 So, before we start our study of differential dynamics and continue with it, I would like to recall to you what we ended with the last time, which was the Hamiltonian for a set of particles interacting with each other by some pair wise interaction between them. The force between any two particles being directed along the line joining these two particles and I mentioned certain properties of the corresponding Hamiltonian. And this Hamiltonian was a function of all the coordinates of the particles, n of them and of the corresponding conjugate momenta p 1 to p n. And it was in the form of the sum of the kinetic energy of the particles and the potential energy of these particles. So, it was a summation i equal to 1 to n p i squared over twice the masses of the particles plus a summation over i j equal to 1 to n no self interaction times the pair wise potential energy, which was a function of r i minus r j modulus in this fashion. And this Hamiltonian has certain properties the first property is that it is invariant under a rotation of the entire coordinate system whatever coordinate system you choose. If you rotated that coordinate system by an arbitrary angle through an arbitrary around an arbitrary axis the Hamiltonian would not change in form at all. This means that rotational symmetry is a symmetry of the system dynamical symmetry of the system. The Hamiltonian itself is a constant of the motion. So, you have h the angular momentum about the origin of coordinates is a constant of the motion because it is rotationally invariant if I call that l total angular momentum of the system is a constant of the motion. There are no external forces on the system. And therefore, the generalization of newton's third law applies the total linear momentum of the system is constant. And there are three other constants of the motion which I will write down very shortly and they go as follows. If you consider this coordinate r which is the coordinate of the center of mass of the set of particles this of course is defined in the standard way as a summation from i equal to 1 to n of m i m r i divided by 1 to n m sub i the total mass this coordinate is the coordinate instantaneous coordinate of the center of mass of this entire system. And then because the total linear momentum of the system is constant it is possible to find what this r is at any instant of time and it would just be given by a simple formula r of t is equal to r of 0 plus since the total linear momentum of the system is constant that would be the way in which this thing would move p t over m where m is the total mass of the system. So, let us call this m we can prove these very rigorously, but physically it is very clear that this coordinate the center of mass would just drift along at uniform speed given by p over m where this is the total momentum because this is a constant this immediately implies that the combination r as a function of t minus p times t over m which happens to be equal to r of 0 is in fact a constant of the motion. So, this set of three coordinates minus this combination p times t over m they are constants of the motion another vector constant of the motion. So, let me write that down here as r of 0 these are also constants of the motion, but they are really dynamical variables and that combination is constant in time absolutely no I do not, but this quantity is a dynamical variable and so is this which happens to be constant in this simple example and the difference of these two is a combination of dynamical variables which is constant. So, it is a constant of the motion this is a different dynamical variable from that. So, a certain set of combination of dynamical variables happens to be constant in time there is. However, something strange about this constant of the motion it is a function of time it is explicitly a function of time. So, that gives us our other lesson constants of the motion may be explicitly time dependent in this case that is true it is just that the time dependence of this quantity this quantity this quantity etcetera are such that they adjust themselves and this combination is time independent. So, what do we have we have 1 plus 3 4 plus 3 7 plus 3 10 constants of the motion in a phase space that is 6n plus 1 because time also now we are talking about an extended phase space since we have got time dependent constants of the motion in this 6n plus 1 dimensional extended phase space we have exactly 10 Galilean constants of the motion these are called Galilean constants and 10 constants of the motion in a 6n plus 1 dimensional phase space is pathetically small it is simply too small for you to make any significant progress based on just these constants of the motion say too much about the motion. And this is as it should be we have after all said nothing about we have said nothing about any of these quantities about the behavior of the system the kind of forces etcetera other than the fact that these are forces which are rotationally invariant. So, it is not surprising that you do not get too much information now, but the fact is that you have 10 Galilean invariance constants of the motion in this large dimensional phase space. So, in general the point I was making was that a given dynamical system generically typically has far fewer constants of the motion which you can find explicitly than the actual dimensionality of the phase space this is the typical case there are exceptions there are integrable systems where you can find all the constants of the motion explicitly, but typically what will happen is that in a given dynamical system even if it is not a Hamiltonian system could be dissipative systems and so on the generic situation is that you have far fewer constants of the motion accessible to you than the dimensionality of the phase space this has to be born in mind to keep this in mind with this now let us go back go back retrace our path a long way and go back and look at general dynamical systems described by a set of n couple first order differential equations and then look at it case by case and see what the importance of various concepts such as critical points equilibrium points and so on. So, I go back now all the way and look at differential dynamics which if you recall was of the form x dot equal to a given vector valued function f of x and x existed x lived in some space which is say n dimensional Euclidean space I write it in this form just as ordered set of n tuples of numbers x 1 x 2 x 3 up to x n and this is the kind of dynamical problem we have to look at a very basic concept in differential dynamics is that of critical points because recall that I mentioned that in the neighborhood of a typical point x naught you could always approximate this right hand side to leading order by f of x naught itself and that was like applying the rectification theorem and you could find the flow locally in an infinitesimal time interval delta t the only exception I mentioned was when this vector field vanished at some point it became 0 then you could not do that you had to go to higher orders you had to compute terms you had to compute the problem once again. Now, what happens then and why is it so important the reason is that when this right hand side vanishes x dot vanishes at that point and since it is a set of first order equations if you give me the initial point I know the future but the derivative at that point if you like is 0 for all the x's and therefore x remains at that point therefore you could define all points which satisfy f of x equal to 0 to be the equilibrium points of the system since equilibrium has this connotation of being a mechanical system and we are not talking about mechanical systems alone we are talking about all kinds of dynamical systems I would like to call it by general name and I would say f of x equal to 0 the roots critical points and I will abbreviate critical points by c p's if you like they are the generalization of equilibrium points nothing has been said as yet about whether it is a stable equilibrium or an unstable equilibrium or a marginal kind of equilibrium or a neutral equilibrium or anything like that they are just the critical points of the system the other statement which is worth bearing in mind which I will make as a statement now and then try to substantiate it as we go along is that by and large the nature of the flow the nature of the phase trajectories is controlled to a great extent by the behavior of the system near its critical points. So, it is important to find where the critical points are after which we have a handle on what the general dynamical system does now let me illustrate this by the simplest of flows a flow in which n is equal to 1 a one dimensional dynamical system and what happens then. So, let us look at n equal to 1 a 1 d system or a 1 d flow given by an equation of the form x dot is f of x where f of x is some scalar function of the single scalar variable x the equilibrium points of this would be given by the roots of the equation f of x equal to 0 and if f of x is a reasonable kind of function its zeros would form an isolated set of points it would not have in general a continuous set of points, but just isolated roots of the equation f of x equal to 0 let us take an even simpler example of this specific case of this to see what happens. So, let us look at f of x equal to x itself. So, linear function of x in this case just x itself what is the flow look like this implies that x dot is x or solving it is very trivial x of t equal to what would you say is the solution to this simple equation just an exponential. So, this is e to the power t x of 0. So, I put in the initial condition which determines the constant of integration that is determined by the initial condition x of 0 and you can check very trivially that if t is 0 x of t reduces to x of 0 in this fashion what does the graph of x of t as a function of t look like. So, here is t and here is x of t it is immediately evident that if I start with a positive value for x of 0 positive initial condition then it starts at this point and takes off exponentially past as a function of time diverges like e to the power t if on the other hand x of 0 over negative it would diverge in this fashion. So, this corresponds to x of 0 less than 0 and this is x of 0 greater than 0 if x of 0 should happen to be 0 you started with 0 as the initial value then of course, you remain right here on this graph itself. So, this would correspond to x of 0 equal to 0. So, it is clear that the critical point at x equal to 0 that is the only critical point is actually separating two different kinds of trajectories those which take you to plus infinity as t goes to infinity and those which take you to minus infinity as t goes to infinity. So, what should we do we draw the phase plane in this case of the phase space and this is one dimensional. So, it is just a phase line if you like. So, here is the phase line this is phase space in this case not much of a space it is just a line here is the point 0 and there are only two different kinds of trajectories possible on this those which would start anywhere here and move off to infinity which I denote by an arrow outwards and those which would start here and move off to infinity which I denote by an arrow pointing leftwards. This point 0 is a critical point. So, this is the only critical point in this problem is this a stable critical point or an unstable one what would you say based on our elementary notions of stability I make a little perturbation away from it and the question is do I go back to it or do I move away from it it is evident and we will make this more rigorous that this is in fact an unstable point. Notice that it is a trajectory by itself because if I start at x of 0 equal to 0 I remain on it at this point. So, there are three classes of phase trajectories here those which start at 0 and remain at 0 that is just a single point a trajectory which represents the positive x axis and another one which represents which is represented by the negative x axis would you call this an attracting critical point or a repelling critical point it repels on both sides. So, it is in fact an unstable critical point and in this case it happens to be what is called a repeller it is immediately clear that yes well we must make precise the notion of unstable which we will do very shortly and repeller at this level is simply because trajectories on either side of it are repelled away initial points on either side of it are repelled away. So, an unstable critical point can be of many types and this instance it happens to be a repeller which I have used in a loose sense as something which pushes away trajectories on either side of it will define the notion of unstable more carefully more precisely. Can I make that quote unquote a stable critical point an attractor something which attracts things points to it phase space points to it can I change this flow to do that 1 by x will have a critical point at infinity I would like to have a critical point in the finite part of the plane of the of the line minus x would do that. So, if I took x dot equal to minus x this would imply x of t equal to e to the minus t x of 0 and the phase the graph of x of t versus t again if I start at 0 I remain here, but wherever else I start if I start at this point or this point etcetera I would exponentially decay towards 0 as time went on. So, here it is that is this point this line. So, this case if I plot a similar phase diagram or phase portrait as they call it in this case 0 is an attractor because the trajectories flow inwards and you begin to see that the phase portrait gives you much more information than specific solutions of the differential equation the point being that there is not much difference between a phase in the behavior of the system for an initial condition which starts here or here or here or here they all form this is all in the same class and the same is true on this other side. So, it is not so important to worry about the specific initial conditions, but we are now dealing with whole classes of initial conditions in one shot and that is it there are only 3 phase trajectories in that problem there are only 3 here and this one is an attractor no matter where you start you are going to asymptotically flow into the point 0 eventually for sufficient if you wait sufficiently long you come arbitrarily close to 0 and this is true for any finite initial condition throughout this phase space every point flows into this origin the critical point at the origin. So, this thing here is actually a global attractor the reason I call this word I call this attractor global is because you can envisage situations where you have more than one attractor and part of the initial conditions would flow into one and another part would flow into another in which case those would be local attractors, but this is a global attractor everything falls into this classification of critical points into attractors and repellers is very specific to the kind of one dimensional systems we have talked about things will get much more complicated as we go along, but already here we can illustrate the fact that more complicated things could happen for instance let us look at the case let us modify this flow slightly this is a good point to actually introduce the idea of bifurcation. So, let us modify this slightly and consider x dot equal to x squared say what kind of solutions do you have what kind of phase portrait would you have where is the critical point it is again at 0 that is the only root of x squared equal to 0, but it is a double root now what kind of arrow should I draw here for trajectories here well one way would be to solve the equation explicitly as a function of t and see what it does for as t increases as a function of t, but we can do that by inspection by looking at just this differential equation itself. If x is any positive number x dot is positive and therefore x increases and since x increases it is clear that the trajectories must move outwards there if x is negative what happens x squared is still positive and therefore x dot makes x increase in algebraic value and therefore it again goes this way is this an attractor or a repeller well it looks like as far as points to the right are concerned it is a repeller, but points to the left are concerned it is an attractor. So, it is neither an attractor nor a repeller it is in fact a higher order critical point, because this is not a linear function of x. So, let me call this higher order or degenerate degenerates critical point the reason we have ended up with the degenerate critical point will be interesting will be important it is because we started with an x squared here if I asked you to write down a polynomial function on the right hand side you would typically start with the constant plus a first order term plus a second order term and so on. The constant can be got rid of by shifting the origin to that value of the constant after which you would have a linear term proportional to x typically, but here you do not have that you started with x squared. So, this is not a typical or generic polynomial on the right hand side a typical one would start with the linear term and then unless there is an accident it would always be present and then go on to a quadratic and cubic term and so on. Therefore, since this is not generic we would like to make it generic by saying this came about this double root at x equal to 0 came about by two roots coinciding with each other. So, let us split one of those roots and consider instead x dot equal to x times let us say x minus epsilon where epsilon is some small positive number to start with where are the critical points of this flow 0 and epsilon. Now, of course we do not have to solve the differential equation that is the whole point of phase plane analysis explicitly we try to get as much as we can from the differential equation. So, here is 0 and here is epsilon at the moment we have no way of knowing which one is the stable one or which one is the unstable one or both whatever we have no way of knowing what the stability is, but that is not hard to deduce let us look at the neighborhood of x equal to 0 that is a critical point. So, I look at it in a small neighborhood of this point so small a neighborhood that finally given epsilon I could neglect the x squared term compared to the linear term in x and then near this in this neighborhood I could write x dot is approximately equal to this is x squared minus epsilon x and I leave out the quadratic term and write this as minus epsilon x. Does that suggest an attractor or a repeller it is an attractor because the solutions nearby would go like e to the minus epsilon t times x of 0. Therefore, the arrows would point inwards here and inwards here by continuity since nothing happens between these two there are no other critical points it is not hard to show that even further beyond it would still have to do this because there is nothing in between to demarcate two different kinds of qualitatively different kinds of behavior in exactly the same way no matter how far you are on the left it would still flow into 0 on this side simply by continuity what happens in a neighborhood of x equal to epsilon one would have to linearize the problem near x equal to epsilon and that is going to be a big tool in our analysis of critical points we would linearize the flow near this point by simply shifting to that point. So let us set u equal to x minus epsilon so x equal to epsilon has been shifted to u equal to 0 and in this plane the equation becomes u dot equal to x is u plus epsilon times u itself is approximately equal to epsilon u near u equal to 0. So once again I neglect the quadratic term and the linear term is epsilon u near u equal to 0 and is that a repeller on an attractor since epsilon is positive that is a repeller. So the picture would look like this and if it is a repeller it is unstable and I use the convention where I mark unstable critical points by crosses and stable critical points by dots in which case this is what the global flow looks like again by continuity this is all it can be. So I delete this and that is the phase portrait of the system you have an attractor and you have a repeller notice the arrows cannot abruptly change direction without passing without going to a critical point in between and I do not need to solve the problem explicitly to do anything else I could I could write down the solution explicitly and then you would discover if you eliminated t that the phase flow does look like this. Now what happened when you ended up with epsilon equal to 0 was that this point and this point coalesced it came in so this region became narrower and narrower and disappeared and you ended up with this with the higher order critical point here. So this procedure by which I took a degenerate critical point and I found its genesis by the coalescence of two separate simple critical points it is called unfolding the singularity this is the most elementary example of this unfolding of the singularity. So I went to what I call a generic or typical case this is generic that is not generic it is degenerate and all this was true for epsilon greater than 0 that was crucial I could now ask the question what if I start with epsilon as a tunable parameter I make it smaller and smaller. So this critical point gets closer and closer it is the origin gives you this picture and moves to the left of it I make epsilon negative what would then happen what would be the picture for epsilon negative you would still have two critical points this would be 0 and this critical point would be at epsilon and remember that epsilon negative now what do you think would happen well near the origin the behavior is minus epsilon x and if epsilon is negative minus epsilon is a positive number. So 0 becomes a repeller and critical point at epsilon becomes an attractor. So you end up with a picture where this is an attractor and that is a repeller and things flow in this fashion. So this 0 becomes a repeller and that becomes an attractor. So while this was the picture for epsilon greater than 0 this was the picture for epsilon equal to 0 and that is the picture for epsilon less than 0. So it is evident from this simple example that at epsilon equal to 0 the systems behavior changes qualitatively it changes from a situation where epsilon was the repeller and 0 was the attractor to the opposite situation where 0 is the repeller and epsilon is the attractor. There is a qualitative change in behavior across this point epsilon equal to 0 in other words a bifurcation of the system happens at parameter value epsilon equal to 0 what kind of bifurcation would you call this. There are fancy names for it but the most illustrative one is the name actually given to this notice that this was an unstable critical point and this was a stable critical point and after the bifurcation the roles have been exchanged what was unstable became stable and what was stable becomes unstable therefore this is called an exchange of stability bifurcation might as well define a bifurcation a bifurcation happens at some critical values of parameters in the problem when the qualitative behavior of the dynamical system changes from one kind of behavior to another. So a small change in epsilon from an infinitesimal positive value to an infinitesimal negative value changes the systems behavior completely quite drastically and that is what a bifurcation is. So one studies bifurcation in parameters bifurcation happen in the parameters so the dynamical variables behave in a very different way from one side of the bifurcation to the other kind of other side of the bifurcation we will study several kinds of elementary bifurcation but for the moment this already serves to illustrate an extremely simple phenomenon. There is just one last point with regard to this trivial example almost trivial example and that has to do with the following if I solve this differential equation for x as a function of t you end up with the traditional exponential behavior in time either exponentially diverging or exponentially converging the same situation is valid here too but here at x dot equal to x squared it is not hard to write down the solution of this equation let us solve it for specific values of initial conditions at this special value 0 of the parameter epsilon if I solve this equation this implies that 1 over x of t well differentiate both sides it says if I separate variables it says this is equal to dt which implies 1 over x of 0 minus 1 over x of t is equal to t itself therefore x of t is equal to x of 0 if I move it to that side divided by 1 minus x of 0 times t I think is right what do the solutions look like as before 0 is a critical point so if I started at 0 I remain on 0 for all time what happens if I start with a positive value of x 0 say here what happens to that solution it is clear that it would diverge at some t it would diverge at t equal to 1 over x 0 so at some finite t which is 1 over x of 0 I start with x of 0 here this solution would in fact diverge to infinity this is this point and it would diverge so this phase point disappears to infinity in a finite time that never happens in first order dynamics it happens in this kind of higher order dynamic x squared it is non-linear to start with intrinsically non-linear on the other hand for negative values of x 0 this quantity never vanishes since t runs from 0 to infinity so it would start at some value here and slowly approach like a 1 over t it would approach the point 0 as t goes to infinity this vanishes like a 1 over t if this is negative so there is again two qualitatively different kinds of time behavior simply because of the non-linear nature of this problem so although it is a very very simple example it serves to illustrate some general issues such as what a bifurcation is and the fact that at a finite time the system may actually go off to infinity the phase point may go off to infinity this is very characteristic of higher order degenerate critical points having seen what happens in first order dynamics we could now generalize this a little bit go on to higher dimensions but before that let me make one more point which is the following what would you say is the dynamical behavior of a system given by x dot is say sin x what kind of behavior would you expect where are the critical points any multiple any integer multiple of pi what kind of critical points would these be by now our lesson with the earlier example tells us that because of the continuity that is involved in drawing these arrows showing the direction of the flow it is quite clear that stable and unstable points must alternate there is no way of having two stable points next to each other because in the system in between would know where to go to what happens near 0 you need to linearize sin x near x equal to 0 what is the leading term in sin x at x equal to 0 x itself and it has a positive coefficient so is x equal to 0 a repeller on an attractor it is a repeller so we know that 0 is a repeller in which case we immediately know that pi is an attractor and therefore 2 pi is a repeller 3 pi is an attractor and likewise minus pi minus 2 pi and so on therefore the arrows should go off like this you could laboriously solve this equation dx over sin x equal to dt and you integrate both sides and so on and so forth you would get explicit solutions but you would have already all the information you need about this simple dynamical system by just writing the phase flow down the complete phase portrait of this kind so you begin to see the power of this phase plane analysis you do not necessarily have to solve the equations completely you can deduce a great deal by figuring out where the critical points are and the nature of the stability of these critical points as an exercise I urge you to try next the following sin squared x and check out what happens in that example what kind of critical points would these be would they be simple critical points now near x equal to 0 this behaves like x squared so right away it is a degenerate critical point therefore the behavior would be very interesting check out what happens in this instance now that we have a little bit of an idea of what one dimensional systems will do let us go over to two dimensional systems and instead of calling the variables by x 1 and x 2 let me call them x and y for short so let us now talk about flows in the phase plane two dimensional dynamical systems and this would be a special case of the n dimensional system I talked about earlier instead of calling the variables x 1 x 2 as I said let us just call them x and y and they are specified by equations of the form x dot is some function of x and y and y dot some other function what would the critical points be given by they would be the roots they would be the simultaneous roots common roots since f of x y equal to 0 is some kind of curve in the x y plane and so is g this is the intersection of the two curves given by f is 0 and g is 0 again the statement is the critical points in the phase plane control the behavior of the flow to a large extent so we would like to analyze what kind of critical points you could possibly have in the phase plane and what would be the way to do this typically typically wherever we are near any point of interest I would do a Taylor expansion of f and of g near that point and study the flow in the neighborhood of that point with the help of this Taylor expansion if it is a critical point and those are the points of interest then these functions are 0 at the critical point and therefore the terms that would you would start with would be linear terms and what would it do in the vicinity of a critical point so let x bar y bar be a critical point Taylor expand f and g what would you then get you will get x dot is approximately equal to the leading term f of x bar comma y bar is 0 by definition because you are at a critical point and what would the first term be it would be linear in the difference x minus x bar this would be the first term so it is x minus x bar times the coefficient what would be the coefficient of the linear term it is the partial derivative delta f over delta x evaluated at x bar y bar plus a similar expansion in y delta f over delta y evaluated at x bar y bar. Plus higher order terms these would be terms which are of the form x minus x bar the whole squared or y minus y bar the whole squared or x minus x bar times y minus y bar times the cross derivative and so on and so forth exactly similarly y dot is x minus x bar delta g over delta x at the critical point plus y minus y bar delta g over delta y at x bar y bar plus higher order terms I could therefore make life simpler by shifting my origin to x bar comma y bar it would just correspond to taking the original phase plane the x y plane and if this is x bar y bar I could just shift origins to that point and work in new coordinates say u and v such that u is x minus x bar and v is y minus y bar instead of cluttering of the place with notation of that kind let me just call x minus x bar x once again and y minus y bar y once again let us just shift origins there and use the same symbols in which case the flow in the vicinity of the origin so let us shift shift the origin to x bar when the flow looks like x dot use the same use the same symbols x and y instead of x minus x bar I just call it x and similarly for y then x dot is of the form a x plus b y plus higher orders and y dot is of the form c x plus d y plus higher orders where a b c d is just the matrix of partial derivatives of f and g evaluated at the origin which is where the critical point is so this is simply delta f by delta x at 0 0 delta f over delta y at 0 0 delta g over delta x at 0 0 and delta g over delta y at the origin so it is neatly this matrix of partial derivatives at any point is called the Jacobian matrix at that point and the system looks like this provided this is these terms are present provided these partial derivatives are not 0 this is what the flow looks like near this point and now the question is we would like to understand what the flow does in the vicinity of this point by looking at the approximation the linear approximation in which you get rid of these terms and therefore sufficiently close to this point sufficiently close to the origin you could neglect these terms and just look at the linear system therefore linearize in the neighborhood let us write this as a matrix x y d over d t of this is l times where l is the Jacobian matrix a b c d now of course you would ask immediately how good an approximation is this statement is that we can estimate that we should need to know how big these terms are therefore to any prescribed degree of accuracy if this matrix is well behaved if these partial derivatives do not vanish then sufficiently close to the origin this looks like a good approximation we will come back and see when it is justified and when it is not but if this were true then this equation is extremely simple to solve because again you have an equation of the form x dot where x is a column vector with components x little x and y this is equal to l times x where l is a matrix of constant coefficients time independent just numbers and what is the solution to this equation the formal solution to this is exactly as if x were just an ordinary scalar function and this would imply immediately that x at time t is e to the power l t x at time 0 so this square matrix e to the power l t acts on the column vector of initial conditions and gives you the value of x and y at any time this is what the formal solution looks like you could of course do the following and I leave this to you as an exercise you could differentiate this a second time and eliminate y from this equation by getting x double dot is a x dot plus b y dot for y for y dot you substitute from this equation but that still involves y what would you do to get rid of y how would you eliminate either x or y from the set of coupled equations I differentiate this I get x double dot is a x dot plus b y dot for y dot I put in C x plus d y how would I write y how would I get rid of the y I would write y as x dot minus a x divided by b and that would give me a second order differential equation for x alone I do the same thing for y give me a second order differential equation and you know you would get a second order differential equation with constant coefficients and how do you write the solution a second order linear differential equation with constant coefficients what is the solution look like it is a sum of exponentials typically what are those exponents you would have to satisfy the characteristic equation corresponding to this differential equation and in fact you can see without much ado that this quantity e to the power l t if I imagine being able to diagonalize this matrix l then e to the power l t would involve in the diagonal form e to the lambda 1 t and e to the lambda 2 t where lambda 1 and lambda 2 are the two eigenvalues so the general solution to this equation you can do this in one step would involve something like x of t equal to some coefficient C 11 e to the lambda 1 t plus C 12 e to the 2 lambda 2 t and similarly y of t would be C 21 e to the lambda 1 t plus C 22 e to the lambda 2 t in general how would you find these constants C 11 C 12 and so on what determines the constant initial conditions absolutely you have initial conditions you know x of 0 and y of 0 but then there is a little paradox here or a little puzzle here I have four constants but I have only two constants I have only x of 0 and y of 0 so I am given x of 0 y of 0 are given how do I find four constants yes how do I get four constants out of two absolutely I need to know x dot and y dot and where do I get those from from the differential equations themselves so in this couple form you would use the differential equations themselves you would set t equal to 0 here and say x dot of 0 is such and such and y dot of 0 is such and such and you would use that in those equations to get four pieces of information and therefore it is completely adequate to specify just these two guys as we know very well we have two couple first order differential equations and therefore two initial pieces of data should be sufficient the fact that I have written it by eliminating the other variable and so on and written it in that form is irrelevant you really can find all the constants you need from the initial data that you have to use the differential equations themselves which are also taken to be valid at t equal to 0 and therefore you can write the general solution down in this form what assumption have I made in doing this what assumption have I made about this matrix that it is not singular that this matrix is not singular in other words I took this set of equations and I said 0 0 is a solution for the critical points namely the right hand sides are equal to 0 and I assumed that the only solution of A x plus B y equal to 0 C x plus D y equal to 0 has a unique solution 0, 0 in other words I said this set of simultaneous equations has only a trivial solution 0, 0 when is that true when this determinant is not 0 so when the acrobin matrix is non singular when L is a non singular matrix then and only then is this true so we assume that we assume that L was non singular we will come back and examine what happens when it is singular but that was the primary assumption it had to be non singular and when is this form of the solution true so L non singular in other words determinant L not equal to 0 that was an assumption and what was the other assumption when I write the solution in this form I have assumed about lambda 1 and lambda 2 I have written 2 terms separately so what is the presumption there it is non degenerate as an eigen value in other words the eigen value is not repeated lambda 1 and lambda 2 are distinct eigen values so you certainly assume that lambda 1 is not equal to lambda 2 what happens if lambda 1 is equal to lambda 2 when the eigen value is repeated what happens to the solution of this differential equation well if I write it as a second order differential equation it is clear that when you have the roots repeated you have a different form of the solution the linearly independent solutions are no longer e to the lambda 1 t and e to the lambda 2 t what are they suppose lambda 1 is equal to lambda 2 is equal to lambda no there is no redundancy in the equations then e to the lambda 1 t e to the lambda 2 t this set is replaced by there is an e to the lambda t and what is the other term t e to the lambda t there is a secular dependence here on t and then an exponential and therefore these equations would be modified to read something like e to the lambda t multiplied by c 11 plus c 12 t and similarly e to the lambda t c 21 plus c 22 we stop here at this stage and then we take up the analysis further analysis of this next time there any questions or comments the final remark is that everything seems to depend on the behavior of these exponentials so this will help us analyze what happens at these at the critical point the nature of the eigen value will tell us everything about the flow and notice once again I am not going to use explicit forms of the solution I do not need to we will define okay next time we define what is meant by the exponential of a matrix if it is a square matrix as this one is then its square is also an n by n matrix an n by n matrix and its cube is also a matrix and so on the exponential of the matrix is defined exactly as e to the power z is defined for a complex number z so it is the identity matrix plus the matrix over one factorial plus the matrix squared over two factorial plus etc and this is guaranteed to be a well defined convergent quantity and I will mention this convergence property next time the exponentials of operators are extremely important and they will appear over and over again we will see how to do that for things even more general than matrices oh yes the domination will appear if lambda 1 and lambda 2 are both positive and lambda 2 is larger or if lambda 1 is negative and lambda 2 is positive it is clear that the divergent behavior the explosive growth would have happened from the positive eigenvalue so sure we would have to worry about which one is positive which is negative interesting things are going to happen because they could either be both positive or both negative or one positive and one negative or complex because the matrix coefficient the coefficients a b c d are real and therefore the roots would appear in complex conjugate pairs this would this would play a role so we will see.