orbit that Poincare would have called an orbit of the first sort. It's approximately circular, and it closes up after one revolution around in this rotating coordinate system. So mathematically, this is a dynamical system of two degrees of freedom, meaning we have two position coordinates. So you're looking at the x and y-coordinates axes here. So that black particle is described by an x and y-coordinate, and also two velocity variables, which I'll call u and v. But those variables are all subject to an energy constraint. So we have a constant of energy, which is this sort of complicated thing here, which constrains those four variables. So since we have one equation, four unknowns, we're working on a three-dimensional manifold. So maybe this is how Poincare got interested in topology. If you do celestial mechanics, three-dimensional manifolds kind of fall in your lap when you start thinking about it. And so that's where the flow takes place really. You're just seeing these pictures in the x, y plane. But really, there's a flow on an energy manifold. So one way to help visualize that is using these so-called hills regions invented by G.W. Hill, an American astronomer and mathematician. And so the energy equation implies that the x and y-coordinates have to satisfy an inequality. Essentially, it means that they have to lie inside some regions determined by the level curves of that function V that I had on the previous slide. So if you look at the level curves of V, and if you fix one of these energy levels, you're constrained to lie inside the shaded region here. The shaded region also has a part outside. So for this fixed energy, you'd be constrained to lie either in this sort of H-shaped region or in the region outside. And you can actually visualize the three-dimensional energy manifold if you realize that for each point in this region, there's a circle of velocities. So the formula that I had on the last page for the energy was like u squared plus v squared equals something. So the velocity is essentially the size of the velocity vector is fixed if you know where you are. So if I'm at this point, I can have any of these velocity vectors, u, v. But if I'm over here, the velocity vectors have to be on this smaller circle. And as you approach the edge of the hills region, the size of the velocity goes to 0. So this is called the zero-velocity curve for that particular energy value. So the energy manifold really consists of all the x, y's, u's, and v's. And you can just get some impression of what that must be like. So here's an orbit moving around inside its hills region. So the hills regions constrain the orbits. So for example, this orbit will never escape to infinity because it's trapped inside this bounded region. The motion inside that region, however, is quite complicated. So you can't really get too much out of the hills region. But this is to give you a feeling for what the energy manifold constraint looks like. And this is an orbit, which I'll call a transit orbit. It has something to do with chaos, but not the kind I'm really going to talk about until maybe you mention it at the very end. But this one is kind of going back and forth between the Earth and the moon, let's say. OK, so to start Poincare's story, he was always interested in the case of letting mu go to zero, where you start out with a simple problem, which is now we have the one big mass at the center. And this mass is also zero now. And so it's not affecting anything. And so this guy can just go around on a solution of the two-body problem. And in particular, maybe you could choose a circular solution of the two-body problem. So this guy is supposed to go around, even in rotating coordinates, it'll still just go around on a circular solution of the two-body problem. And now, so one of Poincare's ideas was to use the implicit function theorem or continuation to prove the existence of similar-looking orbits for mu positive. So here's a picture of what happens as the mu value changes. So these are all periodic orbits which look fairly similar. They're all quite simple. The curve that they describe is just a simple closed curve, which closes up after one revolution in the rotating coordinates. But the shape of the curve changes a lot. So these are just obtained by solving using the implicit function theorem, trying to find a nearby periodic solution to the ones that Poincare would have started with. So the mu value has now changed to be equal to 1 half, which means that we have two equal masses now. This guy is really having a strong effect now on the black mass. OK, so those are the simplest periodic orbits that one could hope for in the restricted three-body problem. And nearby those, in Poincare's classification, we have another kind of orbit called the second genre periodic orbits, which you might just think of as harmonics. So I've put the mu value pretty high. The mu value is 0.4. And so this guy is going around on some kind of closed curve, but it's not terribly circular. I'm just showing you half the Hills region here. So this is what it does. It just goes around this primary and closes up after one revolution. But nearby it, there's an orbit which closes up after three revolutions. So it's very close to the other one. But it has to go around three times before it starts retracing its path again. And so that's like a harmonic of the other one. It's a period would be three times, approximately three times, the period of the original orbit. So actually Poincare has maybe 100 pages in his book about finding these kind of orbits, these second genre orbits, and proving that they exist. And it's nearby there where he was able to find his chaos, his trellises. Okay, so in order to kind of describe that better, we have to introduce another device of Poincare's, which is the Poincare section. And I wanna describe what Poincare section we'll be looking at, because we'll look at it several times. So you have a flow on this three-dimensional manifold which takes place in the X, Y, U, V space, but with one constraint. And it's a flow in a three-dimensional manifold, but you could look at it at a certain fixed times and get a return map of a two-dimensional surface. And we're gonna pick our fixed times are gonna be whenever the third body returns to the X axis. So in rotating coordinates, the third body, if you start it out on the X axis, it may go around and return to the X axis at some other position. And every time that happens, we will just look at what is the state of the system at that time. And so the state of the system is going to necessarily have Y coordinate equal to zero, because you're on the X axis. So you've eliminated one variable. And now, if you remember that the U and V variables are constrained to lie on a certain circle, that you could solve for one of the two using that quadratic equation and just plot two variables instead of three. So we're gonna plot X and U. So like U, maybe I should explain that. So this is, U is X dot, basically. So if U is positive, that means that you come back to the axis and your X variables increasing when you do it. And if U is equal to zero, it would mean you come back to the X axis and meet it perpendicularly. And mu negative would be coming back and heading back in the other direction. Okay, so X and U are gonna be our two variables. And so here's a picture of the Poincare section corresponding to those two periodic orbits that I showed you before, the periodic orbit of the first sort and the one of the second genre. So the first sort orbit is just gonna become a fixed point for this Poincare map. So it starts out at a certain X position and U equal to zero, meaning that it's orthogonal to the axis and it comes back to that same position. And we see that we have two period three orbits around there. So this is the one that I showed you. It starts on the X axis with U equals zero, but then it doesn't come back after one iteration it comes back here, which means it's at a different X position and the U is positive. And then after one more time around it comes back here and then finally at the third time it comes back here. So it's a period three point for this discrete dynamical system that Poincare would have cooked up out of the three dimensional flow. And that's gonna be a hyperbolic or unstable periodic orbit. And then there's another one that goes with it, which is this one. I didn't show you the movie of that one, but there's an elliptic one which looks sort of stable since nearby orbits just kind of stay around. The other orbits are all quite complicated. They look like they're on invariant curves and maybe they are, but one of Poincare's things that he liked to emphasize in his work on celestial mechanics is you should always look for the periodic orbits because they're the simplest. And here you can kind of see that. I mean, these are the only really simple things that are going on are these low period periodic orbits. There's some more periodic orbits maybe around here which are more complicated, but anyway, at least low period periodic orbits are gonna be something to look for, something to understand more easily than the average orbit. I guess I'd also like to point out that this is only a local Poincare section. So not every orbit is actually gonna come back to this axis, especially maybe in the positive direction like I've got it indicated there. So this is actually a fairly small piece of the plane. This is the x direction and this is the u direction and they're only about 0.2. The extent is only about 0.2 in this direction whereas the primaries were one unit apart and similarly about 0.2 in the vertical direction. So it's just a little piece of that, of the x u plane that you're looking at there. Okay, so here's the in Poincare's quote he talked about asymptotic and bi-asymptotic solutions. And so those are just what we would call the stable and unstable manifolds of the hyperbolic periodic orbits. So now superimposed on the hyperbolic period three point, I put it's stable and unstable manifolds, little pieces of it's stable and unstable manifolds. So the stable manifold is gonna be the red, these red curves here are the stable manifolds of those three points and the black curves are the unstable manifolds of the three points. So under one iteration of the Poincare map, what will happen is that this black curve, while this point in the center is gonna go over here and this black curve is gonna go over here and get stretched. So the unstable directions are getting stretched on every iteration of this map. In particular, I didn't draw what happens when they come back but after three steps there's gonna be some exponential factor of stretching associated with that. And if I look at the stable manifolds, this red curve will map to this red curve. So you're kinda mapping and the turning by one third roughly as you go around. So this red curve will map to this red curve and it'll get shrunk a little bit. So those guys are exponentially contracting towards the periodic orbit as time goes on. So we have one stretching direction in positive and if you run it in backwards time, the red curve would get stretched actually. So we have one direction that gets stretched in forward time and one that gets stretched in backward time. So those would be what he would call asymptotic solutions and then by asymptotic would mean that you find a solution that starts here and maybe it goes away on the unstable manifold and then it comes back on the stable manifold of one of the others or maybe on its own stable manifold. So we want to extend these curves and we can extend them by just iterating the Poincare map, they're invariant set. So if you iterate the Poincare map, those curves will just get longer but they'll still be asymptotic curves. So here I have a movie of what happens when you do that. So I'm just starting out with little tiny pieces of the stable and unstable manifolds and the black curves will be extended by forward iteration and the red curves will be extended by backward iteration and it looks like they just coincide. They look like they're exactly the same so that there would be, so that those, all those curves there would consist of these bi asymptotic solutions. So this curve, if you're on this curve, you would think, okay well I'm on the stable manifold of this point so I'm gonna go there in forward time and I'm on the unstable manifold of this point so I'm gonna go there in backward time and that would be a whole curve of bi asymptotic solutions. So if you believe that, you're a lot like Poincare when he wrote his prize winning essay with a famous error in it where he says more or less exactly that, that oh, all these curves they have to coincide and not, or intersect and not only that but they are going to coincide and so they're gonna form little traps and no orbit inside here could get out because these curves are exactly identical. Well okay, so we now know that of course that that's not the case but it's subtle so I'm kind of showing that because it's not easy to see. So here's a picture that cost me a lot of time to make just to see it. So you have to really work hard and blow this up, be very careful about computing these things and you see that they don't really coincide. If you look really close to these points it'll start to get amplified so you can see it better but this would be the unstable manifold of this guy so it's come all the way around and it's kind of building up like that and the stable manifold of this guy has been brought back and it's similarly kind of a complicated curve and they're crossing each other more or less as Poincare eventually figured out they had to do. So in fact he showed that they do have to intersect these two curves but they can cross each other transversely and when they cross each other transversely the hyperbolic stretching is gonna cause them to fold in this very complicated way. So at least according to the computer here these trellises are being formed in this transversal sort of way but yet it's sort of hard to see. You gotta work hard to make this picture and make it look something other than two identical curves. Okay so I wanna move on to some other examples. We'll leave Poincare behind and kind of talk about some more modern examples of this phenomenon and one of the best is maybe the Smale Horseshoe Map which I kind of did a little animation here of the Smale Horseshoe Map. So this is probably the simplest example of a homeoclinic transverse homeoclinic point. So what, when it starts over again. So you start with the unit square and you compress it in the x direction, stretch it in the y direction and then fold it like a horseshoe. And I'll make it start again here. We'll start with the unit square and then it has this hyperbolic stretching and then also the folding. And what happens is that the unit square if you just follow the image of the unit square it just gets folded up into this fractal shape, fractal sort of horseshoe shape. So at first it seems it has not much to do with the previous problem but in fact there is a hyperbolic fixed point and a homeoclinic point. So this picture on the side shows you the origin is a fixed point, a hyperbolic fixed point and the red curve is the stable manifold of that or part of the stable manifold of that hyperbolic fixed point and the black curve is part of the unstable manifold. So as we iterate forward in time this curve is gonna get stretched and folded. Iterate backward in time this curve gets stretched and folded and this point is a transversal crossing of those two curves much more obvious than in the previous example of the restricted three-body problem. So I'll show you how that folding takes place. So that's after one forward and one backward iteration the black curve has gotten longer and the red curve has gotten longer and you see more crossings developing and then after two forward and backward iterations and three forward and backward iterations I might have one more here, yes. Four forward and backward iterations. And so if we reread Poincare's little sentence there you can really see it very clearly in this example. So that red curve is folding back and forth infinitely many times while in the limit but it can never cross itself it's just a single curve basically and so is the black one but then they cross each other in a multitude of these bias and tautic points. And that's probably the simplest trellis that one could make. Okay so to understand why do we call it chaos? Well you can understand that better with symbolic dynamics. So in the horseshoe example just focus on things that remain in the unit square. So they start in the unit square and then they remain in the unit square for all time. Well then you have to be in one of these two boxes or windows, I call them sometimes boxes sometimes windows. So this is window zero and window one. If you're in the middle you're gonna get horseshoeed out. You're gonna get up in the top here and I don't wanna focus on those orbits I only want the ones that stay in the unit square. So anybody that stays in the unit square is just gonna always be in one of those two boxes and you can just describe which box it's in for the sequence of zeros and ones. So this one, the fact that this is zero, dot zero means that right now it's in the zero box and dot one means that right now it's in the one box, the first box. And then you extend, you ask which orbits are, which points are in the unit square now? They're gonna be in the unit square in forward time. They're also gonna be in the unit square in backward time. Then you can describe those. Those are gonna be in one of these four small boxes. And this, for example, this point is right now it's in box number one, but at the previous step it was in box zero. So that when you start to go backwards here you tell where you used to be in the past. And so that's a description of the itinerary or the history of these orbits with respect to these two boxes for two, well one time step plus and minus. And then you keep extending that in forward and backward time so there's what happens, the symbolic coding of some of the orbits that are gonna stay in for two forward and two backward iterations. And so we have 16 little boxes described by the 16 possible sequences of zeros and ones. And then now they got too small to like 64 little boxes now and I didn't label them. But the idea is that if you keep taking this limit you get an invariant set for this mapping. It's a canter set, an invariant canter set and it's in one to one correspondence with these infinite sequences of zeros and ones. So one aspect of the chaos is just that all of these sequences actually occur. So you could flip a coin and generate a random sequence of zeros and ones and there would be an actual orbit of this system which realizes that. So just kind of a random, the possibility of a kind of a random hopping between the two boxes is one aspect of the chaos. So just to bring it back a little to what Poincare was talking about, some of these sequences of zeros and ones correspond to orbits that Poincare would have been interested in. So for example, this fixed point down here, its itinerary is just all zeros. It's always gonna be in this bottom box. And this point, the homoclinic point, transverse homoclinic point, maybe things aren't really showing up there too well. That should be one, one. Yeah, it looks like we have a little glitch here. But anyway, this should be a 0, 0, 0, 1.1, 0, 0, 0. So that's a box, that's in box number one. It's gonna be in box number one in the past and then it's gonna be in box zero in both time directions. So there's zeros on both ends of this thing indicating that it's gonna be converging to this guy whose sequence is all zeros. And other homoclinic points to this point would be sequences like that. Just take any sequence you like and stick it between a bunch of zeros and you're gonna get a homoclinic point. That was another fixed point here, which is at three fourths, three fourths. Again, sort of invisible for some reason. And that's got the itinerary 1111111. So that guy is always gonna stay in box number one. It's a fixed point. So whatever box it's in right now, it's gonna stay there. And it's in box one forever. And it has its own stable and unstable manifolds. So in addition to this guy with its red and black stable and unstable manifolds, we've got another hyperbolic fixed point and it has its own stable and unstable manifolds. So there's like another trellis in the same picture. And if you iterate those trellises forward and backwards, you now have four curves. The yellow curve and the red curve are kind of going parallel to each other. They have to, they can't cross each other. They're both stable manifolds. So they both have to wind around and avoid each other. And whereas the other two curves in the cool colors, green and black are the unstable manifolds and they also wind around without crossing each other. But of course they cross the stable ones and the unstable ones do a lot of crossing. So if you account for all of that behavior, you're still only got a countable set of these infinite sequences of zeros and ones. And again, part of the point is that all of the, all sequences of zeros and ones occur, that's an uncountable set, it's like a canner set. And in fact, all of those sequences do occur. Okay, maybe enough about that. Okay, so here's kind of just a way to think about what was just said. So if you want to find chaos, this kind of chaos, what you should do or one way of thinking about it is you could find some windows. Suppose you have a mapping of the plane to the plane and you have two boxes or windows and they get stretched across each other and it doesn't have to be pretty. As long as it really gets dragged across, this blue box, there's the top of it and there's the bottom of it, it really got dragged across both of the others. And this green box really got dragged across both of the others. That's all you need in order to show that these orbits exist. And you kind of summarize that by saying that box zero gets dragged across itself and it gets dragged across box one. Box one gets dragged across itself and it gets dragged across box zero. And then you can interpret this as a rule for generating which sequences will actually be realized. So all sequences are possible when you have this kind of a graph. And again, all you need is to find some windows for your mapping and verify those kind of crossings and you can prove that these orbits exist. Okay, so now we're gonna go back to celestial mechanics for a while with the Sitnikov problem. So the Sitnikov problem is a special case of the Isosceles three-body problem, which is a three-dimensional problem. So here we have a single particle moves up and down on the z-axis and two other particles are kind of moving around on elliptical orbits in the x-y plane. And if the third guy has mass zero, these guys will stay in the x-y plane and we'll just be kind of influenced by the behavior of the other two masses. So this would be a flow and a three-dimensional flow because it's like a time periodic system. The state of the vertical particle is given by its position z and its velocity, I think I'm calling that v, but it's time dependent because of the behavior of the primary masses there. And since we're interested in things that would z go to plus or minus infinity, it's better to replace z by another variable that stays bounded. So what I picked to set z equal to like a half of tangent theta or something. So now instead of going to infinity, this guy used to go off to minus infinity, but now it just goes to theta equals minus pi over two. So if you see that kind of behavior, it means that really in the z variable, the guy just flew off the screen and never to return again perhaps. And so there's an actual differential equation describing this motion. It's a three-dimensional manifold again. Okay, so suppose you set epsilon equal to zero which was the eccentricity of this binary. Then these two guys are just going on a circle. And so these distances are just depend on z. If you know z, there's no time dependence because these guys are always just at the same distance away from the z-axis. And so it becomes a dynamical system of one degree of freedom without time dependence. So just a flow in the plane. And here's the picture of what the flow in the plane looks like when the eccentricity is zero. So we have some orbits in the middle here which are just going around on these circles and that just represents the particle, the middle particle just going up and down in an oscillation. And then we have a whole line of fixed points at theta equals pi over two and a whole line of fixed points at theta equals minus pi over two. And so those represent going to infinity. So an orbit on this black curve up here would just tend to this equilibrium point over here which means that theta is going to pi over two but z is going to infinity. So it goes off in the upward direction and it reaches infinity with a finite non-zero velocity given by the height here. So this is again like a velocity variable in this direction and this is this kind of compressed position variable. And then in between those two kinds of behaviors we have this so-called parabolic behavior where you just barely get to infinity but you get there with zero velocity. You just have enough energy to reach infinity and it turns out that this is kind of a degenerate rest point as you can see it doesn't look like one of the standard pictures but nevertheless it has a stable manifold, well in this case it's easy to see but in the next also it has a stable manifold and an unstable manifold which I'm going to call just the stable manifold of infinity and the unstable manifold of infinity. Okay so now you perturb that by making the eccentricities of these guys something other than zero and now you do have a time dependence and so you have a Poincare mapping of this plane rather than a flow in the plane. And now you can see that the chaos here or the transversal crossings are much easier to see than they are in the restricted three-body problem that we looked at before. So the stable and unstable manifolds of infinity are what you're looking at now. So the red curve is the stable manifold of this infinity there's two infinities one up and one down and this red curve is the stable manifold of this infinity and then we have the unstable manifold of each of those is in black. So you can see that there are transversal crossings of these stable and unstable manifolds of infinity and this is with like epsilon equals 0.2. And if you keep extending those manifolds you get a picture that looks like this. So this is sort of the extent of the homoclinic tangle for this epsilon equals 0.2. It kind of fills out a little annulus in this plane. This is again the theta v plane. So the region of influence shall we say of these manifolds is some kind of a little annulus here. And you can see the complexity of the way these curves intersect. So it's not really as simple as the horseshoe. You're starting to see things that look like they might be homoclinic tangencies and other more complicated phenomena there. And when you increase the eccentricity things start to get even worse. So this is sort of epsilon equals 0.1 kind of the region of influence of those manifolds. And then I kind of just let the epsilon increase here. So there's 0.3 and 0.4 now those manifolds are getting in towards the center more. And by the time you get up to epsilon equals 0.9 now you could so what this means is you can you could start way in here and still make it out to infinity. So this is part of the stable manifold of infinity but it's intruded way into the center of the phase space. So that's the problem I want to come back to later sort of for the pièce de la résistance. Okay, so what about the symbolic dynamics? I said you should look for windows and watch them stretch across each other. So suppose we set up a window which we'll call plus over here by plus pi over two. So there's my window. Again, it's kind of funny because this is not really a typical rest point or a fixed point and I'll have a window over here for minus. And I want to see that those windows get stretched across each other by the Poincare map or some iterate of the Poincare map for this problem. So I should take say forward iteration of this guy and it should stretch across itself and it should stretch across this one. That's what I want to see. Except it's easier, something somewhat easier to see is if I let this guy go forward a little and then I let this guy go backward a little at the same time. So that's what I did instead. And you can see that they do sort of just barely cross each other. So the image of the blue guy under forward, I think it's seven iterations of the Poincare map gets stretched all the way across like that and it does come out the other side before it starts to turn into a mess. It comes all the way across the green box and comes out the other side and the backward iterate of the green box under seven iterations comes all the way through the blue box and gets stretches all the way across it and then turns into a mess. And it also stretches, each of them also stretches across itself. So what it means is that under 14 iterations of the Poincare map, this blue box would be stretched across both itself and the green box. And then there's a similar story on top. So you have essentially the horseshoe picture where plus means that you're out here and minus means that you're over there. So what Sitnikov was interested in was these oscillatory orbits. So since I'm allowed to pick, I can now pick any sequence of plus and minuses. So suppose I pick a sequence like plus, minus, plus, plus, minus, plus, plus, plus, minus and just keep adding a lot more pluses each time. What that means is that I get really close to infinity over here, but I don't stay there. Instead I go back over here once or after 14 iterations. And then I come back over here, stay here for an even longer time. So you're in the Z coordinates. It means that the Z is going to plus infinity and then it comes back once and then it goes all the way up to plus infinity even closer and then back. So it's one of these oscillatory orbits. Okay, so the last thing I wanna describe is some chaos near triple collision. So there are two kinds of collisions in celestial mechanics. We have the binary collisions, which looks sort of harmless. You kind of have near approaches of two particles, but it turns out you can even get some chaos out of that if you're a Bolatina and Mackay. And Pochere was interested in this. He called them second species orbits, which come close to binary collisions and then do something crazy. But I'm gonna talk about something that I worked on a little bit, which is more like triple collision. So here's the world's simplest triple collision orbit. You take three equal masses, put them in an equilateral triangle and let them go with zero velocity and they will inevitably collide triple collision at the origin. And actually, they don't have to be equal masses. If they're not equal, they'll just collide at their center of mass, it turns out. So that was discovered by Lagrange actually and we'll see that appearing in the Sitnikov problem as well. So I wanna show you that you can construct interesting chaotic dynamics for these near collision orbits. So there's a near collision orbit of the Sitnikov problem. So now I've picked this eccentricity almost one. So these two bytes come really close to each other. The two primaries come really close to each other. And then if you just let this guy go at exactly the right time, it'll go right through the center when those guys are close. And you have a close approach to triple collision. And I've actually done that in such a way that this is a periodic orbit. And it looks a lot like Lagrange actually. It's almost an equilateral triangle and it comes very close and it's almost planar. But then at the last minute, these guys just spin around each other. So this is a very much Lagrange-like periodic orbit. And you could equally well have started the red guy down here and seen something similar. But there's a whole lot more of these orbits. So here's a different periodic orbit which has a close approach. So first it goes down and then on the up stroke it has a close approach to triple collision so it gets thrown down again. And then that's all during one, that's all during one oscillation of these primaries. And so what you can show is actually as the eccentricity gets closer and closer to one on the primaries, there's a whole infinite sequence of these things that's created. And so if you stop at any fixed Epsilon, you're gonna have a lot of hyperbolic periodic orbits of this type. And so here's showing you the Poincare map version of that. So basically I have six hyperbolic periodic orbits. I have infinity. So this means you go to infinity with Z positive. This means you go to infinity with Z negative. And this fixed point here is the one that looks like Lagrange. So you start almost in an equilateral triangle and then you have a close approach to triple collision and you go back. And this here means, that's the second periodic orbit that I showed you that first does a little wobble and then it has its close approach to triple collision. And these are the reflections of the ones over there. So you're starting down instead of going up. And you can see that those manifolds play a big role in the dynamics. If you look at the rest of the Poincare map there, they kind of have cleared out a big area in the Poincare map. So there's the stable and unstable manifolds of these six hyperbolic fixed points and they're just all tangled up. They're intersecting each other transversely all over the place. And so we're gonna get a lot of this chaotic behavior. And you can sort of summarize it with this symbolic dynamics diagram now. So now we have the complete graph basically on six vertices plus some, even some more stuff. I guess it's not quite the complete graph. I didn't, maybe I missed a few arrows here. Actually you can do anything, I should have put the complete graph here. You can do anything you want. You can take any of these six symbols and follow it with any one of the other six symbols. And what it means is basically that, so each of these vertices here represents like a window, a little window in the plane there. This could be a window out near infinity and this is a window near the green dot over here and so forth. And those windows are getting stretched across each other by the Poincare maps in such a way that you can actually go from one to the other according to any path that you want. And so to illustrate that, I kind of tried to realize one of these sequences and these are highly unstable orbits. So it's hard to follow them for too long. But here's a, yeah, I think it should be infinity plus q minus p plus infinity plus. So I wanna actually find you an orbit that does that. So it's gonna start near infinity plus. You can see it over there. It's this, so for this you have to follow the bouncing ball, okay? So here's the bouncing ball and it's starting near infinity plus. And after one iteration, it's sort of still out near infinity plus. At the next iteration, it's over there. It's one iteration of the Poincare map. At the next iteration of the Poincare map, it's over there. At the next iteration of the Poincare map, it's up there. And you can see that it's gonna go to infinity now. So the way I found that was I looked at where those manifolds intersect each other and tried to pick an initial condition in exactly the right place to make it do that. And so here's the actual orbit. So it starts out near infinity plus, which means that it's way up at the top there, slowly moving in. Now it does q minus, so the upside down version of that bouncing orbit. Now it's doing p plus, which is the Lagrange orbit. And now it's going off to positive infinity again. So the point is you can, once you have this kind of chaos, you can pick any sequence you want. You can choose your behaviors. And there exists an orbit which has that behavior. Okay, so just a little bit about why is triple collision so chaotic? And what happens when epsilon equals one, so now these two guys are just going in sort of the limiting case. You actually just have the planar isosceles problem. And that was a problem that's been well studied in the 80s and basically what you have is a bunch of equilibrium points, hyperbolic equilibrium points. So these are Lagrangian equilibrium points and this is, after using the Gehe's method, this manifold here represents triple collision. So this is an orbit which goes away from triple collision and goes back to triple collision. And here's another one with the upside down Lagrange orbit. And the point is there's a lot, these have stable and unstable manifolds which intersect a lot, which means that there's a lot of orbits which there's a lot of rest point connections between these manifolds. You can sort of go away from triple collision, you come back to triple collision in many different ways. I don't expect you to understand that, but just you don't have to see where these things come from. And then, and then there, so this sort of, this schematically represents all of those many different ways, actually infinitely many different ways to go from triple collision to triple collision. And then in the triple collision manifold, there's a connection going the other way. And so that's all for this limiting case of epsilon equals one. And then you perturb a little bit and now you can go past these rest points. So all of those different connection branches can be followed and you get infinitely many, well, for any particular value, you only get finitely many, but as many as you want, hyperbolic periodic orbits nearby. Okay, so just a word about the, a couple of those other kinds of chaos. So this is the orbit I showed you earlier and it turns out that this is also sort of determined by homoclinic chaos. There's a hyperbolic periodic orbit that lives in this little neck and it has transverse stable and unstable manifolds going in the two directions. So an orbit like that is probably caused more or less by the fact that there's a hyperbolic periodic point in there with transverse intersections. And here's just a picture from one of my papers on the planar four body problem, showing a symbolic dynamics connection graph of all the different behaviors that you can get for the low angular momentum planar problem, except you need to use four dimensional windows in that problem. Okay, so just for fun, I thought I would end with by taking Poincare and subjecting him to one of his Poincare maps. So there's my picture of Poincare and so if you take the Sitnikoff problem for epsilon equals 0.1 and apply it to poor Poincare, you get that. All right, thanks. So you basically described the topological chaos. You did not discuss chaos with respect to the Rebecca measure, with respect to the Louisville measure. Indeed, indeed not. I mean, I think that is probably one of the major open problems of Hamiltonian mechanics, I guess, to say something about the measure. Do you have something, some numerical evidence for that? No, these are highly unstable. All of these invariant sets have measured zero for sure. Yeah, I wouldn't want to speculate on that. It's a good, except to say that it's an excellent problem for the 21st century. Sorry, for the picture showed in the very beginning, the separatist is almost coincided. Is there any reason now when we know this picture to explain why they do like this? Well, nobody. So you're near an elliptic fixed point and you can formally basically, to arbitrarily high order, make it look like an integrable problem or else. So I think it's a theorem that if you go close enough to that center point, that the splitting is exponentially small with respect to the parameter. But that's mu equals 0.4. So even with mu equals 0.4 and at a distance of 0.2, you can barely see those things splitting. So it's really pretty impressively close. Thank you. Yep.