 Welcome again. So, in the first lecture in this module, we have introduced the linear systems and we were trying to understand the autonomous linear system where A is independent of t. Autonomous as I mentioned yesterday, the autonomous system has many advantages and we got the existence of solutions in this form. So, we was trying to study this problem x dot of t is equal to A x t with x at t naught is equal to x naught and this have a solution in the form of an exponential representation and solution have this is the unique solution given by e power t minus t naught A x naught. As I said it has disadvantages, one is the other thing computation of an exponential of the matrix is generally difficult and secondly, it does not reveal anything about the solution trajectories. So, today this is a scenario in this scenario, we have introduced what is called the linear equivalence, whether the matrix A can be linearly equivalent to another matrix and this is nothing but the similarity of the transformations. So, when you have A is similar to B via an invertible matrix B, the system corresponding to A as I call the system 1 can be converted to a system corresponding to B something like y dot equal to B y t and if the system B is corresponding to system corresponding to B is easy to the exponential is easy to computer then the solutions will change and this change is only a coordinate change and the nature of the equilibrium point which we have introduced yesterday will not change. If the equilibrium point is stable it will be stable for even for the transformer equation. So, what we are going will be going to see a detailed analysis of 2 by 2 systems which is much more simpler in particular if we can get the matrix B to be diagonal. So, an immediately solution can be written and this is called the diagonalization of the matrix, but as we know from the linear algebra not that every matrix can be diagonalized and that is what we have seen a particular matrix lambda 1, 0 lambda cannot be diagonalized that is a simple one example, because the diagonalization is something like equivalent to the existence of an independent diagonal vectors which may not be available to you. What best we can do is what is called the Jordan decomposition, but this Jordan decomposition in 2 by 2 systems is much more easy and we will have a complete analysis today. So, in this scenario we are going to introduce what is called a phase plane, we introduce the phase portrait, we will do all this in the form of examples and we will introduce after that what is called a dynamical system corresponding to this one dynamical system. These notions we will see through examples of the system, we will also the concept of flow and vector field, you see all these notions quickly I will recall, but you will also see these notions in more detail when we study the non-linear system. So, as I said yesterday these modules is a precursor to the non-linear system and that is what more interesting problems you will see more interesting applications we will be able to see these things. So, let me start with the example that is why the best way to easy. So, let us look at a very simple example x dot of t is equal to a of x t with x at say you can put at any point in particular you can put at 0 because of the autonomy autonomous system what is a I am taking to be a very diagonal matrix of the form 1 0 0 minus 2. So, this is a decoupled system. So, the this system is nothing but you have x 1 dot t is equal to x 1 t and x 2 dot t is equal to x 2 dot t is equal to minus 2 x 2. Now, you know how to write the solutions either you can directly use this formula because it is a diagonal matrix. So, you know how to compute that or you can solve here whatever it is the solution will look like this the solution of x t is nothing but it is at the origin say it is a 0 we are trying to do it and hence the solution written as e power a t e power a t is nothing but e power t 0 0 e power minus 2 t x naught. If you want it in the component wise you will have x 1 t. So, what if you want to learn this course you have to work this kind of more and more examples here. So, you will have the solutions e power t x naught 1 what is the first component of x naught x naught has 2 component. So, this is x naught 1 and x 2 t is equal to e power minus 2 t into x naught 2 t. Now, here is where I want to understand you something the as I say t is a time which is treated as a parameter, but this motion of x x 1 t x 2 t the x 1 t the curve x 1 t x 2 t. If you write it this is equal to your x or t this moves in the plane x 1 x 2 moves in the x 1 x 2 plane. So, what is happening a is that this is can think it as a motion of a particle in the plane x 1 x 2 and this plane x 1 x 2 plane is the phase plane called phase plane associated to the system. And you want to understand as this is the motion of the particle and x 1 t x 2 t is the parametric representation with the parameter being time in the x 1 x 2 plane. So, you want to see the picture the trajectories in the x 1 x 2 plane you see you have x 1 here you have x 2 here and then you want to see how the trajectories will move along that. If you look at as you say that a is an invertible matrix in this case the origin 0 this is this origin 0 0 is the 0 equal to 0 0 is the equilibrium point is the equilibrium point which is a solution. So, if you any solution starting with the origin will remain there forever, but now look at the trajectories x 1 t which I have introduced for this x 1 t is equal to e power t x naught 1 and x 2 t is equal to e power minus 2 t x 0 2. And this one if you look at it tower be the x naught 1 or x 2 depending on the sign of x naught 1 this will go to plus x or minus infinity as t tends to infinity. On the other hand irrespective of which coordinate it belongs to this will go to 0 as t tends to infinity. So, here is a situation wherever be the initial point the first component will go to in one of the component will go to plus or minus infinity and the other component will go to 0. So, you want to plot this graph in the x 1 at 2. So, what you do is that to plot this one you eliminate this one eliminate t that is what you do it you want a closed representation if you eliminate here the best way to eliminate square x 1. So, you will go back to at 1. So, if you square it x 1 square will be e power 2 t, but e power minus 2 t. So, you can put it the denominator these are all will be constant x naught 1 and x naught 2 will be constant. So, you will get basically c by x 2. So, this is the closed form of this solution. So, if you plot here suppose you started a trajectory at this point if you start that trajectory. So, this is the point nothing, but some x naught 1 0. Since the second component is 0 these are all these equations is a decoupled system x naught 2 will be 0 the trajectory will remain x 2 component will remain to be 0 all the time. So, the trajectory will be here. So, this will be a trajectory basically. On the other hand if you start a point here. So, this is also a trajectory only thing which direction we will move the direction of the motion is only important thing you have to see and the direction will prescribe whether it will go to plus or minus. So, in this particular case. So, if you start with a point here and as x 1 component will go to 0 infinity the motion will be in this direction. So, if you start from here the motion from if you start from here this will move in this direction on the other hand if you look at it here if you start a trajectory from there the again the trajectory will remain in the x 2 itself because x 1 will be 0 all the time, but the x 2 part is going to 0. So, the trajectory will move towards here. If you start from here the trajectory will move here. So, this is at the time t equal to 0 or any other time and as t this arrow represents as t tends to infinity all the time go to understand this arrow as t. So, if you move from here it will move along this direction all that equilibrium point, but if you start from here it moves away from the equilibrium point. Now, let us start from here. So, suppose you take an initial point here this is your point x naught. So, if you start from there your trajectory will the graph tells you is something like this kind of thing is something like a x 1 square equal to x 2. So, it will the graph if you plot here. So, this is a best example to see it will plot here your trajectories if you plot because this equation is defined for not only for t positive it defined for t negative as well. So, if you start here your trajectories will be here. So, all your trajectories will plot here. So, if you start. So, I have your thing. So, if you have your trajectory you have your trajectories like this if you have your thing you will have your trajectories here if you plot here your trajectory all the trajectories. So, only thing is that where is the motion in which direction this will move. For example, if you start from here as you know that x 1 coordinate go to infinity and x 2 coordinate will go to 0 as t tends to think it moves along this direction because this is the 1 x 2 component is going to 0 x 1 component going to infinity. So, if you start. So, it will match with this it will move in this direction if you start from here. So, this is the part where t tends to greater than 0 part. So, if you move here it will move in this direction if you have initial point it will move. So, everything moves away from that. So, it is t. So, the phase plane together with all the trajectories is the phase portrait of the system. So, the phase plane together with the trajectories together with the trajectories is called the phase portrait of the system called the phase portrait of the system phase portrait. So, this is the phase portrait phase portrait it is a portrait. So, this gives you the complete analysis of your trajectory and such an equilibrium point. So, this equilibrium point because the trajectory is to one components pose. So, if you start a trajectory anywhere here it is moving away as t tends to infinity it moves away you see that is what. So, move away from the equilibrium points whenever that happen such things are called unstable. So, this is an unstable equilibrium it is not stable. So, however close it is it will eventually move away from that unstable equilibrium unstable equilibrium point. And in this particular thing these are there are different types of stable and unstable equilibrium and this is something like a saddle you see it is sitting something like that a trajectory is moving and it is called the saddle point actually saddle point. This equilibrium is a saddle point see this is the situation about the. So, you have the phase plane where your trajectories are moving in R n space in n dimension this phase plane is called the phase space for R 3 the phase space is x 1 x 2 x 3 phase and R 4 is a 4 dimensional phase plane. And so, if you are studying a system in n dimension and it is you want to see the motion of the trajectories in the and that is the corresponding phase portrait of the system. So, with that so, we have other another notion called dynamical system. You can also call the given system to be dynamical system but we can also give a more thing consider the map p from R plus or R if the solution x is R plus cross R i to R i given by phi at t and x naught is nothing but x t x naught. And so, what is your system x dot t is equal to a x t x a let us start with 0 does not matter as far as this system is concerned whether you start with a t naught or start with a t does not matter. So, this is nothing but the solution at the time t starting from the point x. So, if you start in R n R 2 so, you will have an extant here the solution will move along this as thing and this is nothing but the. So, you can imagine the dynamical system is nothing but the motion of the particle once you fix x naught is nothing but the motion of the particle along a trajectory. And it gives you the position of the particle trajectory at time t starting from x naught at time 0 for this particular system at dot t is equal to a x t with x at 0 is equal to t. So, it is a in fact it is the motion of the particle dynamical system is the motion of the particle in the phase play motion of the particle in the phase play in the phase play that is what you have to understand this is an another related notion this gives you a better feeling about this what is what is called a flow of the system flow of the system. Now, what you do is that we will have a physical motivation for this you look at fix t you consider phi t phi t is nothing namely e power a t t a e power t a for the system this is a mapping from r n you can view that this is a mapping from r n to r n for any dimension. So, you collect everything so this collection is called the flow actually the collection phi t is equal to e power a t as mapping t a such that for all t in the system you have as you see that not only for t r plus you can concern is called the flow. What is the flow doing what is the advantage of the flow flow here for example, the main advantage when we look at it a fluid flow you can think that a fluid at time t equal to 0 in one position then you view the fluid as particles and then for each particle after some time the particle will be at another point but this does not give you this view of a motion of the particle if you view just a particle in the fluid for every particle you move to some other position at after some time t. But for a layman point of view or a natural point of view when you look at a flow we never see the flow as particles are moving rather we see a collective motion of particles. So, this forces you to understand the motion of just to one point we want to see the motion of that point together with its neighborhood. So, if you start with at x naught this is a different v x naught and look at this neighborhood some neighborhood v and this is nothing but at the position phi h 0. So, phi h 0 of v is v that is because t 0 it moves there then for each t for look at all the particle in the neighborhood and you collectively try to see this will be moving here and this will be moving here will be moving something. So, you want to see this motion and all this will be at a different positions this may be at a position somewhere here and you will get a neighborhood. So, this is nothing but your phi t at v. So, if you look at that not only the point x naught and all the neighborhood you try to see the how the neighborhood together moves and that gives you the better feeling of the motion of the particle and that is actually what you are doing trying to do that through the flow of the system you see that a column or a neighborhood of the fluid moves in that direction. So, that is what the flow gives you the concept of flow and flow has some nice advantages like kind of semi group properties and all that which you may probably learn in some other thing phi naught of x is always x the more important thing is that suppose flow move the from at time flow from x this is essentially tells you that the flow from x phi t of x gives you from the position x it moves to the position t at thing and then composite with that phi thing will be the same as phi composition s plus t of x this is a very important thing. So, you flow moves from one position to another position and then it moves is the same as flow moving from to other position and another interesting property this also has a group structure. In fact, for this particular system you will have phi t of phi minus t of x is equal to x in general many of the flow is the reversible systems you may not get the last property, but for this systems are. So, you can also reverse the flow that is why it exist for all t in R, but in a semi group when you go to partial differential equations etcetera when you try to understand this kind of motions you may not get a group structure you may only get a semi group structure that is anyway not the topic of discussion here and one more notion which you probably would like to think. So, these are all some geometric way of understanding vector field. So, to understand the thing is not as you see now is not only understanding the solution how to interpret the solution how to view the solution all part of the study vector field for given x in R n A x is another point in R n, but in the beginning of our discussion long back in the previous lectures we said in R n you can view either a the points of R n as points or you can view also point as a vector just like we have seen. So, either you can view this point x and you can view that point as a vector here the advantage of this vector view this as a vector I can put this vector with the same length parallel to that. So, you can get that view. So, for example when you have a fluid flow of a particle the position of the particle you would like to view it as a particle, but on the other hand the velocity at that point is also a point in R n, but then is better to view the velocity is a vector placed at that point. So, when you given a dynamical system x dot is equal to A x, x t you view it as a position and A x you view it as a vector. So, that way any matrix A any matrix A gives a vector field structure in R n gives a vector field structure vector field in R n. So, what you can saying is that for all the points here x you have a point here x you can associate a vector this is vector is nothing but A x. If you take another point you will have another vector all the points can be associated with vectors. So, want to know for a given dynamical system. So, if you look at it here clearly what are we trying to associate we are trying to associate x dot equal to A x. So, when you are looking for a solution x t to this dynamical system x dot equal to A x you are looking for this trajectory A x. So, that x dot x dot is equal to nothing but your tangential vector field this tangential vector field should be A x that is what you are looking at it. So, the vector associated to this point each point should be each point. So, we are looking at a trajectory. So, solution associated to that one you want it these are all should be your A x you see. So, that way a solution a trajectory x t is a curve x t is a trajectory to the solution at each point the tangent is given from the vector field. The moment A is given to you you have a vector field given a vectors are associated with all r n. So, you are trying to find trajectories in such a way. So, that each point on that curve the tangential vector coincides with the vector field given to you. So, the moment you are given a linear system like A x x dot equal to A x you want to plot your vector field in that way. So, for the given example given example earlier given example earlier. So, if you try to plot your vector field you can see that the vector field you already know the trajectory. So, your vector field will be like this. So, if you plot your vector field it will be something like that. So, if you get all your vector field like this this is your vector field associated. So, if you plot it here you see. So, you plot here. So, that is a vector field let me plot again the vectors field. So, if you have your vector field to that example are something like that this is for the given example. So, it may vary if you plot from here it the curve that is how curves similarly, if you plot here like that these are the this point this is the vector field. I know these directions I have changed that is not the direction. So, the direction is in this direction. So, if you plot here the direction will be like this if you plot here this will be the direction you get each point you will have the corresponding this will be here because it is a stable one. So, this is the vector field. So, every system you will have a vector field associated with that. So, what I would like those who are learning here as an exercise plot these things exercise plot the curves plot the curves and vector fields curves and the curves means solution trajectory I mean trajectory and vector field for the above and other examples for the above and other examples. So, what we have seen is that when you have the solution trajectories and if you plot all the tangent vectors you will get your vector field on the other hand if the vector field is given you take any trajectory. So, that the tangent vector to that curve should be from the vector field it will be a vector field associated to that and that curve will be the solution to the system with this basic notions we will see more and more examples as we go along what we have seen is only a saddle point example. Now, we are going to see the entire analysis for the 2 by 2 systems which we will do that and so basically we want to do a phase plane analysis of 2 dimensional system that is what we are going to do now phase plane analysis analysis of 2 by 2 systems. So, you have some terminologies of phase plane and you are going to see not only here even the other module of non-linear systems. So, what are the things now we have to recall from the linear algebra and we want to know if a is 2 by 2 a is 2 by 2 we want to know what are the possible 2 by 2 linear equivalent matrices. Typically, this can be classified into 3 different categories what I am trying to see that the linear algebra tells you that every 2 by 2 matrix is linearly equivalent to one of the 3 categories according to the existence the Eigen values whether it is a real Eigen value and distinct it is a real Eigen value but multiple real coinciding Eigen value or the Eigen values are complex. So, I am going to write the 3 things which I will not do it probably some of them you will learnt in the basics. So, that is what we are called so that type 1 we want to call it type 1 in this case a will be let me put a notation this is for linear equivalence to b 1 b 1 is of the form lambda 0 0 b when I say that this notation means that a linearly equivalent that is the meaning of this linearly equality means b 1 can be written as p inverse of thing what is b that means your b 1 is of the form p inverse of a p for some invertible matrix p that is the meaning of a when is this happens this happens when a has 2 distinct real Eigen values if a has 2 distinct Eigen values lambda that means lambda not equal to b this is the type 1 matrix and what are the matrix of p in this case is easy you look for since there are 2 distinct Eigen values it has 2 Eigen vectors which are independent put that Eigen vectors as column vectors of p you get your this thing. So, p also can be constructed by obtaining the Eigen vectors corresponding to lambda n mu because it is independent it is invertible. So, that is the case type 1 situation so what is type 2 type 2 a will be linearly there are 2 cases in type 2 also it can be of this form lambda 0 0 b 1 b 1 b 1 b 1 b 0 0 lambda there are 2 cases. So, in type 2 there is a case 1 and this happens if a has a double Eigen value double Eigen value double Eigen value means Eigen value with multiplicity algebraic multiplicity 2 double Eigen value 2 that is the meaning of double Eigen value means Eigen value with algebraic multiplicity 2 with that is the thing now when has an Eigen value a coinciding A D M value there are 2 cases there can be 2 still it can have 2 independent Eigen vectors or it can have only 1 independent Eigen vector. So, that is the problem when you have 2 distinct Eigen values you have 2 distinct independent Eigen vector when it is a coinciding Eigen value it is it depending on the matrix it can have 2 a independent Eigen vector and that is refers to as the geometric multiplicity. So, the classification within this comes because of the geometric multiplicity. So, this is the case where it gives you 2 independent Eigen values if it has 2 independent Eigen value and still it form a basis and you have the diagonalizability and with 2 independent with 2 independent Eigen Eigen vectors and this will be equivalent to b 3 in that case b 3 takes the form and non diagonalizable matrix a diagonalizability in general is not possible 0 lambda lambda again double Eigen value with algebraic multi Eigen value lambda, but only 1 independent Eigen vector independent Eigen vector you see the whole trouble in diagonalizability is the lack of independent Eigen vectors 1 independent. That means it is the first case you have algebraic multiplicity 2 geometric multiplicity 2 in the second case algebraic multiplicity 2, but the geometric multiplicity is 1. Hence it will give only 1 column vector for p the remaining vector 1 has to construct to find the equivalence you want to write a is equal to 2 you have to find p all the time in the other cases getting p through Eigen vectors. So, that is the concept of generalized Eigen vector and that is the kind of the whole linear algebra and the Jordanic composition theory. In higher dimension there will be many complications some of the Eigen values are real some are complex some are multiplicity there are certain things have the independent certain things are simple Eigen value. So, all complications happen. So, you have to distinguish all that that is why you do not get the full diagonalizability you get block diagonalizability. And type 3 is the last case which you know already now you know type 3. So, the type 3 this is the case in this case a will be linearly equivalent to a minus b b a, where a has complexity. So, complex Eigen values, complex Eigen values lambda is equal to a plus i b and the other Eigen value lambda bar is equal to a minus i b. So, you have two Eigen values I will be linearly equivalent to a minus b b a and this let me call it b 4. So, our aim now. So, if you want to and as I said again and again the if you want to understand the linear system x dot equal to a x especially the nature and the behavior of solutions it is enough to understand the systems corresponding to b 1, b 2, b 3, b 4. And then we can recall everything therefore, in conclusion we only need to study the systems y dot equal to b corresponding to each of this four b i's b i of y where for all i equal to 1, 2, 3, 4 with some initial values say y at 0 is equal to y naught. Once we know that we understand the nature and the corresponding a will have the same nature of that one and then there will be a coordinate change which we will see how these things are happening. So, we are going to study this 1 by 1. So, we are going to do type 1 case now we start with type 1. So, in type 1 a will be let me recall again will be equivalent to b 1 is equal to lambda 0 0 mu this is the case where a has to again recalling a has two distinct Eigen values lambda mu not equal to mu and real. So, we want to understand this thing now again we will split into various cases. So, again we are putting the various k and we are assuming here also determinant of whole analysis right now determinant of equal to 0. If the determinant of a not equal to 0 lambda and mu cannot be a. So, the only equilibrium point is the origin no other because of the invertibility of a if one of them is 0 or two of them 0 then there will be many equilibrium point. So, that are all special degenerate cases we will give another two examples of degenerate cases later. So, we want to understand the situation where determinant of a not equal to 0. So, in the type 1 we have two classifications case 1 this is within type 1 where lambda mu the product lambda mu less than 0. So, we will start with the examples and then you will see what happens thing. So, let us take we already seen one example say similar example we will try to see. So, example and then we will see that all will work same way. So, we want to understand the phase portrait. So, we will take lambda is equal to say minus 1 we already consider such an example, but let me do it in a slightly sign change how the arrows will change earlier we considered a saddle point situation this is a saddle point situation with 1 and minus 2. So, it will behave same thing. So, what is the corresponding system you will have y 1 dot is equal to y 1 this we already studied. So, that is nothing much to do it minus y 1 y 2 dot is equal to 2 y 2 you see and you have your solution I can immediately write my solution y 1 t is equal to e power minus t y naught 1 and y 2 t equal to e power 2 t y naught 2 this goes to 0 as t tends to infinity this will go to plus or minus infinity depends of this I know y naught 2 y naught 2 is positive go to plus y naught 2 is negative go to minus infinity as t tends to infinity eliminate x t we have done exactly the same t eliminate t you will get it you square it you will get y 1 square is equal to some constant with y naught by y naught 1 by y naught 2 constant into y 2. So, this is the this is called the saddle point equilibrium saddle point which is an unstable equilibrium saddle point equilibrium is always an unstable saddle point equilibrium. So, if you plot your curve here is an exact thing if you plot here. So, this is a equilibrium point and y 1 is always going to 0. So, these things are here y 1 part this will be here and y 2 going to infinity. So, it will be here it will be here. So, if you plot your curves here now. So, this will go to y 2 is going to infinity. So, it will be here. So, this is your phase property. So, if you plot here this will be like this this will be like this. So, you have only certain cases I will draw the entire thing after that you should plot it accordingly. So, if you look at it the x axis x axis is called stable subspace. So, this is stable subspace and y axis and. So, you will define E s the stable subspace E s set of all elements of the form x 0 with x in r and y axis unstable subspace is called unstable subspace that is called E u is equal to set of all elements of the 0 y y is equal to r and this will give you this together will give you your r 2 can be decomposed into E s plus this is a general feature which we will see later. Every r and for a given dynamical system we can classify into stable unstable plus one more thing will come node when you go to this thing for the shaded point this is the case you can classify thing. You can also you also know how to plot your thing how to plot the vector field here. So, this is the phase portrait in this situation by taking lambda equal to minus 1 and mu equal to 2. What I am going to say tell you is that this from here is not anything specific about lambda and mu what you need is a difference sign. So, this is the case this is the case for any lambda mu with lambda mu negative that is what I am saying. If lambda and mu have opposite signs this is case 1 within type 1. So, whenever you have two eigen values real eigen values this thing, but it will have different sign you can do this one instead of y 1 t power minus t y naught you will get e power lambda t y naught 1 and y 2 t will be e power mu t y naught 2 and then if you eliminate t you will have an expression of the form some y 1 power alpha is equal to c by y 2 power. So, the trajectories will be the same it will be remain the same way only thing depending on this sign which one is negative which one is positive the arrows will change you will have one directional arrows with lambda negative and mu positive and the arrows will change if lambda negative and mu positive and the stable axis either x axis will be the stable in this case and in that case y axis will be unstable or you will have x axis unstable and y axis stable. So, this gave the case for all type 1 with distinct eigen values with different opposite sign will have this case and all this equilibrium points in this situation of lambda mu negative is called the saddle point equilibrium. So, the behavior will be the same as saddle point equilibrium and one more point I want to remark here before going to the next case if you make this linear equivalence. So, if you have this particular matrix B 1 which is came from a linear equivalence A 1 I told you this is just a change of coordinate system. So, instead of having this thing corresponding to A you will have a new coordinate system something like that you may have a another coordinate system need not even be perpendicular, but again this corresponding point origin will still remain because under linear equivalent there is only one equilibrium points which is the origin and the origin will still remain because as the equilibrium point in that case and there will be a coordinate change and your trajectories will still be like that you see the nature of the trajectories will be like saddle point equilibrium with appropriate the arrows you have to put appropriate arrows that is all that will give you depending on the sign and thing. So, all here and this will be in this direct you see. So, that is what I said there is no difference in the nature of the saddle point nature of the equilibrium point for all the matrices all coming from B 1 or B 1 corresponding to the matrix A with lambda mu less than 0 will have the same nature of the solution. So, now go to the type 1 and type 1 again with lambda mu case 2 with lambda mu positive you see that is a key what we had seen is that with lambda mu negative now within this there are 2 cases you see there are 2 cases with lambda positive mu positive and there is another case with lambda negative mu negative. So, to understand both these cases. So, the best way again is to take up the example it will behave in the same way. So, let us take start with an example. So, this is the case again B 1. So, you start with example then you will see that there is no difference example you take B 1 to B of the form it is all in the case of B 1 type 1. So, you take to be 1 0 0 2 you take it you want the case. So, if you do these things what will happen if B 1 will have you can write down the your solution your y 1 t will be e power t y naught 1. And y 2 t will be e power 2 t into y naught 2 see in either case this will go to plus or minus infinity because e power t minus depending on the sign this will also go to plus infinity. Now, if you eliminate here your y 1 square will be constant into y 2 this will be the this is something like a parabola you see. So, you have a parabola thing. So, if you plot this curve here here again look at it here let us look at our first the equilibrium point this is an equilibrium point this is a decoupled system. So, if you start from here it will remain in the same axis it cannot move because if it is in x axis your y naught 2 will be 0 and y 2 t will be 0 and hence it will remain in the x axis. But then y 1 t goes to infinity because y 1 t is equal to e power t into y naught 1 and. So, the arrows will move along this direction if you move from here it will move away from here if you start from here the same situation, but both eigen values are positive it will move in this direction in the saddle point equilibrium if one axis moves away the other axis moves towards it, but in this case both moves here. So, if you move here. So, it will be something like that. So, it will be like that what about if you start a point here if you start a point there both y 1 and y 2 goes to infinity, but it moves according to this rule and that something like a curve which is something like a parabola. So, as t tends to infinity it will move along this direction and this portion will be the t because the solutions are defined even for the minus infinity. So, it moves along these curves. So, if you start from here it will again move here for this is for the t positive side and this is for the t negative side here at this point is your y naught at t equal to 0 y naught at t equal to 0 you see this is for the part t negative. So, if you from here if you move again it will move along the parabola this is the region. So, if you start from here it will move the parabola may change it what I am trying to see that this is the same situation for any lambda mu positive same case for any lambda positive only thing that e power t you will have e power lambda t y naught 1 y 2 t will be e power mu t y naught 1. So, you may not get a relation y 1 square is equal to c y 2, but you get a relation something like y 1 power alpha is equal to c y 2 power beta thing, but the curves may change the behavior will be same such a point is called the equilibrium point is called a node equilibrium equilibrium is called I do this one node n o d e and this is unstable is called a is an unstable node because whatever be the solution any point you start a solution here solution will move however, closety does not matter it will move. So, of course, the trajectories do not intersect. So, whatever be the close point y naught closer to the origin the solution will trajectory will think and this equilibrium is referred to as the node and what I am saying that this is the same situation with any lambda mu for all lambda positive mu positive this curve will be the behavior will be the same thing. Now, let us look at this case. So, in this case if you look at it you start with b 1 in this case say both with minus 1 0 0 minus 2 you want a system. So, the corresponding system you can write. So, you will have your y 1 t is equal to e power minus t y naught 1 y 2 t is equal to e power minus 2 t y naught 1. So, if you eliminate you will get again y 1 square is equal to some constant into y 2 the only thing is that this will go to 0 both will go to 0 in this case. So, if you plot this curve the curve will be the same. So, you have your equilibrium point. So, in this case if you start from here only the arrows will change because it is going to 0. If you start here you will have here. So, the curve from if you start here now this will come towards the origin along that you see this is the negative part. So, this is the positive part now. So, if you start from here. So, it will here this is the t positive part. So, this is. So, if you start anything from here any point does not matter if you start from here it will go to the it will only go as t tends to infinity. So, it is the same thing it will go to infinity. So, this is your phase portrait as a and this is again a situation of a naught in this case we call a stable naught. You will have this stable naught and you have this same same for all lambda mu with lambda mu. So, what we have done is that in the type 1 when it has two distinct Eigen values lambda mu negative you have a saddle mu negative point equilibrium which is unstable one trajectory go to 0 and trajectory go to infinity or minus infinity. When it is lambda mu positive either both trajectories will move away from the equilibrium point going to infinity giving an unstable equilibrium. On the other hand when lambda mu positive with both lambda and mu are negative both Eigen values are negative then the solution trajectory will go to in fact it goes to the origin in a asymptotic way giving a stable equilibrium. And we refer to this case with lambda mu positive is a equilibrium point is called a node and your subspace in the unstable case the whole r 2 is an unstable space in the stable thing whole r 2 is a stable case. Now, in the next class we will continue with this for the other two cases of type 2 and type 3 what you are going to see is that type 2 more or less behaves like in this fashion with slight change in the shape of the trajectory, but the behavior will be like a node behavior. But a type 3 case where the Eigen values are complex you will have a different behavior you will also see some periodic behavior which you are going to see anyway in the non-linear systems more the periodic trajectories with this we will end this particular talk. Thank you.