 So, welcome everyone to the next lecture. So, today and over the next two lectures, we will cover a topic called dynamical systems on manifolds. So, what is a dynamical system and what is a manifold? Dynamical system, of course you have seen, there is nothing but a differential equation, but today we will see in little more detail what is a manifold and that arises in non-linear dynamical systems and hence that is relevant. I would say this is one of the ways that people work in non-linear dynamical systems to be more precise. So, when we take x dot is equal to f of x, may be x dot is equal to f of x comma u if there is an input. Of course, this f and this f are different clearly. This f has only one argument x. On the other hand, this f has two arguments x and u in which it is implicit that u is an input to the system, but then the values of x itself at any time instant are where is the question. So, can x t take any value in Rn or is it that x of t is required to be in a subset of Rn? Is this subset a subspace or is it a more general set? Does it have some notion of dimension? When we speak of Rn, when we speak of the vector space Rn, we speak of it being n dimensional, but when x of t takes its values in not necessarily the whole of Rn, but in a subset, then what is the meaning of dimension? For those purposes, we will speak today in more detail about something called manifold. Manifold generally speaking are smooth manifolds. What is smooth about it that we will see soon? In other words, it is also called regular. What is regular about it? At every point its dimension is fixed. As we change the point, the dimension of the manifold, once we give the notion of manifold property called dimension, once we give that, we can speak about locally is the dimension constant as we change this point. So, that is what we will use to define a regular manifold. That is what is also smooth manifold. Then on a manifold, we will speak about tangent space. Why is tangent space relevant? Because it is in the tangent space that the vector field lives. Sorry for this bad handwriting. So, tangent space. So, we will speak about the notion of tangent space. If time permits, we will see some examples of manifolds, some equations today only. So, why is this relevant for dynamical systems? Suppose we say dynamics on circle or dynamics on sphere. So, it turns out that when we are speaking about the rate of change of angle, for example, we know that angle varies from 0 to 2 pi. Agreed, but not just that the angle equal to 2 pi is same as 0. So, in that sense, it is incorrect to view this set, this angle as a interval like this. Why I am not making it closed at both sides? Because the angle equal to 2 pi same as 0. So, we let angle to be equal to either 0 or 2 pi. We cannot let it be equal to both 0 and 2 pi, because they are actually the same. But then it appears, if we write it as an interval like this, it appears like some point, some angle value here and some angle value here, let us say 0.1, this is radians. When we say 0 to 2 pi, clearly the angle is being measured in radians. But suppose angle equal to 0.1 and angle equal to 2 pi corresponds to 2 into 3.14, which is let us say 6.28. Suppose 6.27 is slightly less than 2 pi, these two values of angles are not actually very far. So, one should note that this particular point is actually the same as this given that these two angles are the same and hence an open interval is not a good way to picturize this particular set of all values where angle, the set where angle takes its values. On the other hand, if we let that particular angle be denoted like this on a circle and we say that this is theta is being measured like this. So, we know that as theta increases and 0.1 angle and 6.27 angle indeed are close. So, here it is also very explicit that angle 0 and angle 2 pi are the same. So, when angle, for example, when dealing with angle, it is reasonable to think of the angle as angle theta takes values on a circle. At any time instant, we can say that theta of t is some particular point on this circle. So, that is one example where we like to think that our theta of t does not take arbitrary real values. Even though it is mod 2 pi, we know that angle is the same when if it is differing by a integral multiple of 2 pi. In spite of that, this is not a good set because it does not suggest that 0 and 2 pi angles are the same, for example, or that 0.1 and 6.27 angles are actually very close. On the other hand, instead of this, if we let the theta of t takes its values on a circle, then this is a correct representation of this set where theta takes its values. Now, we can ask that this particular set, the circle, is it one dimension or two dimensional? So, the question, the next question that arises is that what is the notion of dimension for such a set which is not r? If it is a vector space, if it is rn or rm or r or r2, the plane, clearly the dimension is here it is nm, here it is 1, 2, but then for such more general sets. So, this is what we will like to call a manifold. The circle is an example of a manifold. But for such sets, what is the notion of dimension? So, very soon we will make this more precise. So, we will like to say that if we are sitting at a particular point on the circle, then locally this just looks like a line. For example, when we are on this earth or when we are on this planet earth, that time we know that actually the earth is a very big globe, it is a big sphere. But at the particular place where we are, it looks like r2. We like to think that we are on this particular sphere which is r2. So, sphere. So, let us draw a sphere like this. So, when we are at a particular point on the sphere, then when we draw a tangent to this, at that particular point, tangent plane, we are not, at a very, at a particular point, we are not very concerned that far away this particular plane indeed gets rotated and becomes a sphere. Locally at a particular point it looks as good as a plane. So, that is as far as local, local view point. So, we will like to say that the dimension of a set is what it is when viewed very locally. So, how do we make this more precise? So, for that purpose we will speak about embedding, embedding a manifold in Rm. So, manifold just loosely speaking a set that looks like Rn locally. Locally meaning wherever we are at a particular point when we view around it, then it looks just like Rn, but not necessarily globally. Not necessarily globally. When we view the entire set together, then it need not be like Rn. It is only locally that we think that it is Rn. More generally such a manifold M might have to be embedded in a dimension Rm vector space and clearly in that case, M will be greater than N. So, as extreme case it might be in fact equal to M in which case Rm was equal to Rn. That time the manifold itself was Rm, except for that or it can also be a special case Rm equal to Rn is also manifold. The entire Rn is also a manifold. Open subsets is also a manifold. So, clearly manifolds can be bounded. It need not be unbounded like Rm is. So, one can have smaller subsets also now. So, how does one characterize it? So, typically manifold described by equations. For example, circle. If you take the unit circle, as far as the angle is concerned it does not matter what the radius of that particular circle is, but let us consider that the radius is equal to 1. So, x square plus y square equal to 1 is that particular circle that we already drew, a circle centered at the origin and radius equal to 1. So, we can view this as f of x, y equal to 0, where f of x, y is defined as x square plus y square minus 1. So, it turns out that manifold can be written as solution to a system of equations. In this case, there is only one equation and two variables. One can write f of x, y equal to 0, where f of x, y is defined like this. So, now what is a particular point? A particular point will be in the manifold. It will be on the circle if it satisfies f of x, y equal to 0. So, let us take a point is 3, 4 on manifold. What was the manifold defined as? This particular manifold, the circle was defined as the set of all x, y in R2 such that f of x, y equal to 0. So, the definition of our set, as far as this definition is concerned, 3, 4 we can check, 3 square plus 4 square minus 1 is equal to 9 plus 16, 25 minus 1, that is 24, that is not equal to 0. Hence, it is not on the circle. So, let us consider 3 by 5, 4 by 5. Is this an element of that manifold? Is this in that set? Set of all points will satisfy that equation. So, we can check this. It turns out that this will indeed be equal to 0. 3, 5 square plus 4, 3 by 5 square plus 4 by 5 square minus 1 is that equal to 0? Yes, it is equal to 0. This will turn out to indeed be equal to 0. So, we see that this particular point is on the manifold. Now, what we can do is we can take the so-called Jacobian, del f by del x, del f by del y. We can construct this particular matrix. That particular matrix turns out to be equal to 2x, 2y and this matrix in general will have x and y because f was dependent on x and y, f was the function of x and y. So, this one we will evaluate at a particular point. For example, 3 by 5, 4 by 5. At this particular point on the manifold, when we evaluate it, we get 8 by 5 and we get 6 by 5 and 8 by 5. This is what we get as del f by del x, del f by del y evaluated at a point, at a point P in the manifold. For the point P equal to 3 by 5, 4 by 5, we get it equal to this. Now, we are able to speak about the dimension of the manifold more concretely using this. So, let us come dimension of the circle manifold at P is equal to 3 by 5, 4 by 5 equal to dimension of null space. Null space of which matrix, of that particular matrix that we obtained, 6 by 5, 8 by 5. So, this is a particular matrix that we get by evaluating del f by del x, del f by del y and we can speak about its null space. Why we have to speak about null space? We will see in some detail very soon. So, f depends on two variables, del x, y rank of this particular matrix, del f by del x, del f by del y evaluated at a point P, at which point P equal to 3 by 5, 4 by 5 turned out to be equal to 1 and hence dimension of null space. Null space of a matrix is set of all vectors that go to 0, where that matrix acts on it. We will give a formal definition in the next slide. This is equal to minus rank, rank of this matrix which was equal to 1, this is equal to 1. This is the dimension of the null space at that particular point. So, notice that the matrix del f by del x, del f by del y depends on x and y and when you substitute different points, you get different different matrices and in general the ranks might change even though the number of columns is the same and hence the dimension of the null space might change in general. But one can verify that at every point P on the manifold, the dimension will indeed be 1. The rank of the matrix will be 1 and hence the dimension of the null space will indeed be equal to 1 and hence the circle manifold is what we will like to call as a regular manifold. This is what we will see in more detail now. So, before we see in more detail, we will just give a formal definition of a null space of a matrix. Suppose we are given with a matrix P, capital P this is different from the point P that we just now saw with n rows and m columns, then it is null space. Null space of P is defined as set of all vectors v sitting in R m such that P v equal to 0. So, take a matrix with all real entries, R for real entries with n rows, m columns, then its null space is defined as set of all vectors v such that P times v equal to 0, the matrix P times the vector v equal to 0. So, some other words for this purpose is called kernel, kernel of P also means the same thing. So, this is set of all vectors that go to 0, null space and kernel both mean the same. They are both in general a subset of R m. So, if P is a map from R m to R n, then null space and kernel both mean the same that is what is defined here are subsets of R m. Why do we say P which has m columns maps R m into R n? Why? Because it has m columns when it acts on a vector the way it is written here, it will require the vector v to have m components and hence the vector v is an element in R m. So, null space is an element of this. So, come back to our particular problem. So, del f by del x, del f by del y that particular matrix when we evaluate at a particular point small p equal to 3 by 5, 4 by 5, then we had got that this one is equal to 6 by 5 and 8 by 5. This particular constant matrix we can look at the set of all vectors that go to 0 and that turns out to be nothing but null space of p is of let us call this particular matrix as capital P. It is null space is nothing but the span of 8 minus 6. So, span means you take linear combinations of this particular vector and that particular that set which you get by linear combinations of this is precisely equal to the null space of this. They are precisely the vectors v which get sent to 0. This you can verify by just plain multiplication. So, what is the dimension of the span of this? Exactly one. We have only one independent vector and any linear combinations will all generate a one-dimensional subspace. So, this is dimension 1. So, that is how we conclude that the circle locally at every point gives you a null space of dimension 1 and hence it is a manifold of local dimension 1. The next question arises is at the point p we verified what about other point? The other point also will it be indeed null space dimension equal to 1 that is indeed the case that you can verify yourself, but we will right now define dimension of a manifold little more generally. So, suppose f is a map from R m to R n and we say f equal to 0 is a system of equations. So, please note this x here is different from the x that we wrote in the previous example. Why? Because f acts on R m and gives you R n because of that if f acts on x, x has to have m components already and f of x itself has n components. So, more precisely we can say f 1 of x 1, x 2 up to x m equal to 0 f 2 x 1, x 2, x m equal to 0 like this up to f n equal to 0. So, there are actually n equations that is why I wrote system of equations, this system of n equations to be precise and each equation involves m variables. So, this system of equations may or may not have a solution in general. So, suppose you take a particular point x 1 up to x m that satisfies all these n equations that particular x point you will include into the manifold. So, more generally manifolds are defined like this, large class of manifolds are all defined as solution to a system of equations, solution to a system of n equations and this already makes that manifold a subset of R m. So, what is our manifold? Manifold was a subset of R m more precisely it was set of all x in R m such that f of x equal to 0. So, n equations are satisfied. Now suppose we define del f by del x, this we can evaluate at a particular point small m in m. When we evaluate it at a particular point then this matrix that we get after evaluating becomes a constant matrix with how many rows it has exactly n rows because f 1 up to f n, n functions are getting differentiated and how many columns will it have it gets differentiated with respect to m components. Hence, it will have m columns. So, this particular matrix this matrix that we have has n rows and m columns one can speak of rank of this particular matrix del f by del x. After evaluating we speak of rank of constant matrices as far as this course is concerned. So, we will find out the rank of this matrix only after evaluating it at a particular point m on the manifold. Of course, in principle this matrix is defined for any point in R m. We can evaluate it at any point in R m, but then we are interested in what happens to this matrix at a point on the manifold. Hence, we are going to evaluate it at a particular point m inside the manifold m, capital M. So, this particular rank that decides what is that will help in finding out the dimension of the null space. Suppose this rank is equal to R, suppose R is that particular number then what is the dimension of the null space in that case by this according to this dimension of null space of del f by del x after evaluating it at a particular point m on the manifold will be equal to m minus R. So, I should point out a few things about the notation here. This m was because capital M is a subset of R m, it is a integer. So, this m is because of this particular m, it has this particular matrix has m columns while this m is a particular point p on the manifold. So, it is better that I change this to a particular point p on the manifold where dimension of the null space at point p in the manifold. At point p in the manifold, what is the dimension of the null space? m minus the rank of this constant matrix. Which constant matrix? The matrix that you get by evaluating that matrix at this particular point p on the manifold. So, this is the dimension of local. What is local about it? Because we have evaluated this matrix at that particular point, local dimension of that manifold at point p. So, now we can ask, when you go for different different points p, does the dimension change? Does the dimension, what is the dimension? m minus R, does the rank change? m itself will not change because this entire manifold is a subset of R m. So, m itself will not change, the number of columns of this matrix will not change, but the number of the rank itself might change depending on the point that you substitute. So, does the rank R change with the point p where you evaluate it? This matrix of functions, you can find out once and for all, but depending on where you evaluate it, its rank might change. If the rank does not change depending on the point p of the manifold, then the dimension of the null space will also not change because m minus R is the dimension of the null space at that particular point p. So, we will call this manifold. Manifold is called regular if dimension is constant. What dimension of what? Dimension of the null space. Dimension of the null space will be constant if the rank of this particular matrix is constant. So, such manifolds are called regular manifolds and they are the ones that are easiest to study and we will study only them. So, what are examples of such manifolds? Circle, sphere, all the ones that we can think of. So, circle is a one-dimensional manifold embedded in R2 because it is embedded in two-dimensional plane. Sphere, so this circle, this sphere, this sphere is also called S2. This is called S1. So, S2, the sphere S2 is a subset of R3 while S1 is a subset of R2. So, this is also a regular manifold in the sensor. At any point, we can evaluate the particular function. How is S2 sphere defined? It is defined using the formal x square plus y square plus z square minus 1 equal to 0. In R3, in three-dimension y, x, y, z, three components, if you take one equation, then if that equation, unless that equation is trivial, unless it does not set any constraint, we expect that a two-dimensional degree of freedom is there and this degree of freedom is exactly the dimension of the null space that we were talking about. So, at any point, there are two local directions, one can move and those two dimensions are indeed the null space of this particular del f by del x, y, z that we get by using this equation. So, that, hence, the sphere we will say is of dimension 2. So, what is an example of an irregular manifold? So, look at this particular set, the interior of this set. Here, it looks like R3. Here, it looks like R2. At this particular point, you can go anywhere in these two directions, but as this becomes like this, the same set, when we are here, there are only one independent direction. Either we go here or the negative of that gives the opposite direction. So, there is only one independent direction. At this particular point, on the other hand, in the interior here, one can go in two directions. Similarly, if we have a circle and its interior, on the interior, we have two dimensional. Here, we can go anywhere in two directions independently, but on the boundary, we have a problem. We cannot go here, we can go like this and the inside. In that sense, there are some constraints where all we can go on the boundary. So, these are situations where we say that the dimension is not constant. This is an irregular, this both are irregular manifolds. So, with that, we will not look into more detail about how manifolds are defined and what is the meaning of its dimension at a particular point because all our examples will have manifolds with constant dimension at every point and they are the regular manifolds. So, what is the vector field defined for a manifold? So, take a manifold m of dimension say r. So, this r is not to be confused with the rank r that we had in the previous few pages. Suppose, its dimension is r and this m we will like to embed it in rn. So, it is embed. What is embed about it? Even though as I said, the sphere itself is dimension 2 manifold, the sphere is physically being placed in r3. So, we embed it in a larger dimensional vector space if required. So, sphere S2 is a manifold of dimension 2. Dimension 2 manifold is embedded in r3. r3 meaning x, y, z our space has dimension 3. So, one can think of the sphere S2. Even though it is a manifold of dimension 2, it cannot be placed in r2. One has to embed it in a larger dimension vector space r3. So, it is an important question about manifolds, about what dimension vector space you have to minimum go larger and embed. So, such theory is explained in more detail in books by Spivak. He has one book on calculation manifolds, one thin book, but he has many more volumes which speak about such questions in much detail and also more complex questions about manifolds. So, as far as we are concerned, we are dealing with dynamical systems where the variable x evolves on a manifold. So, consider x dot is equal to f of x. For the time being this is a time invariant system and x of t takes its values in a manifold m and we do not want this manifold m to have a dimension that is varying. So, we will call it a regular manifold. Regular manifolds are the ones which are also called smooth manifolds. One can speak of C infinity function defined on manifolds, tangent spaces defined on such manifolds in a more general setting. So, let us take an example. We will like to say that while x evolves on the manifold, the vector field itself x dot, this function f, this f is different from the f that we had used for defining the manifold. There f was such that its solution set of all solutions was defining the manifold, but right now the manifold is already defined and if required it is also been embedded in a larger dimension vector space. Right now this f is defining the vector field, it is defining the dynamics. So, take a sphere and this x is evolving on the manifold only. Now we will like to say that x dot is a vector in which in which set we will like to say that it lives in the tangent space to the manifold at that point. So, take another, let us start with a circle. So, x of t takes its values on the manifold and suppose the manifold is a circle and suppose at some time instant it is here, then x of t, the fact that x of t has to remain on the manifold, x dot itself takes its values in a tangent space, in a tangent line to this manifold either positive or negative. That is where the rate of change can be. Why it is important to note that the vector x dot itself cannot be out going out. Why? Because you see if one is required to be on this manifold, the circle, then the rate of change cannot suggest that we go here. It will clearly come out of the manifold immediately, but if we say it has to go in this direction, then it will go little in that direction and then one gets a different point on the manifold and one evolves like this. At this point, we might say we have to go like this, here like this. Of course, we are not going to move here in the next time instant, infinitesimally after a little amount of time we will reach here and there the tangent is at a different point. So, x dot is equal to f of x is a differential equation at a particular x on the manifold, f x is a vector in the tangent space to the manifold at that point. More precisely, tangent space to the manifold of dimension r at point p of manifold m. So, it is a tangent space to the manifold. It is tangential to the manifold, but tangential at each point at the point p of that manifold. This tangent space is if the manifold itself was dimension r, locally it looked like r r. So, dimension r m of dimension r means locally it looks like r r is r dimension locally at point p. It is like r r, for example, a sphere. As I said, let us take a circle for example, at a particular point we said that it looks like a line. So, when we draw the tangent line to this particular circle at a point p, then the tangent space, this tangent space itself is certainly r r r r equal to tangent space to the manifold at the point p, where p is in the manifold. So, what is this tangent space to the manifold at point p? Of course, p has to also be in the manifold. We do not consider tangent spaces to the manifold and the tangent space is itself tangential at some point p, not on the manifold. No, that is not going to happen, because x of t lives in the manifold at any time instant. At this point, suppose this was x of t, then it can move either here or here. So, it is forced to be tangential to that particular manifold at every time instant. That is the rate of change and hence that particular set of all vectors, where the rate of change can belong to, that is called as a tangent space to the manifold at that particular point p. That tangent space not just looks locally like r r, which the manifold was looking like. Manifold was locally like r r at every point p, because it was a r dimensional manifold. This tangent space on the other hand, in fact, is equal to r r. It is equal to r r, where what about the origin of this particular vector space? The origin of this vector space is exactly the point p. That is the important thing that there are different copies of this tangent space. T tangent space to the curve at the point p is r r with origin of this vector space as point p, p of manifold. What do I mean by this? Let us take this circle. Let us take this particular point. This is one dimensional manifold, the circle and this is a point p. This is the origin of this particular line. We speak of line as r 1. This is the origin. We speak of this as r 2 and this is the origin. Now, when this r 1 happens to be the tangent space to this particular manifold, this manifold m itself is embedded in r 2. The circle itself is embedded in r 2, where this perhaps is the origin. So, origin for r 2 in which circle is embedded. So, this is our coordinates and our particular circle is here. So, notice that this particular circle does not have its center at the origin like the earlier circle. This is some other circle whose center is somewhere else radius is not necessarily 1. This is just a manifold example of a manifold. This also can be written by some such system of equations like we have written before. But now, on this manifold the point p is here and its tangent space is here. The origin of the tangent space itself is also an r 1. That is the tangent space to that particular manifold at the point p and the origin of that particular vector space r 1, its origin is this 0 which is exactly the point p. So, if you have another tangent space at this particular point, this is the origin for this, let us call this p 1, this p 2. Similarly, if you have this as another point p 3 and this is the origin of that. In that sense, we have plenty of tangent spaces. We do not have just 1, 2 or 3 tangent spaces. This circle has been embedded in r 2. There is a origin of r 2 in which this manifold has been embedded. That origin is a origin of r 2, but we are speaking of this manifold which is one-dimensional manifold. And for this manifold, this point p 1 at which that tangent space is tangential to the manifold at that point p 1, that p 1 is itself the origin of that tangent space. As I said, the tangent space is not just locally like r 1, it is in fact r 1. So, where is the origin to this particular of this vector space? Its origin is exactly p 1. What is the significance of this point p 1 being the origin that we will see when we actually consider the differential equation? So, let us consider a circle in which theta dot equal to 1. So, at every point, we will like to say that this is how theta increases. So, one may say, why is this not anticlockwise? Normally, we take anticlockwise as positive. This is just a manifold and one can have any convention as far as this manifold is concerned. So, now, at every point, rate of increase of theta is equal to 1. So, the rate of change is equal to a vector 1 in that direction. What about theta dot equal to 0? Where would that vector be? At every point, it is a vector of length 0. So, it would just be there, neither left side or right side. Where would theta dot equal to minus 1 be? Theta we have said is increasing like this, increasing clockwise. So, theta dot equal to minus 1 would be that theta is decreasing. So, it would be in the opposite direction. At every point, this one would correspond to, this was our theta, theta was increasing like this. At every point, the vector would be pointing like this. It is decreasing at a particular rate. What about the length of the vector? The length of the vector indeed denotes the rate of change. And length of the vector has actually units theta by time, the rate of change of angle with respect to time. And hence, the length of the vector itself cannot be directly related to the coordinates R2 in which this circle has been embedded. So, except for the length of the vector, the direction itself has lot of significance. But it also has some relative significance in the sense that we know that if this is theta dot equal to minus 1, then this vector corresponds to theta dot equal to minus 1.5 because it is longer than this vector. So, the vectors within the tangent space can be compared with respect to each other. But a length of a vector in the tangent space is not directly comparable to the length in the manifold itself because elements in the tangent space have dimension, value divided by time. So, in that sense, the units are different. So, notice that we can see that theta dot equal to 0 means that at that particular point that arrow has length 0, it neither increases nor decreases and that corresponds to the origin. The 0 is in the tangent space to that particular curve at that particular point and hence, it is exactly the origin. So, it is a point P neither increasing nor decreasing. So, important conclusion is origin of tangent space, origin of the tangent space Tm at point P is exactly at point P of manifold. So, as an example, x dot is equal to f of x, f of x equal to 0 at every x in the manifold. So, this is like constant solutions or solutions are constant. x of T is identically equal to x0 for x at time t. At any time, if at t equal to 0, it is equal to x0, it will identically be equal to x0 for all time t, for all t greater than or equal to 0 also. Why? Because the rate of change is 0. So, the tangent space at every point, which vector is been picked. So, this differential equation, from all the vectors that are possible in the tangent space at that particular point at an initial condition, f of x picks a particular vector in that tangent space. It tells us which direction it will evolve and if you say constant vector field, it means it will pick the 0 vector. So, it will just remain there that there is no arrow, there is no arrow because the arrow has length 0. So, that is an example where the origin is in fact the vector that the vector field has picked. So, x dot is equal to f of x on a manifold tells which vector in tm at a point p is picked for x dot. This picking is what f of x is doing, f of x decides. So, this is the way of understanding a vector field. There are vector field, a particular differential equation tells that at a particular point x on the manifold, f of x tells you which particular vector in the tangent space to the manifold at that point p has been picked and has been defined as x dot. When you integrate, you go to a particular future time and then f is evaluated at a different point of the manifold. But the fact that at each time instant, the vector belongs to the tangent space ensures that the vector does not, the rate of change does not make the x go out of the manifold. The fact that the dynamics are constrained to be in the manifold that is guaranteed by the fact that f of x is an element of the tangent space. f of x is not suggesting a vector outside the tangent space. That ensures that the dynamics remain on the manifold. To the manifold, dynamics remain on the manifold. So, this set of three lectures, this and the next two are not intended to be into a lot of depth about this particular way of viewing non-linear dynamical systems. It just suffices that we take the union. We just introduce some words. So, if you take the union over all points m in the manifold of the tangent space to the manifold at the point p, the union of this is called tangent bundle. What is a tangent bundle? It is a bundle, it is a collection of tangent spaces. How is the collection being defined? We said t m, p is a tangent space to the manifold at the point p, but you can vary this point p for all p in the manifold. That union, that collection together is what defines the tangent bundle. One can speak what is the structure of the tangent bundle itself. These are indeed questions that are asked from a research viewpoint. Since many years, what about control? Control of dynamical systems. So, how is control viewed here? Till now we had been viewing x dot is equal to f of x. At every point p, we pick only one vector from the tangent space, but if you have x dot is equal to f of x, u. So, as I warned in the beginning of today's lecture, these are different. There is only one argument. This is a different system of equations. Only one argument to f, while here we have two. When you see various papers, one should note that this f has two arguments. Two arguments and we would like to say that u is input. u is an input to the system. Now here, different u values helps pick different vectors in the tangent space. More precisely, it helps you pick a whole family of vectors in the vector field. So, what family? Let us take an example and see. Suppose, we are on a sphere and at this particular point p, this is a tangent space to the vector field, tangent space to the manifold at the point p. Suppose, this is how f of x dot is equal to f of x. When 0 input is given, this is how it is, but when some particular value u1 is given, it helps to say here. So, it is possible that this is a whole class of inputs that you get for different different values of u1. So, different u1 values, different u values fetches all these vectors in this, whatever has been shown, fetches a subset of TMP. One can speak whether this, it fetches a subset for different values, this f of x, u will be different vectors, you see. So, it will help in picking not just one vector in TMP, but a whole collection of vectors. In that sense, this whole family of vectors from the tangent space can be pegged by taking different values of u. So, now we can ask what about controllability? So, control itself means that you pick a whole collection of vectors, that whole collection is defined and now inside the collection which vector you pick is about choosing a particular value u. Controllability is about whether that collection is enough that you can go anywhere in the manifold. So, global controllability is that you can find some trajectory to go to every point. Perhaps you need lot of time to go there. On the other hand, we can ask that by just very small quick manipulations can we go to every point nearby. So, this is what we will say small time local controllability. So, given the fact that different u values might give you more than just one vector in the tangent space to the manifold at that point p, but by different choices of u's, can you go around some open interval, open neighborhood of that manifold at that point p? Can you go if that open neighborhood is made smaller and smaller? That is what is called small time local controllability. On the other hand, global controllability asks easier question, can you go from any point to any point by some choice? When you go to different points, you get a different family of vector fields which is a collection of the tangent space at that point. So, by such careful choice, can we go from any point to any point? That is what is global controllability about. So, this is as far as the different questions that are asked using this long way using this notion of tangent spaces. So, we will see some more properties of tangent spaces in a few minutes in the next lecture. Thank you.