 So, we begin today with an example of when you have a center manifold in a dynamical system and how linearization could lead to an erroneous conclusion and again I do this with the help of an example. So, let us consider the following system consider x dot equal to minus y and y dot equal to x as we know this linear system has a critical point at the origin which is a center because the Eigen values of this matrix of plus or minus i the linearized matrix and you would expect the center it is like the harmonic oscillator problem, but now I add non-linear terms to it. So, suppose I add x plus x times this plus y times x squared plus y squared or this clearly a critical point at 0 0 and the linearized system x dot equal to minus y y dot equal to x implies a center you therefore expect the trajectories to go around in small circles or ellipses about the center which is a stable critical point on the other hand we can solve the entire system the full non-linear system because of the form of the non-linearity it is straight forward to solve this problem completely and what would the solution be all I have to do is to use the fact that RR dot is x x dot plus y y dot if I do that I multiply this by x and that by y and add these cancel and I end up with an R to the power 4 which implies that R dot is R cubed which immediately says since R can only take on non-negative values as soon as you have a finite R0 which is non-zero R increases and keeps increasing indefinitely and the question is you could also find out how it increases as a function of time that is straight forward because this says dr over R cubed is equal to dt and that of course implies that one half 1 over R of 0 minus 1 over R of t this quantity is equal to t itself R0 square and it is easy to see from this if you solve for R of t so this implies that R of t tends to infinity at some more over if you do this in plane polar coordinates and look at what theta dot does then theta dot is equal to if you recall this is d over dt of tan inverse y over x and of course we already have a formula for it which is equal to x y dot minus y x dot divided by x square plus y square and in the present problem theta dot works out to so I multiply this by x and this by y and subtract these terms go out and you end up with an x square plus y square divided by the same thing so this is equal to 1 so in this particular problem we see that linearization has led to a completely erroneous conclusion whereas if you restricted yourself to the linear part of the system you conclude that the origin was a stable center it turns out that is not so at all and wherever you start you are actually going to flow outwards in this fashion such that you hit infinity at some finite value of time because of this specific form of nonlinearity so the origin is not a center it is an unstable node in this case and things flow out and this shows you the perils of linearization when you have a center manifold very clearly shows you that the nonlinearity has completely changed the behavior of the dynamical system from what the linear one would predict the linearized form would predict and this is the reason I said early on that when you have a center manifold there is no guarantee that linearization produces the true flow in the vicinity of the critical point and this is a simple example which illustrates this point let us go on to the next topic that we had started namely we were looking at higher order critical points and I made a statement that these higher order critical points are generally formed by the coalescence of simple critical points and indeed we saw in the case of a saddle node how a bifurcation occurs and you get a higher order critical point at the point of an exchange of stability bifurcation. Now let us look at some further examples of this let us look at for instance x dot equal to x squared minus y squared and y dot is twice x y it is immediately clear that the origin is a critical point but there are no linear terms on the right hand side and therefore this thing is intrinsically a higher order critical point at the origin the question is what kind of critical point is it what does the flow look like in this instance what would you suggest looking at this function what does it suggest to you let me give you a hint suppose you wrote z equal to x plus i y and regarded this as a complex variable x plus i y as a complex variable what does this suggest to you so if you set z equal to x plus i y this set of equations implies z dot equal to so it is equal to z squared itself so it is immediately clear that the origin the singularity at the origin is such that if you took a circuit once around the z plane once around the origin the function on the right hand side its argument changes by 4 pi rather than 2 pi because it is z squared so this brings us to the concept of the winding number of a singularity of a planar vector field and let me explain what that is the winding number or Poincare index of a planar vector field namely a vector field in a plane as a function of x and y we will come back to this we will come back to this the statement I made was if in the z plane there is some singularity at z equal to 0 at the origin but it is such that if I move around once in the z plane and increase the argument of z by 2 pi the argument of this vector field on the right hand side increases by 4 pi because it is the square and this has a specific implication for what I am about to say in the statement is the following suppose you have a singularity of a vector field somewhere in the x y plane say the origin and the field lines around the origin look in some complicated fashion they perhaps look like this these are perhaps the field lines this is how it would look for instance near an unstable node in this fashion at every point other than the origin which is taken to be a singularity of this vector field the vector field is unique and by vector field I mean the set of equations which we wrote down once again x dot is f of x, y y dot is g of x, y and you recall I combined these two into reading as a vector equation f of x and this vector field f has two components f and g so let me write that explicitly and write f of x, y is times the unit vector in the x direction plus times the unit vector in the y direction this field vanishes at the origin both f and g vanish at the origin and you have a critical point of some kind and now I would like to characterize this singularity by the concept of what is by the concept of the winding number of this singularity which is defined as follows if I start at any point here any arbitrary point there is a unique direction to this vector field f and it is evident that if I write this vector field f itself as a modulus times an argument this is equivalent to writing if you like this vector field as a complex number so I could write w equal to f of x, y plus i times g of x, y and this w is a function of x and y instead of a planar vector field I represent it as a complex number for a two dimensional vector field this is only true in two dimensions to start with if I took this w which is a function of x and y and wrote this as some r a to the i psi where r is the modulus of w and psi is the argument of w then if I start at any point in the plane and make a circuit of some kind and come back to this same point it is evident that psi should return to its original value because the field at every point is unique therefore this argument psi must come back to its original value it is quite clear or it must increase by a multiple of 2 pi so that e to the 2 pi ni is unity and you do not see it at all it is therefore clear that the integral of psi around any such closed circuit must be an integer times 2 pi if I call the circuit C therefore assert that integral around the closed circuit C d psi this must be equal to 2n pi where n is some integer simply from the single valuedness of this number of this argument psi which implies I go back here that the combination 1 over 2 pi integral d psi must be an integer but what psi itself this is 1 over 2 pi over C d tan inverse g over f because that is the definition of the argument of a complex number whose real and imaginary parts are given by these two fields scale of fields but this in turn is equal to 1 over 2 pi over C and if you simplify this exactly as we did tan inverse y over x it is immediately clear that you get fdg minus gdf over f square plus g square where f and g are functions of x and y and therefore in principle you could write dg and df in terms of dx and dy but if you integrate around any closed path in this vector field you are guaranteed as long as this function is well defined you are guaranteed to get an integer now what would happen and we can see this geometrically what would happen if I took a circuit which does not enclose a singularity of this vector field namely a point where f and g vanish or the vector field is not defined what would happen then that is easy to see because if I simply took a region of the space where the vector field is well behaved and has no singularities of any kind when I start at some point where the vector field points in this direction and took a closed path and came back in this fashion we can track what happens to the argument of this vector field by looking at what direction it points in by pretending there is a little umbrella which you hold as you move along this path and ask what happens to this umbrella's direction I start here in this fashion and I move up there so it perhaps tilts in this fashion and then I come down and it perhaps moves this way it comes here it comes back and then it goes out like this and then when I come back it slowly comes back to its original value so all it has done is to take a little perambulation of this kind and back to its initial value therefore this n has not increased at all first you had a small increment in the argument and then it decreased back to its original value went back oscillated and came back to 0 so in this case n was 0 immediately on the other hand if there is a singularity of the vector field and you went around a path which enclose the singularity then this is no longer true let's take a simple example let's look at a vector field which perhaps is in this fashion it kind of radial field in this fashion and they'll say there's a singularity somewhere there what would happen if I started and enclosed this path once in this fashion now let's follow it once again I start here in the vector field points like this in the tangential direction and let me just look at instead of moving along with this curve let me look at simply keep it fixed here and turn this arrow around to reflect what it does at various points in its path so it starts in this fashion by the time it comes here it's pointed like this by the time it comes here it's pointing in some direction like this like this out here it's definitely in this fashion and here it's like this here it's back there and it's back here and back to this so in this path as I go around the singularity once in the counterclockwise sense this arrow this umbrella has also gone around once completely in the same sense counterclockwise sense and the n is equal to 1 in this case because the total angle is 2 pi by which this vector field is rotated and I say that the singularity of this vector field at this point is the winding number is 1 plus 1 this is the definition of the Poincare index or winding number of a vector field around a singularity if it doesn't enclose the singularity if the singularity is somewhere here and this path doesn't have the singularity inside then of course the winding number is 0 so in a region where a vector field is completely well defined on a contour on which the vector field is well defined if that contour does not enclose the singularity of the vector field then the winding number corresponding to that contour is 0 because the argument doesn't change at all it might oscillate but never completes a complete 2 pi yes if I start here and I don't enclose this singularity it's evident that all this vector field can do is go up like this go down like this again and come back it doesn't turn once upon around its center completely so the argument starts at some value theta 0 increases to some theta 0 plus alpha comes back to theta 0 goes to theta 0 minus some beta and comes back to theta 0 it doesn't complete a circuit therefore the algebraic sum of all these increments is 0 and the winding number is 0 so you play around with this and you convince yourself that the only way in which the winding number is going to be non-zero is if this contour C encloses at least one singularity of the vector field and for this radial pattern we discovered that the winding number is plus 1 and now we can begin to ask what is it for other kinds of singularities what's it if you had a saddle point or a center or a node or a spiral point of various kinds those were the kinds of critical points we had for two dimensional flows and we could ask what's the singularity of the vector field look like but before that I'd like to point out to you that this number which we got as one for this kind of field is independent of this contour C I could have started here or I could have started there and gone around the fact is independent of this contour C as long as it encircles this singularity once in the counterclockwise sense the increment in psi is also guaranteed to be 2 pi it's therefore a topological property which is not a property of the specific contour C but rather a property of the singularity itself it's very similar to Cauchy's theorem in the calculus of residues it simply says if you have a closed contour which encircles a pole a simple pole of a function and the function of a complex variable Z then the line integral f of Z dz of this function around the singularity is reduced to 2 pi I times the residue at this point and it's independent of the actual contour as long as the contour encircles the singularity once in the positive sense now this statement is independent of the contour also the direction of the contour because if I started here at this point you could ask what's to stop me from doing this I go around once but then I do all kinds of meanderings here and come back and hit this once again it still doesn't matter all these increments would cancel out and the net result would again be 2 pi as long as you enclose this once in the counterclockwise sense what happens if instead I started with this kind of radial field moving outwards and I decided to traverse the contour in the counterclockwise sense I did this instead what would happen now well it's clear that again if I go around in the counterclockwise sense in the clockwise sense sorry in the negative sense then this vector would also rotate in exactly the same sense as the contour and the winding number is again plus 1 so it's a property not of the specific contour but rather of the singularity that's being enclosed that's why it's so important so it's remarkable that if you have some functions of X and Y which are well behaved in a certain region except for a singularity at some point where F and G vanish then this combination of algebraic quantities is guaranteed to be independent of this contour C provided it encloses the singularity and is an integer yes it doesn't take it itself so the statement is if I go around once in the positive sense in what direction does psi increase it increases by plus 2 pi if I go around once in the negative sense it increases by minus 2 pi therefore the statement is if psi increases by the same amount as what happens when I go around once then the winding number is plus 1 it should be independent of the C so I change by going around once like this if this is a complex z plane the complex plane in which X and Y are the real and imaginary parts the argument of Z increases by 2 pi once in going around this way and the argument psi also increases by plus 2 pi so the winding number is plus 1 if I went around like this the argument of psi decreases it changes by minus 2 pi and so does psi therefore the winding number is again plus 1 it's independent of the sense in which I describe this what happens if I took other kinds of singularities well let's look at some of them what if you had a singularity which is like what happens in the case of a center so we have field lines which go around like this this is what the field lines are for this vector field and let's put a sense on it what would be the winding number corresponding to this singularity at the origin well can you do exactly the same thing start at some point the vector field looks like this and go around and follow this trajectory all the time I start here in this fashion when I come here I am like this when I come here I am like this when I go here I am like this back to this back to this therefore the winding number is again plus 1 n equal to 1 what if I had a radially outward field well n was equal to plus 1 we saw but what if I had a radially inward field what if I had something like this these are the field lines what happens if I did this what would happen now again we do the same thing I start here at this point the field points so then I am going to go around in this fashion so it's when I am here it's like this when I am here it's like this when I am here it's like this like this this this and then when I am here I am back here and when I come back to this point I am back here. So as I go around once in the positive sense this field also goes around once in the positive sense and the winding number is again plus 1. So we see that at a node the winding number is plus 1 regardless of whether this node is stable or unstable. Therefore the idea of this winding number does not say much about the stability of the critical point but it says something about the local geometry of the vector field at this point for this as well it is plus 1 so is it for that and it is so even if the arrows are pointing outwards does not matter at all. So for a spiral point a node and a center it is easy to check regardless of the stability of the center of this of the nodes or the spiral points the winding number is plus 1 always what if you had a saddle point at a saddle the vector field let us say this is a saddle point two lines in two lines out and let us say these guys go out in this fashion in this fashion what would be the winding number around this saddle point well I start with the same trick as before I make a contour of this kind now look at what happens when I am here the field points so when I come to this point what does the field do so it starts here and by the time I come here to this point the field does this so please notice I am moving in the counter clockwise sense along this contour but the field on the other hand is moving in the clockwise sense so it does this does this by the time it comes here it is done that and by the time it comes here back again it is done this therefore it goes in the other direction so while the argument of z increases by 2 pi the argument psi decreases by 2 pi it becomes minus 2 pi is the change therefore the winding number in this case n equal to minus 1 for a saddle but it is plus 1 for a center node or spiral point regardless again of the stability of these points what that suggests is that what looks like a radial flow in one region can actually be uniformly deformed to look like a tangent flow as you move out this is certainly possible because this is what topology is all about you could start with a flow which perhaps near the origin is radially outwards let's say and as you go up it starts curving and as you go further out it starts getting more and more curved completely smoothly so that it is clear that the flow eventually could even look tangential if you are far enough away from the origin so this suggests that flows which look like sources or sinks could be made to look tangential by smooth changes in a smooth manner without actually crossing any singularities and that's the reason why the concept of a winding number didn't distinguish between centers nodes or spiral points on the other hand a hyperbolic point the saddle point is very different and there is no way in which one of these flows flows which correspond to any of these singularities can be deformed by a smooth change of variables to look like this not possible so it's of limited use but it gives us some hints as to how vector fields behave what would happen if you had more than one singularity inside I made a statement that the winding number if you enclose a singularity you are guaranteed that the winding number corresponding to the singularity is not zero on the other hand if I have a closed contour and I discover the change in this argument as I come back is zero I can't conclude that there are no singularities inside for exactly the same reason that I can't do that in complex variables because you might have two poles whose residues cancel each other just as you might have two charges in Gauss's theorem it simply says the total flux across a closed surface of the electrostatic field is equal to proportional to the sum of the algebraic sum of the charges inside and you could have two charges whose fluxes could actually cancel as you took the full integral in exactly the same way it's possible that if you had a closed contour in which you had both for instance a node as well as a saddle point then the minus one of the saddle point and the plus one of the node could add up to give you a zero and this is not difficult to see let me give you an instance right away where this happens we looked at the example of a saddle node which was x dot is x squared y dot equal to minus y and if you recall the flow here was along the positive x axis here and along the y axis it was inwards in this fashion and this side looked very much like a saddle point so the flow was like this and out here the flow flowed into this point in this fashion and the question is what kind of winding number does this field have around this point of course we unfolded the singularity and we discovered it really came about by coalescence by the coalescence of a saddle point with a node but you can unfold to put this back together in this fashion and it's not hard to see that if you took a contour and went around this contour see the net change in psi is in fact zero and what's happened here is that the plus one of the node and the minus one of the saddle have added up to give you a net change in the argument psi equal to zero you can't conclude based on that that there is no singularity of the vector field there very much is a singularity but the sum of these winding numbers has added up to zero in this case so the statement is if you discover that the winding number around a closed circuit is nonzero there exists at least one singularity inside on the other hand if you discover that the winding number as you do this integral around a closed circuit is zero you can't conclude there are no necessarily conclude there are no singularities inside there could be a set of singularities whose net winding number is zero and now let's go back to the example we started with which was essentially z dot equal to z squared and now this will make sense right away I don't even have to draw a picture because the flow x dot is x squared minus y squared y dot is 2 x y could be combined into z dot equal to z squared and the statement I made was simply that if you went around once in the z plane in the counter clockwise sense then the argument of z increased by 2 pi but the argument of z squared obviously increases by 4 pi which means that this vector field which was my W has a winding number at the origin of 2 because the angle increases by 4 pi and what is the field itself look like what is this field look like there's a complicated singularity here and it's a dipole field in this case so the field lines look like this it's exactly what a point dipole would do and I leave you to verify that if you took a closed circuit around this origin here then the net change in the argument psi would be 4 pi provided you traverse the circuit once in the positive sense this is what a point dipole does this is what the magnetic field lines of a point dipole of the electrostatic field due to an electric dipole look like and what it is and why is it to physically why is this to can you tell me this from your experience with electrostatics why is this to exactly there are two charges in there one of them acts like a source and the other acts like a sink if you take a single point charge if it's positive then the field lines are radially outwards that looks like an unstable node and if you took a negative charge the field lines go directly inwards and that looks like a stable node asymptotically stable node you put the two together arbitrarily close to each other such that the distance between the two vanishes and the product of the distance multiplied by the charge is finite the charge becomes infinite such that the product is finite you get a point dipole which looks exactly like this so this is characteristic of a dipole field and the winding number n equal to 2 in this case well you are sealing fan acts like a dipole field for the velocity field if you assume this fan the just the central portion of this fan is like a point source then it's sucking in air from above and pushing out air from below and there is circulation in this pattern so that is a simple example of a dipole source for the velocity field but the charge example is more familiar to you from electrostatics this is exactly what it looks like so this is one way in which you get some handle on higher order singularities by examining what the Poincare index of the vector field looks like gives you some hint as to what's going on let's go on now to another model which is very useful and very common and this helps illustrate a little theorem I want to talk to you about regarding limit cycles I pointed out that limit cycles don't exist in conservative systems but only in dissipative systems the reason is if this limit cycle is for instance stable then it says a whole lot of points in its basin of attraction get attracted to it exactly as in the case of a critical point which is an asymptotically stable critical point and such attractors don't conserve phase space volume because the whole area falls in into a line or a point and therefore cannot occur in conservative systems which we've defined as those systems for which the divergence of F is 0 everywhere measure preserving flows now let's look at a famous example of a nonlinear oscillator and ask whether in the presence of nonlinearity and dissipation you might perhaps have limit cycles and this model is called the duffing oscillator and it goes as follows it's a second order differential equation that specifies the duffing oscillator but we'll interpret this in physical terms and the equation of motion is x double dot plus some friction term gamma x dot and then terms which would correspond to nonlinear oscillations and this corresponds to oscillations in a double well potential and the potential looks like this versus x and as we know this potential is described by a fourth order potential function which has a maximum at this point at the origin say and two equal minima on either side so it's like having an inverted parabola near the origin and then moves out on either side so a model for v of x would perhaps be minus x squared over 2 plus x 4 over 4 with some constants multiplying these two cases and with that kind of choice of origin picture actually looks like this so that the potential vanishes at the origin and is symmetric has two minima on either side of it so little inverted parabola and then a fourth order term which takes you up in this fashion what would the phase trajectory is here look like we'll come back to this in a second that correspond to oscillations here or here or oscillations over both across both potentials both wells so this thing here has a term which is minus some constant times x which would come by differentiating this point I put an alpha there let's put an alpha here plus perhaps a beta x cubed which would come by differentiating this term and putting a minus sign and this in general could also be driven you could also take this system and put an external force upon it which could for instance be sinusoidal so perhaps some a cos omega t and you could now ask what about the dynamical behavior of this system and let's interpret these terms once again I've divided through by the mass of this oscillator so this is just x double dot this term represents linear damping proportional to the velocity instantaneous velocity of the particle these two terms come from the restoring force represented by the potential the fact that you have a minus sign here comes from the fact that this is an unstable point the origin is actually unstable and this beta x cubed with positive beta comes from the fact that you have two stable minima on either side it comes from this potential and that is driven by some external force with frequency omega with amplitude a how many parameters are there in this problem well there's one here two three four the amplitude of the driving force and five with five parameters you have an extremely rich set of possibilities for the dynamical behavior of the system but we could choose the scale of time so as to get rid of one of these constants and let's choose the scale of time in such a way that this thing here is unity you still have four parameters gamma beta a and omega and in the parameter space of these four constants these four parameters you actually have many many possibilities and the full set of possibilities of the duffing oscillator can only be understood numerically and it includes chaotic behavior of various kinds it's extremely complicated is this an autonomous system or non-autonomous it's non-autonomous you're right because there's this forcing term here so there's explicit time dependence so very complicated things could happen in this system let's look at the simplest version of this in some form in which I don't have a forcing at all but let's look at the case x double dot plus gamma x dot minus x plus let's just put this beta equal to one this equal to zero and ask what about the unforced nothing duffing oscillator that corresponds to motion in this potential but in the presence of damping and now I could go ahead and ask all right suppose you didn't have this damping at all what does the what do the phase trajectories look like and that's very straightforward because here's what it does here's the potential and if I plot x here versus x dot for instance on this side then the phase trajectory is a very straightforward you have a little center here you have a center here and you have a hyperbolic or saddle point out there for small enough energy just above the minimum of this potential you could have stable oscillations about either of these centers and therefore you'd have trajectories of this kind for slightly larger amplitude oscillations you'd have non harmonic oscillations you'd still have this kind of behavior on the other hand if you had enough energy and you oscillated with this total energy it's clear that the particle could cross this barrier go to the other side till that point come back and oscillate with this amplitude and it would do so with something which looks like this this amplitude here so it would certainly do something like this it encloses both the wells the minimum and what's the critical value or the separatrix value of the energy in this case it's zero itself at this value it's evident that if I start here and give it a little perturbation to the right it would start here go down like this and come back and form a homo clinic orbit which is a separatrix on which the time period would diverge similarly on this side you'd have a symmetric thing which looks like this and you have your saddle point and these are the separatrices we can easily find the tangents to these separatrices in this model but the actual stable and unstable manifolds are like this it's the really homo clinic orbits and outside you have oscillation across both wells and inside you have oscillation around this well this is what would happen if you had this system without the gamma without any damping at all but now the question is interesting question is I switch on the gamma I have a finite positive value of gamma what do the phase trajectories look like it's not easy to draw this immediately because I could start at some point in the space and then it would start by doing this but because of the damping the amplitude would keep reducing and eventually it would find itself stuck either at this equilibrium point or at this equilibrium point here this separatrix is no longer there once you have friction that's the whole point this is only true if you have no friction but once you have friction this is no longer true so I start with this energy perhaps it goes around the first time and then it comes back it slows down it's below this and then it's stuck here and oscillates here on the other hand depending on what it does whether it crosses this for the last time to the right or to the left it could any starting point would fall in the basin of attraction of either this attractor or that attract and these would actually be spiral points asymptotically stable spiral points so unlike the non-dissipative system the absence of gamma where you have two centers and a saddle point in between in the dissipative system you have two asymptotically stable spiral points and where a given initial point falls finally is dependent very crucially on where you start and it turns out actually that in this model subsequently I try to show you some pictures of this the basins of attraction of this point and that point they riddle each other the kind of interleave with each other so you could perhaps start here and end up there but you start here and you could end up here in this fashion they actually fold around each other in a very intricate fashion and I can only do this numerically at some level but you also can ask the interesting question can this system with the damping switched on have a limit cycle somewhere what's to stop me from doing that after all I start by saying look there's a nonlinearity in the problem due to execute and there's dissipation so is it not possible that there exists actually a stable limit cycle there's some isolated trajectory on which you have periodic motion and everything within it falls into some asymptotically stable spiral point this could happen and we'd like to find out if this is true or not so the question is is there a limit cycle in this system is there a limit cycle and there's a criterion called the Bendixson criterion which says there isn't in this problem and it goes as follows let me stop with that so you have the Bendixson criterion which reads as follows I start with x dot is f of x, y, y dot is equal to g of x, y which is exactly what this system is like because recall that this implies x dot equal to y and y dot equal to minus x equal to x minus x cubed minus gamma y so y double dot x double dot is y dot the same as y dot and that's equal to this and the criterion says the following it says if you have a vector field of this kind a dynamical system of this kind and you consider this in some domain in the x, y plane some domain D which is simply connected no holes there are no holes in this no singularities in this domain and in that domain D f and g are continuous and have continuous partial derivatives first partial derivatives and moreover if delta f over delta x plus delta g over delta y has a definite sign at every point in other words is always positive or always negative in this region and this remember is just the gradient of our vector field f in this case so if this has a definite sign always positive or always negative then there can be no closed trajectories lying in this region in D therefore it would say there can be no limit cycles either which is a closed isolated periodic orbit the proof is very simple if there is such a trajectory then on that trajectory y dot is g of x, y and x dot is f. Therefore dy which could be written as dy over dx times dx would be equal to well dy is g over f dx and on this trajectory it immediately follows that f dy – g dx equal to 0 on the trajectory and if I integrate over this trajectory this must be equal to 0. So if you had a closed trajectory of this kind inside this domain D that must be true but by Green's theorem in the plane this quantity is also equal to over the surface s bounded by this C it is also equal to over s delta f over delta x plus delta g over delta y dx dy but that contradicts our initial assumption if this function never vanishes has the same sign throughout s then this can't be 0 on the other hand it must be equal to 0 if you had such a closed trajectory. So this forbids you from having a limit cycle if the Bendixson criterion is satisfied what's the gradient what is del dot f in this problem it's the derivative of this with respect to x which is 0 plus the derivative of this with respect to y which is minus gamma and that's not 0 therefore we conclude that by the Bendixson criterion the unforced duffing oscillator linearly damped duffing oscillator cannot have a limit cycle it's an illustration of a fairly powerful theorem which works for these planar vector fields otherwise you'd have to examine numerically and we're never sure whether there could be such a case or not a limit cycle or not some isolated periodic orbit but this tells you no matter what your initial conditions are you're not going to have a limit cycle you're always going to flow either end to this or this attract attract the point attract eventually a very useful criterion and we'll see more properties of this next time thank