 Hello everyone, I am P.S. Dutty from TIFR central for applicable mathematics, Bangalore. I will be covering some portion of this COD course along with my colleagues Nanda Kumaran and Raju K. George. Today, I will discuss this qualitative theory of differential equations. So, let me first explain my notation and some notions and what is the main objective of this part of the theory of differential equations. So, we will be studying a system of N first order equations. So, I will write this system in the following form x j dot. So, dot for me is d by d t. So, let me write that. So, this is same as d x j by d t and that is equal to f j of x 1 to x n. And so, there are n equations. This is 1 less than or equal to j less than or equal to 1. So, let me number this equation for this class. So, I will call this. So, you can use the shortened vector notation. So, the same thing. So, vector notation which will be using more often. So, this whole system just written as x dot equal to f of x. So, this x is for me this n vector and each is a function of t. And similarly, this f is a vector of this n functions f 1 f 2. So, this t is the independent variable. So, usually we call it time variable. So, we would like to think the system one as a dynamical system. And this x 1, x 2, x n are positions of the particles. And we want to determine given some initial position how the system evolves given that law of the differential equation system of equation. So, this x j's are real valued. So, what about the functions f 1 f 2 f n? So, we need some assumptions on that. And to start with at this stage the assumptions on f 1 f 2 f n will be that the system one the corresponding initial value problem has a unique solution given an initial vector in R n. So, that is those are the assumptions. So, f 1 f 2 f n are functions either from an open set in R n. So, u is an open set it could be R n at this stage again. So, for example, if f 1 f 2 f n are globally lift change. So, if f 1 f 2 f n globally lift change that you already learnt in the existence theory of differential equation then I v p for 1. So, initial value problem I v p for 1 is values. So, what does that mean? That is let me. So, given a vector x 0 in R n t 0 is R there exist a unique solution x t of 1 satisfying x t 0 is x. So, for the time being these assumptions on the functions f 1 f 2 is sufficient in order to introduce some notions. And later on in fact we need more and we will add those hypothesis on these functions as we go along. So, we denote this solution this. So, this is guaranteed this is guaranteed for us from the existence theory once we assume this global list conditions on this functions f 1 f 2 f n. And most of our examples though I am stating it for a general n most of our examples are n equal to 1 that is 1 d n equal to 2 d and n equal to 3 that is 3 1 2 3 dimensions. So, n is referred to as dimension of the system. So, before I introduce some notations. So, let me also just. So, this notation this solution we denote it by denote this solution x of t. So, this I am emphasizing that initial time and the initial data and when they are not changing in a discussion then we might as well ignore those t 0 and x 0 just simply write x t. But when they are important in a discussion then we will emphasize the dependence on t 0 and x 0 and usually we take t 0 equal to 0. System 1 is called an autonomous system for the reason as the functions these are right hand functions f 1 f 2 f n do not depend on t explicitly. So, for a so when they do depend on t explicitly these functions depend t also the system a simple example in 1 d. So, d this capital D refers to dimension. So, this is just a single equation. So, this x plus t. So, is an example there is only one. So, let us call it equation not system and any non autonomous system just as a remark I will any non autonomous system system may be converted into an autonomous system. But then we have to pay a price for this by increasing the dimension by 1. So, I will quickly explain this. So, for example, you have if you have a 2 d non autonomous system and if I want to convert that to autonomous system my system will now has dimension 3 and this is quite significant as we will see as we go along. So, how it is done? So, let me just quickly mention that. So, x so we have a non autonomous system. So, x j dot f j x 1 x 2 x n and now these f j is also depend on t also this is. So, I have again system of n equations now the right hand side explicitly depend on t. So, I will make convert this into an autonomous system by introducing another. So, these are only n equations now I introduce another equation x n plus 1 dot and whole thing now I can rewrite as as an autonomous system x j dot equal to let me write j x 1 x 2 x n plus 1 now. So, 1 j less than n plus 1 and if you look at this system here. So, f n plus 1 is identically 1 that is constant function. So, we restrict ourselves to a study of autonomous systems and they enjoy certain properties. So, that we are going to list now if you nice properties and one more thing I would like to mention at this stage. So, if you have a nth order equation. So, this also we have already learnt nth order equation of the form d n y by d t n plus a n minus 1 0 y equal to 0 for example, where this coefficients a n minus 1 a n minus 2 a 0 can be functions of y and its derivatives up to the n minus 1 order. So, in that sense it is Cauchy linear we call it Cauchy linear this can be converted into a system of first order equations autonomous first order equations by simply introducing the new variable. So, x 1 equal to y. So, you already learnt it. So, just mention it as x 2 is equal to y dot etcetera. So, x n is equal to. So, this converts into first order system x 1 x 2. So, the consideration of first order systems in a sense more general than even considering any nth order equation. So, that is also makes it important. So, just we have to concentrate just on the first order systems. So, that will also cover in fact most of our examples are second order equations and we will be writing as first order system. So, that just concentrate the study on first order systems. So, recall that we have denoted by this x t t 0 x 0 is the solution of the system 1 with x of t 0 equal to x 0. So, x 0 again remember it is a given vector in R n. So, you can also write it as x 0 1 x z by x 0 n. So, n n vector. So, since we assume nice conditions on f 1 f 2 f n. So, the solution is defined for all t the solution defined for all t in R. So, we have already seen examples that this may not be the case for all differential equations. So, there is for certain differential equations the solution does not exist for all t in R. So, in that case one can consider the interval maximal interval of existence, but for simplicity I just assume that we will see those examples again and that will not affect the qualitative analysis we are going to do. So, next we define two notions these are important notions as far as the autonomous systems are concerned. The first one I will not give any notation that just passing I will mention that the graph of x t t 0 x 0 e z. So, this is a subset of R n plus 1. So, I write the coordinates as t x. So, t belongs to R and x belongs to R n and what is this x to do with t and the solution. So, I have start with a solution the initial value problem of 1 and now I define its graph. So, x is nothing, but x of t and t belongs to R. So, we will have some simple examples later. So, this is nothing, but suppose n equal to 1. So, this is just I am instead of drawing the graph I am just writing it as a set. So, that is a and the next notion is very important that we are going to. So, this graph is also important, but more than graph we want to suppress t we want to suppress t. So, we want to project this graph on to R n and that is the name we give for that that what is that projected set is. So, the orbit of 1 so, that always you remember that system orbit of 1 through x 0. So, this is a given vector in R n. So, this we denote by x 0. So, this O for orbit and x 0 is the point through which the orbit is passing through is defined by and now this is just a subset of R n this is just x belongs to R n such that x is you see if you compare this definition of the graph that is the set and now I have just suppressed that t. So, that means I am projecting this graph on to R n and I am calling that as orbit through x n and if you do not restrict this t belongs if you just restrict it to only certain set then the positive orbit I denote it by O plus x 0 again x 0 e the set. So, it is a subset of that full orbit. So, again x is in R n x is x of t t 0 x 0 now I restrict only. So, what happens in the future I am not interested in suppose I would not interested to know what happens before I start I am interested only in the future. So, I start at time t equal to t 0 and just would like to know what happens after that. So, I just study this positive orbit. So, the objective of this qualitative theory. So, the main objective now I can after introduction of this notions we can simply say that. So, given a system given one. So, that is describe all the orbits or just positive orbits. So, at least objective is very easy to state. So, I we are given a system one and just describe the orbits and positive orbits. So, easier to say, but very difficult to do. So, will be trying to do in some situations especially in 1, 2 and 3 dimensions. So, before I go further to describe the properties of these orbits and other things. So, let me just take some simple examples. So, simple examples. So, the first one is 1 d. So, one just already seen this many times. So, just let me write it in a different way that is all. So, this is just one equation lambda x. So, I take t 0 equal to 0 and x since it is one equation. So, let me write it as x 0 say 1. So, just so the solution is x t 0 1 in our notation and this is just simply e to the lambda t. So, at time t equal to 0 it takes the value 1. So, whatever may be the lambda. So, the graph. So, let me just the usual graph. So, for example, let me take lambda positive and we will see what happens to lambda negative also. So, this is let me describe it graphically. So, this is our t and this is x. So, I am taking lambda positive. So, it looks something like this and if I take for example, x 0 minus 1 that also I can just do it may be with different color. So, that is still lambda is positive. So, this is minus. So, this is the graph of this solution and this is the solution and these are also referred to as solution curves or even phase curves. So, that word phase I will explain in a minute that is not there is no mystery about it. So, phase curve phase plane analysis phase line. So, since we are in dimension we can call it this phase plane phase line analysis and what about orbit. So, let me just in this case for the same. So, this orbit passing through what is this if you just look at. So, I have to suppress t. So, this what you see. So, this even lambda is positive the solution is always positive. So, you see that it is just 0. So, this is the orbit and similarly minus 1 you see this is minus 1. So, again graphically we will be doing that most of the time. So, this which you can just represent it on the line. So, this is the origin. So, this we can see in fact it is one is no specialty here is true for all positive values and similarly here. So, one more thing we have to do with respect to orbits here. Since in case of graph we have the notion of positive t axis and positive axis at least there is a direction of t. So, we see that by looking at the graph whether t is positive or not and here since we are suppressing that t we have to draw a line to represent the increasing values of t in this orbit. So, as you see here. So, this is important. So, this is. So, they always put arrow in this direction in indicating that is if you go along this thing that means, we are going along the increasing values of t and similarly here you see that this is the. So, increasing value. So, whenever we draw orbit we will be doing this. So, that will indicate the increasing values of t. So, in this simple example. So, similarly if in lambda is negative you can do that thing. So, that is a no problem. So, what about suppose we are interested only in the positive orbit. Suppose just we are interested in positive orbit in this case. So, this is just select me in the next page. So, O plus 1. So, since we are taken t 0 equal to 0. So, we are only interested in t bigger than equal to 0 and that you easily see that it is just 1 and similarly here O plus minus 1 it is minus infinity. So, in simple examples you if you get used to this notions of orbits that will be useful. So, whenever you are in doubt you do a simple example and notions will be clear. They are not very difficult ones. So, just always try to do some examples some simple ones just like I did. And let me now take another example. This is a 2 D example. Again this we have seen many times during the course. So, now 2 D. So, there are 2 unknown functions. So, let me just write H dot. So, this is linear pendulum equation. This we have seen many times. So, we can write down the explicit solution in this case. So, again let me just. So, take t 0 equal to 0 and now this x 1 0 we need the initial point x 2 0. Let me take for simplicity just 1 0 passing through 1 0. So, you can immediately write down the solution. So, you have just let me just write x 1 t. So, suppress that everything. So, this will be cos t. So, it is very difficult to draw the graph even in the simple case. So, let me try to do that. So, graph. So, this is the t axis and let me draw this plane here. So, this is. So, you will see that. Pardon me if I am not drawing it correctly, but that is. So, it will be a spiral kind of thing. It goes on like that. You will see that as t moves on. So, if you draw that cos t and minus t, but orbit is very easy to explain. So, just you project everything on to this plane. So, orbit is very simple. It is just. So, let me just write x 1 x 2 and this is just 1 0. The orbit is passing through 1 0. It is circle and again want to indicate the increasing values of t and if you look at the equation, it will be in this direction. So, it just goes around. So, if you change the signs in the equations, for example, if you take x 1 dot equal to x 2 and x 2 dot equal to minus x 1, then we will be reversing the orientation instead of counter clockwise. We will be clockwise. We will be going counter clockwise. So, these are two simple examples giving you the idea of graph of a solution curve and then it is orbit. Now, we are going to discuss some important properties of autonomous systems. So, let me begin with that thing. So, before that, so again consider. So, now we are going back to system 1. So, remember the system 1 is that x naught equal to f of x and we have a solution. So, this is our system 1. So, we have this x of t t 0 x 0. So, solution which takes the value at given initial point t 0 and we define. So, the orbit. So, I will ask a simple question. So, now I pick any vector. So, remember this is an R n in this orbit. So, for just pictures x. So, suppose this is O of x 0. So, it is passing through x 0. So, let me just say this is x 0. So, that is a vector in R n. So, I am just trying it in the plane and x 1 is here somewhere. This side that side it does not matter. So, this means what? If you look at the definition of the orbit. So, x 1 is equal to x of t 1 t 0 x 0. So, remember this x of t etcetera is a solution of this system of differential equations. So, that you remember it is not just any ordinary function. So, it is a solution of that differential equation that you remember. And now I ask the question what is the orbit of x 1. So, here is the picture. So, I am considering an orbit through a given point x 0 and now I am picking a point in that orbit on that orbit x 1. And by definition x 1 is this for some t 1 or some t 1 in R 1. So, that is now I am asking the question what is this O x 1. So, your guess is very right and it has to be just the same. Just by looking at the rigorous mathematical proof comes from the uniqueness that you should remember. So, now you see that x is a solution of this x is a solution of the given differential system system of equations. And at t 1 it takes the value x 1. So, you interpret this one as the solution is taking the value x 1 that x 1 is given to us at time t equal to t 1. And by uniqueness there is no other solution that is the only solution passing through x 1 at given time t 1. So, the same x the same solution works for this point also since it is already on the orbit and hence the two orbits coincide. So, this is an important thing. So, if I start with an orbit and then I take any point on that orbit. So, orbit through that another point is the same as this orbit and that is coming from uniqueness. So, this remember that comes from uniqueness. So, this simple things, but you should know that. So, now we will go to describe some simple properties. So, simple properties simple, but important. So, I will state these things in the form of lemmas. So, lemmas I will put one lemma 1. So, if I emphasize the dependence on t because we are going to change that t if x t is a solution of 1. So, is x t plus c for any fixed and this is a very special property for the autonomous systems. So, it is certainly not true for the non-autonomous systems. So, simple examples you can try. So, what does this mean? Let me explain this thing. So, just let me that is if y is defined by y of t is equal to x of t plus c then y also satisfies. So, this is direct differentiation. So, just differentiate y with respect to t and then that will translate into the derivative of x. So, this proof is just one line. So, direct differentiation. So, I am not doing anything here and you the fact that this x dot t is equal to f of x 1 t etcetera x n t for all t. So, in particular if I replace t by t plus c this will be true and that is what we want for y. So, this you remember this is for all t. So, this equation is satisfied for all t. And simple example if you want to try this just again 1 d let me mention that 1 d. So, when I change t to t plus c you see there is a dependence on t. So, that also will be change to t plus c. So, I will not have this if I define again y t equal to x plus t c it will not satisfy y dot equal to y plus t. It will satisfy y dot equal to y plus t plus c. So, that is the explicit dependence on of the right hand side on t. So, this lemma 1 is enjoyed by only autonomous systems and from there those. So, simple a very useful result follows that I. So, you pick two points in R n. So, let x 1 and x 2 be in R n. So, I am writing superscripts. So, that you know that they are vectors they are not just real numbers. So, if you want me to stress that. So, x 1 or x i x i x i 1 x i n. So, they are x i equal to 1 2 and consider this orbits through x 1 and x 2 and this lemma 2. So, let put that 2 the lemma 2 says either this is equal to that or. So, this is empty. So, this is a very remarkable result. This follows immediately from lemma 1. So, that in the remaining 5 minutes I will just indicate the proof of this. So, this is remarkable thing. Remarkable thing it is slightly more than uniqueness. Uniqueness is certainly there, but it is slightly more than that. So, we are not using the graph here. We are projecting the graph on to R n and there also we are claiming the a kind of uniqueness. So, either any 2 orbits are the same or they are disjoint. There is no common point between 2 orbits. If they are distinct then they will remain distinct forever. So, proof is very simple and this one we will be using throughout this discussion on the qualitative theory. So, suppose they have a common point this orbit y x 1 and y x 2. Suppose they have if they are already disjoint I do not have to prove anything suppose they have a common point. So, there is a point here and there is a point there and they are equal. That means what suppose x 1 sorry for this x 1 t 1 is equal to let me not bother about that orbit passing. So, this is orbit passing through x 1. So, let me just suppose x of t 1. So, let me just write it t 0 x 1. So, this is a point in y x 1 orbit through x 1. So, this is x of t semicolon t 0 x 1 is the solution curve passing through x 1 at time t equal to 0 and suppose that is equal to x of t 2 t 0. So, in fact we already seen that is why that remark was there. So, if I take any point on the orbit then orbit through that point is also the same as the original thing. So, that is now define x of t. So, I am just remember these are all x x x. So, let me this is a new thing. So, let me may be I will define y of t y of t is x of t. This we can figure it out plus or minus we will see that this is for some t 1 t 2. Then I would like to conclude that the orbit of x through x 1 and orbit through x 2 are the same. So, this either of this thing. So, let me take this solution may be I should have called it x 1 that is fine. So, now by lemma 1. So, I am just translating this t 1 minus t 2. So, this y t is also a solution y is also a solution. So, now you just compute y at t 2 just let me figure it out t if I compute at t 2 then this is will be x of t 1 t 0 x 0 that is fine. Important thing now y of t 2 is just you substitute t equal to t 2 and this will just become x of t 1 t 0 x 1 and that is same as according to our hypothesis you just see there this hypothesis x of t 2 t 0. So, we have obtained a solution of again system 1 who is satisfied this y of t 2 equal to x of t 2 x 0 x 1 and now you use uniqueness. So, that is by uniqueness y is identically equal to y of t x of t t 0 x 2. So, this solution. So, this implies the orbit of the solution y the two orbits are same and by if you look at this one this says the orbit of y is same as orbit of x and now with this thing this orbit of y is also same as this one. So, this let me just so this complete this implies y of x 1 and completes the group and we will elaborate this in the next class. Thank you.