 Okay, so good morning everyone. So we'll start with the, we'll start, this is the dynamical systems and differential equations course. So let me start immediately with the definition of a differential equation, which I think most of you are probably familiar with. So as you know, definition, an autonomous ordinary differential equation is an equation. Have you all seen equations like this? Has anyone not seen an equation like this? I will explain what it is, okay, but where V is a generally continuous function, continuous map. A solution to this equation, let's call it star, is a map, so the solution to star for a certain initial condition with initial condition x0 is a map x, such that x dot of t equals V, a local solution is defined only on an interval minus epsilon, epsilon, greater than 0. Okay, so what does all this mean? So what we have is we're working in Rn, so imagine this is Rn, for example R3. So to understand what this equation means, let's try to understand what the solution, what is the solution. So what is the map x from R to Rn? Or so this, when I say a solution, I mean a global, sometimes this is called a global solution as opposed to a local solution. The local solution is the same thing, but just defined on a small interval instead of all of R, okay? So we take some point x0 and imagine we have, let's think of just a local solution, the global solution is the same. So we have the interval minus epsilon, epsilon, and we have this map x that maps from minus epsilon, epsilon into Rn, okay? So what does such a map look like? So if x minus epsilon, epsilon to Rn, I should have said this should be a C1 map, a C1 map, so differentiable, continuously differentiable map. So what does this mean? What is the derivative of this map? So how do we formulate this map? This is a map from Rn, so this can be written in coordinates, right? So t goes to x of t and how do we write x of t? We can write it as x1 of t, x2 of t, xn of t, where xi are the coordinate functions, okay? So this is a standard way to, standard notation for a map from some interval to Rn. So what is the derivative of this map? What do we mean by the derivative of this map? Then x dot of t, which is a notation for the derivative, this is a parameterized curve, this is a parameterized curve, this is x of t. Take a point t and this is x of t, right? And x of t has coordinates x1 of t, x1 of t, x2 of t, x3 of t, right? These are the coordinates of this point. So what does it mean to differentiate this parameterized curve? I'm assuming this is all review for you, okay? This is basic stuff about calculus, so if you don't know it, you should review what it means, okay, you should review these things. I'm just assuming this is at least the notion of a parameterized curve in Rn and of the derivative of a parameterized curve with respect to the parameter. So what is the derivative? Each of these coordinate functions is just a function from R to R. It's a scalar function, right? So differentiating this means just differentiating each component. So if you want to know the derivative of this, you just take the vector, the derivative of this is a vector. This you can think of as a vector. The derivative is a vector whose components are the derivative of x1 with respect to t. The derivative of x2 with respect to t and so on, right? So x dot of t is just equal to the vector, which is x1 of t, x2 dot of t, where the top is just the derivative with respect to t, xn dot of t, and so on. This is the vector, the derivative of this parameterized curve. And geometrically, it is very useful to think of this vector as being based at the point x of t. So at this point here, we have a vector which is in coordinates. So we have a copy of R3 here and we have this vector here is x1 dot of t, x2 dot of t, x3 dot of t, and this gives the vector, right? So this is the vector x dot of t. Everything finds so far? Does this make any sense? Yeah? So what does this mean? This is the definition of a solution of the differential equation, okay? So not every parameterized curve is a solution. What is this definition of solution saying? So x dot of t is the vector I just drew there, right? It's the derivative of the parameterized curve. So I'm saying that if I have a c1 map like this, this is a solution if the vector x dot of t coincides exactly with the vector v at the point x of t, which is given by this function here. So this is a slightly different way of looking at differential equations, maybe than what you're familiar with. It's a more geometric way, but it is what really a differential equation is. So what, notice that this here is exactly the same as this. The only thing is the dependence on t is implicit. This and this are exactly the same. It's just sometimes you forget that this actually means this, okay? So when you're saying you want to solve a differential equation, what it means is I want to find a function x which satisfies this equation, okay? And satisfying this equation, this dot means derivative with respect to t. T is implied in both sides, okay? You just don't write it down because you suppose to understand that it's implied. And when you write it explicitly, it means exactly this, okay? And when you write it like this, you see exactly what it means. Because when you write it like this, you see that x is a parameterized curve. It has its derivative vector which is defined independently of the function v, okay? And this vector has to coincide with the vector that is defined at that point. So when you're given a differential equation, what you're given is this function here. And what this function does is it defines at each point of Rn a vector in Rn, right? So really the problem of differential equations is the following. You are given v, you're given a vector at each point. That's what v does, okay? v is a function from Rn to Rn which means you choose a point in Rn and it gives you a vector in Rn which you think of as the vector sitting at that point, okay? So sometimes we think of v, we write v is called a vector field. And the problem of differential equations is do there exist solutions to this vector field, okay? So the problem is can you find a curve that has this property? Can you find a curve, a parameterized curve that is everywhere tangent to this given vector field? This is the problem of the solutions of differential equations, okay? So it is not at all obvious that this is true. So solution x is sometimes also called an integral curve, integral curve of v, okay? So this is the problem of differential equations. The motivation of course as you know is that the derivative is a kind of velocity field. So the idea of differential equations comes from modeling physical phenomena and the idea is that if you know some physical laws then you know that given a certain system in a certain state the system because of the interaction between the particles will tend to change in a certain direction with a certain velocity. So in some sense you can model in certain situations systems by knowing which forces act on them that make them change. You model by a vector field, okay? And the idea is that if you know at each point what is the force acting on the system can you describe the evolution of the system. So can you describe how the system changes in time, right? So you want to choose a certain initial condition and you say does it exist a solution? A solution is a solution that has the property that exactly at each point the rate of change is given by the given vector field. So that's why it's a solution to that vector field. So let me describe this. Now the basic problem sometimes also called the Cauchy problem, okay? That one generally wants to solve is you're given a differential equation, some x in Rn and you're given some initial condition. So you want the solution to satisfy x of 0 equals some x0, okay? For some initial condition x0 in Rn, okay? Question, does there exist a solution to this equation? So the problem is not at all trivial. Like you say, okay, just give you a bunch of vectors. Can I find for any point a curve that is exactly a smooth curve that is exactly tangent to these vectors at every point? Naturally, once you find a solution you say, okay, is this solution unique? Okay, the usual question is you're asking mathematics. So if you find a solution at this point, maybe there's another solution, a different solution that goes through that point is also tangent to this vector and is also everywhere tangent to vectors here. This one is also everywhere tangent to vectors here. So it could be that both of these solutions are solutions. They satisfy these conditions but they go through the same initial condition so there might be two solutions or infinitely many solutions and maybe many solutions, okay? And the third question is you want to know how, if it's a local solution or a global solution. So notice that a local solution, so I wanted to make a couple of remarks actually that I forgot. Before we, so there is a basic theorem for this problem but first let me just make a couple of remarks. So remark, local and global refers to the domain of definition of x, of the function x. It does not refer to the fact that the solution is small geometrically, okay? You can have a function and this is one of the exercises. So I will give you at the end of the class a sheet of exercises which cover them related to the material we'll do in the first couple of lectures. So one of the exercises or one of the examples in fact I will give also now shows that you can have a local solution, okay? Which means a map that is only defined on minus epsilon to epsilon. It cannot be defined on anything larger but the image of the solution geometrically is infinite. That may seem strange but I think you all know functions that map a closed interval to the whole real line, right? It's not difficult to have a function like that, okay? And similarly, a global solution refers to the fact that this solution is defined on the whole real line but its image geometrically might be very small, okay? So this is something I will come back to during the course of the lecture today. There's two different elements that you should think of in this solution. The solution is a parameterized, the solution, this is a parameterized curve but the solution is the function that defines the parameterized curve. The parameterized curve is in some sense geometrically the solution, okay? But the definition of the solution is the function not the image. You might, we'll see some examples and I will come back to this. I just wanted to mention this issue. The second remark is that sometimes we can have non-autonomous differential equations. A non-autonomous ODE is of the form x dot equals V of xt. So what this means is that the vector field itself changes with time. So you are starting from some initial condition and at time zero you have a certain vector field and so the curve has to be tangent to a vector field that is changing all the time. So it is, when you just think of it like that you think this is extremely complicated. In fact, all you need to do is add another dimension of time and this becomes an autonomous vector field in Rn plus 1. Okay, you can think of it. But the theory of non-autonomous ODE has some differences and some additional complications so we will not discuss this. I just want to point this out that often in text you will find in general text of differential equations that we will often discuss the more general notion of a non-autonomous vector field. Okay, in our case we are concentrating on autonomous vector fields where the vector field is fixed once and for all. Okay, so these were just two remarks. So what I wanted to do is to discuss this Cauchy problem and it turns out that this existence of the solution is often true as long as the vector field is continuous we always have existence of solutions but we do not always have uniqueness of solutions or globally defined solutions. So to have globally defined solutions we need the vector field to be Lipschitz continuous and then we have the following theorem which is the fundamental theorem of the theorem of differential equations. So let X dot equals B of X where V from Rn to Rn is Lipschitz then for every X0 in Rn there exists a unique global solution which depends continuously on X0. So this is a classical theorem. I'm not sure if we will get to prove it or not. I don't want to prove it straight away. We might prove it depending on how much time we have towards the end of the course because the proof is not particularly complicated but it does take a little bit of time and it uses some ideas and techniques which are not the ones that I want to concentrate on in this course. I presume that several of you should have seen this theorem already in courses you've done and maybe even proved this theorem. But what I will rather do is just give some additional examples to try to explain it a little bit more. So first let me give some remarks again about this theorem. First of all there is a weaker version of this theorem assuming only that V is locally Lipschitz. So let me just remind you in case you forgot what Lipschitz means. So we say that V is locally Lipschitz if for all X0 in ln there exists a neighborhood U of X0 and a constant K greater than 0 such that V of X minus V of Y is less than or equal to K X minus Y for all X, Y and U at every point. Globally Lipschitz means that you can take as this neighborhood your whole space ln in this definition. What is the difference between locally Lipschitz and globally Lipschitz? Which one is weaker? Why is that? That means it's stronger. Globally Lipschitz is a stronger condition because if it's globally then of course it's locally. What about the converse? Is it false? If it's locally Lipschitz why is it not globally Lipschitz? Because here I'm saying at every point at every point it slipschitz. Sorry? Exactly. The constant K depends on the point that you choose and in particular it may become arbitrarily large. So what we're saying is that given any K there might be some points where this is not holds because you need to take a bigger K. So you cannot find one K that is uniform in the whole space. That's the difference between locally and globally Lipschitz. And what happens if it's only locally Lipschitz it turns out that we still have existence and uniqueness and continuous dependence of solutions but they're not global. Then we can only guarantee local solutions. And I will give an example in a second to show why we can only have local solutions. It's very interesting actually to also to understand the difference between local and global solutions. Another remark before I give these examples is what do we mean by continuous dependence on X0? What does it mean that two solutions depend continuously on X0? Of course if you take one solution X0 here and you take one point Y0 here and you look at the solution to this point. Even if X0 and Y0 are very close these solutions geometrically after a while this one may go somewhere and this one may go completely somewhere else. They don't stay together. So what do we mean by continuous dependence on solutions? Any ideas of how we can define this continuous dependence? What does it mean? X are two solutions. They may diverge even if X0 so both of these satisfy X of 0 equals some initial condition X0. Y of 0 equals some initial condition Y0. Even if the two points are very close X0 and Y0 this could do something like this and after a while this could do something like this. So how do we define what does it mean continuous dependence? I will also give an example of that. Yes? That's right exactly but what do we mean exactly because eventually this distance diverges so how do we define it exactly? Intuitively it means that the solutions are close but they cannot be close for all time in general. I will show some examples again. So now just on an abstract level assuming these two solutions however close you get eventually the solutions diverge and do something completely different which happens. What do you mean? Try to be more precise. You need to be more precise. You need to say how do we define. Well I will tell you so what this continuous dependence mean what you need to do is you need to fix a time. What does this mean? So continuous dependence on initial conditions means that for all t in R X of t minus Y of t goes to 0 as X0 minus Y0 goes to 0. Which is what we all have in mind but I just want to remark the difference in the formulation. To formulate this you need to fix a t. So the fact that it holds for all t does not mean that they stay together. You have to understand the subtle difference between these two things. You fix a t so now I fix a t. So this is Y of t and this is X of t. Now when I fix a t so I fix an initial condition X0 and I want to know if this depends continuously. So I take Y0 very close to X0 and then Y of t will converge to Xt. If I fix the initial conditions then after some t these will diverge. But now suppose I fix this t here. So now I fix another t a bigger t. So this is Y of t prime. Then I say ah but there's a problem that these diverge. Of course they diverge but that's not what the statement says. Sorry I'm studying continuity X0 so I need to take it here. So suppose I take some Xt prime here. What does this say? Now I fix Xt prime and now I need to take initial conditions arbitrarily close to X0. And now if I take an initial condition closer to this it will do something like this. And then if I take it sufficiently close it will stay close for a very long time and then afterwards it will diverge. So if I take after I fix the t I can clearly find initial condition that is sufficiently close so that it stays close up to that time. But then afterwards it will diverge. So for every t I need to take closer and closer. I'm looking at the limit and then this is true. So you cannot even say that these two maps from R to R are close for all time because it's not true. You fix the time and then there's an initial condition that's sufficiently close so that they stay close. So spend some time. I will give you some exercises discussing this. I will give you some notes. Spend some time thinking about this because it's a subtle point but it's important to learn to formulate these concepts clearly. Because continuous dependence we always say okay continuous dependence and it's a little bit more subtle than it seems at first sight. So let's do some examples now to illustrate this. So first really basic example maybe the simplest example of all. We have AX in R and let's define the differential equation X dot equals AX. So let's study a little bit the geometry and the analytic properties of this differential equation. So A is a constant. A is a scalar constant. But this is the function. So what I mean here is that the function V of X is equal to A of X. That's what this means. This is the function V of X in the abstract definition. And this is just a scalar multiple of X. So first of all what is this as a vector field? So is this a vector field? Can we think of it as a vector field? Can we think of it as a vector field? What are the vectors here? So this is a differential equation on R. This is zero. V of X is equal to A of X. What does that mean? So we have vectors at each point X. We have a vector that is given by A of X. So of course it depends what A is. If A is negative or positive or zero, one of the exercises asks you to study all the different cases. So now I will not go into the details. I will just give the abstract formulation. But for example, if A is positive, then you can see that the vectors will be in the positive direction. So here you will have a vector like this, proportional to X by the factor A. So as you go, here you have a bigger vector. Here you have an even bigger vector. So at each point here you have very small vectors because they're proportional to the distance. And here of course because X is negative, A is fixed and positive. But if X is negative, this is all negative. So you will have vectors going in the other direction. I'm drawing them below the line, but you should think of them as sitting on the line, of course. So this is the vector field. Now what are the solutions? Well, the solutions, we actually know them analytically, right? It's very easy to calculate them because the solutions are X of t. So solutions for initial condition X0. So if you choose some X0 here, the solutions are given just by X of t equals X0 e to the at. This is just a linear differential equation, which I'm sure you've all seen before. So let's check that this satisfies the condition for solutions, right? And let's see what this means. So first of all, let's check analytically what this means. So we need to differentiate with respect to t, right? X dot of t, you differentiate with respect to t. This is just a X0 a e to the at. This is just the definition of this function. This is a function from r to r in this case, right? Because a solution will just be a function from r to r, X. And what is this? So this is equal to a X0 e to the at. And what is this? X of t, right? So this is a X of t, which is equal to v of X of t, right? v of X is equal to ax, v of X of t is equal to a of X of t, X dot of t is equal to v of X of t, okay? So this is exactly the definition of the solution. What does it mean geometrically? Geometrically, we have a map that goes from r. We have a map X. What is the geometry of the solution? What is the image of the solution here? What happens when t is equal to 0? Oh, I forgot, of course, this solution not only has to satisfy this, but it has to satisfy the initial condition, right? So what is X of 0? X0, right? Because t equals 0, you just get X0. So this is a solution with initial condition X0, which is what we were looking for, okay? Is this a global or local solution? It's a global solution because it's defined on all of r. What does define on all of r mean? It means it's defined for every t in r. This is well defined for every t in r. What is the geometry of the solution? What is the image of r in r? What happens when t is 0? So this is a copy of r. When t is 0, it maps to X0. What about when t is positive and large, for example? Sorry? It's close to infinity, exactly. So when t is very large, this is very large. So when t is very large here, it maps along here. As t increases, you can see that 0 maps to X0. And then as t increases, it kind of moves along here, okay? What velocity does it move? If a is positive, that's right. We've already, in this picture, we've assumed a is positive and we've drawn the vectors in this direction, right? The velocity with which it moves is exactly given by the vectors. That's what we've shown here, okay? That is what the solution means, is that the velocity with which it moves is exactly the same. What about as t goes to minus infinity? Is t becomes negative? It is going to 0, exactly. As t becomes negative, this goes to 0. So what is the image of r? What is the geometrical solution? What is the solution geometrically here? Exactly. It's the half-open line 0 to infinity, okay? So this already gives you a bit of an example of saying why the... So the left-hand side of the line from 0 to minus infinity here maps just to this finite segment, okay? Again, distinguish between what we mean by the domain of the solution and the image of the solution in the phase space. They may be different, right? The solution is global because it's defined on all of r. But for example, this left-hand side, which is an infinite domain, gets mapped to this finite interval in the phase space, okay? This is the comment I was saying before about local and global. Global does not mean that it covers the whole phase space. It just means that the domain of definition is all of r. Okay, so this is also a very nice example to study exactly this comment about the continuous dependence of solutions that I was talking about before. So what is... Let's check what happens if we take two initial conditions, y, x0 and y0. And we take the two corresponding solutions, right? If we have x of t equals x0 eat and y of t is equal to y0 eat. These are two solutions with two different initial conditions, okay? So there's several comments to make about this. First of all, what is the... geometrically, what are the curves? These are parameterized curves, okay? As we just said before, the image of this curve is the open interval 0 to infinity. And the image of this curve is the same. It just has a different initial condition, but the image geometrically is still the open interval 0 to infinity, okay? So these are two distinct solutions, but geometrically they coincide in the phase space, like their images coincide. But they are distinct solutions because they have different initial conditions. There are different places at different times, but they overlap. In fact, they are distinct because the only thing that distinguishes them is the initial condition. In fact, they are one on top of each other. You've just taken two initial conditions, okay? This does not contradict the uniqueness, okay? The uniqueness that for every initial condition, there's a unique solution that has that point as its initial condition. Now, this solution here does for some t map to y0, but it does it not for t equals 0. It does it for t equals something else, right? So the uniqueness of solutions, when you have a solution geometrically, there are infinite number of solutions inside it with the same geometric shape because you just take different initial conditions. Okay, maybe I'm insisting too much. It's fairly elementary at this point, but it's important. So what about the continuous dependence? What can we say about x of t minus y of t? So this is equal to x0 e to the at minus y0 e to the at, okay? Which we can write as e to the at times x0 minus y0, right? So what happens to this as t gets large? Exactly, okay? I'm just emphasizing this. So this is a specific example of what I was saying before. So if you fix the two initial conditions, then this will go to infinity, right? So in some sense, you do not see the continuous dependence by looking at it like this. This is the long way to look at it. What we want to look at is for each fixed t, okay? You fix the t and then you can find two initial conditions sufficiently close so that this is sufficiently close. That is obvious because of this, right? For each fixed t, as long as you find the initial condition sufficiently small, this would be sufficiently small, okay? This is just an explicit example to show this. Okay, so that was one example. Now, example number two. Consider the Cauchy problem x dot equals 3x to the 2 thirds and x0 equals 0. So this is again a differential equation on the real line, right? What is this? This is the cube root of x squared, the positive cube root of x squared. So what does this look like? What does the vector field look like corresponding to this, roughly? So we can write this as cube root of x squared. So when x is big, what will this look like? When x is positive, what will it look like? It's also big, so it looks very similar to the other one. I mean, it's different magnitudes because it's not proportional to x. It's given exactly by this function, but you still get something like this. When x is 0, this is 0. The vector at the point 0 is 0. And the vector at other points are increasing. The bigger x is and the bigger the vector is, like this, okay? What about on this side? It's still positive because we take the positive because, yeah, because it's positive, it's the cube root of a positive number. So it's still like this. And it's again, the further away you are, the bigger it is, right? So here it's very big. Here it's smaller, here it's smaller, here it's very small. It's getting small. So if you think of it, what should be happening is a solution is going very fast here, and then it slows down, it slows down. Here the vector field is 0. So in principle, a solution with initial condition 0, if the vector field is 0, generally intuitively it stays where it is. It doesn't move. It's just always constant 0. In fact, in this example, of course, I did not remark, but clearly if you take x equal to equal to 0, then this is always 0 and everything is still satisfied and this is a solution, right? So geometrically, I forgot to... well, in the exercise I asked you to think of all these cases, but think of the case that if x is 0 is equal to 0, then you have a solution that's just identically 0 and still satisfies this condition. So that's still a global solution, right? You have a solution that maps for every t just maps to the point 0. Okay? This is a solution. It's called a fixed point or a stationary point solution. Okay? Here you also have a point where the vector field is 0. Now, the curious thing about this, however, is that we have two solutions. In this case, we do not have uniqueness of solutions. So we have one solution is given by x of t just identically 0. This is clearly a solution because of what I just said, right? Here the vector field is 0. If you take x t identically 0, then it clearly satisfies the initial condition and it satisfies this condition here because you have x dot equals 0 and here you also have x dot equals 0, right? Here you have x dot of t is equal to 0, which is equal to v of 0, which is v of x of t. It clearly satisfies the conditions for the solution plus the initial condition. The interesting thing is that we have also another solution, which is y t equals t cubed. And if we check, if we differentiate, so we get y dot of t is equal to 3 t squared and we can write this. So how do we check that this is a solution? We want to show that this is equal to v of x of t. So we can write this as 3 t cubed to the 2 thirds and this is the same as... this is t cubed is actually y of t, right? So this is equal to 3 y of t to the 2 thirds. This is the point y of t, right? Which is exactly equal to v of y of t, which is exactly the definition of solution, yeah? So what is the geometrically the solution y of t? This is r, okay? y of t maps 0 to 0, so it has the correct initial condition, y of 0 equals 0. It has the initial condition that we need. And then what does it do? As t maps to plus infinity, it maps here and as t maps to minus infinity, it maps to minus infinity. So this actually maps bi-jectively onto the whole real line. So why does this not contradict the fundamental theorem on existence and uniqueness? Yes? We have a lack of uniqueness. Two solutions with the same initial condition. Sorry? The Lipschitz condition is not satisfied. Why not? So the derivative here goes to infinity, the derivative of this function. So this derivative is not Lipschitz continuous at 0, right? If you calculate the Lipschitz at points close to 0, it is locally Lipschitz continuous, but it's not Lipschitz continuous at 0 because the derivative is infinity. As you know, the Lipschitz constant is a bound for the derivative, right? So if the derivative goes to infinity, it cannot be Lipschitz at that point, okay? So you can easily check that. And the fact that this is not Lipschitz is what makes the uniqueness fail at this point. In fact, you can also check that if you take a different initial condition, you will get uniqueness at every point, okay? Because you have locally Lipschitz at each point and therefore you have a locally defined solution, but at 0, it's not even locally Lipschitz. So you don't get that. Okay, so that's the example to show the lack of uniqueness. Now, one more example to show the difference between local and global solutions. So here we have this Cauchy problem, x dot equals 1 plus x squared, x of 0 equals some arbitrary point x0, okay? This is again the differential equation on R. So solutions are given by x of t equals tangent of t plus c, where c is equal to inverse tan of x0. So, of course, I'm not explaining here any theory about how to find these solutions and so on. I'm just saying that these are, and we verify that this is true, because x dot of t is equal to, by differentiating tan, gives 1 plus tan squared of t plus c. c is just a constant, right? It depends on x0, which is fixed initial condition, because we want a solution to this problem here. So this is equal to 1 plus tan of t plus c, all squared, and tan of t plus c is exactly x of t, right? So this is 1 plus x of t squared, which is exactly what we want, right? 1 plus x of t is equal to v1 plus x squared. 1 plus x squared equals v. This is exactly, again, the definition of the solution. So what is the problem with this solution? What is the domain of definition of these solutions? What is the domain of the tan function? What happens with tan function? It blows up to infinity minus infinity, right? So tan function is only defined on minus pi over 2, pi over 2 on that interval. But in that case, it blows up to infinity. So in this case, this is defined. So each solution, each solution defined interval i equals c minus c minus pi over 2 minus c plus pi over 2, right? Because minus c minus pi over 2 gives exactly minus pi over 2 here. Minus c plus pi over 2 gives exactly plus pi over 2 here. So that's the domain of definition. So you see that the domain of definition depends on the constant c, which depends on the initial condition, the finite domain of definition, right? So what happens, so x is actually a function from this interval into r, right? r is our dynamical space where we have our... So we forgot to look at the vector field. What does this vector field look like? Here, 1 plus x squared. So at 0, what's the vector at 0? It's just 1. And then as we go bigger here, they just get bigger and bigger, right? They get bigger and bigger, the vectors here. And here, they also are positive. They also look like this, right? And they get bigger and bigger like this. And here it's 1. So there's no zeros of this vector field, okay? So it's intuitively what happens is you choose some initial condition here. x0, okay? It's going... its velocity is pointing to the right. So the solution is going to be pointing to the right, like this, okay? So what does this map do? This takes the interval i, the interval i, and it maps it... this is the solution. It maps it to the real line. What is the geometry? What is the geometric image of this interval of the domain of definition? All of the line. That's right. Because the tan function maps this interval to the whole real line, okay? So once again, this example is a simple but it's very useful related to the comment I made before about local and global solution. This is a local solution because it's only defined on this, but its image is global in the sense that it covers the whole real line. So what this means is that this... if you think intuitively, it's inevitable to think of these solutions as points moving in some sense in space. There's a kind of dynamics involved. So what this point is doing, this initial condition is going to infinity in a finite time, right? Because this is a finite time. This is... this is a finite interval of time and in this finite time, as you approach the boundary of this interval, the solution is going to infinity. Because this is the domain of definition of the solution and we think of it as time. We think of it as time. Maybe it can create some counter-intuitive conclusions. If you don't think of it as time, everything makes sense, formally everything makes sense. But we think of T as time. We think of these solutions as time because when we have a differential equation we have a vector field and we think of an initial condition as a state of your system and we think of the solution as describing the evolution of the system in time if you want. And as the system evolves in time you look at the time x of T, that's where the system is at time T. That's what x of T means. Now this solution is only defined on a finite time interval. It's defined on an interval and since I think of the domain of definition of the solution as time, x of T, right? And then what it tells me is where the point is at time T. So as I increase T here this point is moving and it's moving so fast that whatever point I choose here you tell me one point as big as you want I will reach that point in a finite time and in fact I will reach arbitrarily large points in a uniformly bounded amount of time. That's what I mean by I reach infinity in finite time. I don't really reach infinity I just mean this, okay? Okay, so I think this is all I wanted to say in terms of these examples and counter examples. So often in the class I have just some very short breaks so I think it's useful even though we don't have that much time left we just have literally two minute break just for you to stretch your legs if you want and relax your minds and then we will finish the next 20 minutes, okay? Okay, so now that we've given the basic theorem and the basic examples I want to describe a very powerful way to look at differential equations and introduce the concept of a dynamical system. So suppose x dot equals v of x is an ODE with v Lipschitz continuous so that we have globally defined solutions to every point. Then for every t in R we can define a map f t to Rn which associates to the point x0 the point x x of t where x is a solution with initial condition x0. So what does this mean? This is a shift in the way to look at these equations which is really simple but quite fundamental. So what we've done until now is we've taken initial condition and looked at the solution for all t. Now I'm going to fix t and I'm going to look so what I'm doing here is I look at my space okay, I fix t and I look for every initial condition I look at where this initial condition is after time t. So this initial condition will map to here this is x0 this is xt y0 will map to yt z0 maps to zt every point maps to some other point it's like considering all the solutions simultaneously at the same time but for a fixed t so you look at all the points through every point there's a unique solution I flow all these solutions for t time t and then I look at where everything ended up this gives a map from Rn to Rn it's called the time t map time t map of the ODE so this map is a bijection because through every point there's an initial condition so it's a bijection so of course from every point you end up because every point is reached somewhere by this map if you think about it a little bit you see easily that it has to be a bijection by the uniqueness by the existence and uniqueness of the solutions and it's also continuous by the continuous dependence of the solutions because for a fixed t the image xt depends continuously on x0 every value t what we get is a family of maps so we get a family of maps ft t in R and this family has some interesting properties for example what is f0 f0 of x0 is x0 because time t equals 0 by the way this is defined this is x0 what is a map called that looks like this yes if it's true for every point then it's called the identity map so f0 is the identity map then there's another property suppose you take f of s composed with f of t of x0 what does this mean so this means that we take x0 we choose an initial condition x0 and we look at the solution the unique solution through x0 and we look at time the point xs ft xt which is ft of x0 which is x of t by definition and now we take this as a new initial condition and we look at the solution it goes through this point after time s so this is fs of x of t this is f so I've taken x of t as my new initial condition and I flow it for fs so the notation starts getting a little bit complex here let me I think to make it simple it is useful to write this as x of t let me this is often useful to write this as x of t and it's a very consistent notation because x of 0 equals x0 and x of t equals x of t this can be useful because you think of it more as a specific point x of t so now we take we have this new point x of t we think of this as a new initial condition and we look at this image the solution that has as initial condition x of t we call it y of 0 if you want we can call this y of 0 and then we take fs of y0 ok and this as you can see I'm not going to give a formal proof here but I think you can see particularly by the existence and uniqueness and by the way solutions are defined clearly this is the same as if you had a point for x0 because this here is just a continuation of the solution here if you didn't stop at time t you would have continued exactly along this direction by the uniqueness of solutions it has to be the same one so this is exactly equal to f of s plus t of x0 although the definitions of this and these are independent of each other there are two different definitions this you look at the initial condition this is the same as flowing for time t and then using this as a new initial condition and flowing for time s and therefore obviously this is also the same as f of t composed with f of s in x0 in particular this is true if s so notice that s and t are not necessarily positive I might be coming back so if s was negative if s is positive if t is positive I flow out for t and then if s is negative I don't move forward but I come back along the same way I came everything is formally the same so I might be t and s some of this might be in forward time some of this might be in backward time I might go forward and backwards in particular if s is equal to minus t I have f of minus t composed with f of t f of x0 is equal to f of 0 of x0 which is equal to x0 which is the identity map as well ok so what do these properties say about this family they say that it has a specific structure under composition it has an identity it has an inverse if you compose these two maps this by this you get the identity which means that this is in some sense the inverse of this what is a structure like this called a group exactly this is a group under the operation of composition because you're composing it's closed on the composition because if you compose two maps you get another map it's closed on the composition it has an identity and an inverse it's a group so this is a group so f of t is a group of transformations and this is very interesting this is what I was saying is a different way of looking at a different way of formulating the structure of solutions of another differential equation ok notice that what I obtain is not just with a differential equation and the fact that I have existence and uniqueness of solutions and then I said ok instead of looking at just one initial condition and the solution for all time let me do something slightly different which is fix the time and look simultaneously at all the solutions what they do after time t and that suddenly gives me a completely different way of looking at the system ok and I look at the system and I see that I have a group of transformations I will explain this a little bit one of the way that is useful is first of all it allows us to generalize this notion to other groups of transformations so it introduces this notion of a group of transformations which is actually much more general than the differential equation so in this case we sometimes say this um sorry this is often called the flow of the ODE um so groups under groups of transformations so there is there other groups already embedded inside this here so for example if we restrict ourselves to just integer values of t ok notice that we can we can restrict to t in z and we get fn or restrict to z it's often just formally often for integers we use letters like n to denote an integer so we use n but it's the same map here ok so what happens in this family I just choose only some of these maps of this family so my question is this also a group of transformations of course these conditions will basically be satisfied we need to check the closure we need to check that the composition of these two remains inside that group and does not come out of this because I've restricted myself but this is obvious because if you compose n you get fn plus m which is still an integer what does this group of transformations look like what does it mean so we still have the identity element what this means is that you only you're jumping instead of flowing as you vary the parameter t you get this continuously varying family of transformations which means that if you fix one initial condition and you apply you change the parameter t you recover the original solution of the differential equations in this case you fix some parameter one initial one point x0 you apply the different maps of this one parameter group as t varies in r you recover exactly the solution x of t what this means is that we restrict ourselves only to t integers we're still looking at points that belong to the same curve but we're only looking at integer points x1 x2 x3 and so on x4 x5 so far I've done nothing particularly amazing I've just reduced in some sense I've simplified even more the important observation however here is that to reconstruct this group all I need is one map which is f1 ok so the key observation here is that I can reconstruct z from f1 why is that in other words all I need is f1 what sorry what do you mean by cyclic group can't use words just like that what is f1 let's remember what is f1 f1 means you take all your points in your space you look at where they are after time one and you define the map that's a map that's a bijection from rn to rn for each point you look at x1 you take another point look at x1 you do that how do I reconstruct f2 from that I just apply f1 again how do I reconstruct f3 I just apply f3 times so if I have f1 I know that f2 will just be applying f1 and then using that as new initial condition I'm applying f2 because of the properties that I've defined so I cannot reconstruct the I cannot necessarily reconstruct the whole continuous time flow just from f1 but I can continue this in some sense this does not contain all the information of the flow in some sense you can reconstruct it too but just like that you cannot reconstruct all the information of the flow but to reconstruct this family all I need is f1 since all I just iterates I just compositions I just compositions fn is just equal to f1 composed with f1 composed with f1n times and if n is negative of course f1 means I also have f-1 because f-1 is just the inverse of f1 so I have f-1 is also well defined and so f-n is equal to f-1 composed with f-1 composed with f-1 and so what is the next step and then we will finish for today but what is the obvious next step now that I've told you that we can reconstruct this family just from f1 where did I get f1 from I got it from my differential equation because my differential equation gave me my family of maps here my solutions my solutions give me these family of maps then I just take t equals 1 and I get my f1 but do I really need a differential equation all I need is f1 forget about the differential equation suppose I just take a map let f rn to rn be a bijection whatever be a bijection f-1 the inverse of f which is well defined fn is equal to the composition of f n times f-n is equal to the composition of f-1 n times okay exercise for you to take home exercise the family fn n in z is a group on the composition very easy to check so we're really coming to the definition of a dynamical system is simply a group of transformations okay this group can be parameterized by continuous parameter like this or it can be parameterized by discrete parameter like this and what we have what I have tried to lead you to is the fact that a discrete time dynamical system you no longer need can arise out of a differential equation so if you have a differential equation you naturally have this discrete time dynamical system but you can have much more generally you just take any map and this gives you a discrete time dynamical system so this gives a generalization of the dynamical system of the group of transformations that you get from the flow of a differential equation and it turns out that this is a larger class of systems so not every map can arise as a time one map of a differential equation there are some restrictions because the differential equation has this in the case of differential equation this is inside a continuous flow so the points have to lie on some curves that have certain properties in general if you take a map in this map you could have points that are jumping back and forth in such a way that if you take all the points along one if you take one initial condition here and you iterate it you might not be able to put a smooth curve through it this is also a little exercise that is in the curves so this will be the starting point of the next lecture and this will be also the focus of the rest of the course we will concentrate mainly on these discrete time dynamical systems which are at one and the same time a kind of special situation that arises out of ordinary differential equations but also at the same time a more general setting because we don't even need not only we don't need a differential equation but we don't even need vectors or vector fields so we don't need to define this on Rn we can define this on any set we can take any set instead of Rn and we can define a bijection on that set and we can define this group on the composition okay so it allows us to generalize significantly the setting of these systems and this is what we will study okay, thank you very much