 Now let us consider the possibility of critical points which are like lines now in some sense and I am going to do this with the help of an example. We look at a simple non-linear system where this happens again in 2 degrees of freedom 2 dimensions. So let us look at x dot is equal to say mu x plus may be a y plus x times x squared plus y squared and y dot equal to mu y. So let us put an x here plus y with a mu where mu is a real parameter. So this portion is a linear part of it and that is the non-linear part of it and it is immediately obvious from here that you have critical points you have a critical point certainly at 0 0 both these guys vanish and what sort of critical point is it if you linearize near the origin this L matrix is mu a 1 here a minus 1 here and a mu here which implies that the trace is equal to twice mu and the determinant is equal to mu squared plus 1 and remember that lambda 1, 2 is equal to minus t plus or minus square root of t squared minus 4 delta divided by 2 plus t. So this is equal to twice mu plus or minus square root of t squared which is mu squared minus mu squared minus 1. So it is 3 mu squared 4 mu squared minus 4 delta. So that is 4 mu squared minus 4 divided by 2. So this cancels out here and this just gives you 2 i mu plus or minus i. So what kind of critical point is this depends on the sign of mu but since it is complex conjugate as a pair of complex conjugate critical points it is eigenvalues it is clear that the critical point is a spiral point. So this immediately says that it is an asymptotically stable spiral for mu less than 0 and an unstable spiral for mu greater than 0 at the origin and the phase picture as you can imagine is a spiral going inwards into the origin for mu less than 0 and away from the origin outwards for mu greater than 0. We should like to know what the direction of the flow is as well. For that let's go to polar coordinates plane polar coordinates. So I set in the usual way x is equal to r cos theta y equal to r sin theta which immediately implies that r squared is x squared plus y squared which implies by differentiate both sides r r dot equal to x x dot plus y y dot and that becomes in the present instance I multiply this by x and this by y add the 2 and I get mu times x squared plus y squared so mu times r squared plus r to the power 4 which comes from these 2 terms and then this gets multiplied by x and this by y when you add the 2 it vanishes. So this gives me r squared into mu plus r squared in this fashion which implies that r dot is r times mu plus r squared. Therefore the picture which says that for mu less than 0 this is an asymptotically stable spiral also shows that for mu less than 0 there exists a trajectory corresponding to r equal to square root of minus mu on which there is no change of r at all and since r dot is 0 the trajectory is a circle of some kind. On the other hand if mu is positive this is an unstable spiral nothing happens everything flows off to infinity we'd like to know the direction in which things flow and it's evident immediately that we have to draw two different phase portraits one for mu less than 0 and the other for mu greater than 0. So let's do that let's plot this for mu positive on this side and on this side for mu negative what kind of behavior do you have for mu negative for mu positive this is an unstable fixed point a critical point there on the other hand it's stable so mu less than 0 but it's also clear for mu less than 0 r dot equal to r times r squared minus modulus mu which is equal to 0 for r equal to square root of modulus mu. For mu less than 0 there exists a circle this kind on which you don't change r at all therefore this is a trajectory by itself on the other hand you have asymptotically stable spiral here and outside here it's evident from this picture that if r is greater than square root of mod mu r dot is positive therefore the flow is outwards off to infinity on the other hand if r is less than square root of mod mu the flow is inwards r decreases towards the origin towards a stable fixed point at the origin the picture is very different for mu positive there when mu is positive this can never vanish so you don't have such a root at all such a solution at all and it's clear that r always increases the only question is in what direction do things happen and to find that we need to find out what theta does so let me write that down retaining this equation theta is tan inverse y over x which implies that theta dot is equal to 1 over 1 plus y squared over x squared multiplied by the derivative of this which is y dot over x minus y x dot over x squared and that simplifies to x y dot minus y x dot divided by x squared plus y squared if I go back to these equations of motion I multiply this by x and this by y and subtract if I multiply this by x I get minus x squared so let's write this down so in the present problem theta dot is equal to minus x squared minus plus mu x y plus x y times x squared plus y squared minus this is a got to be multiplied by y and subtracted so minus mu x y minus y squared minus x y times x squared plus y squared that's the numerator and the whole thing is divided by x squared plus y squared and it's immediately clear that these terms cancel out and you end up with is equal to minus 1 which means on that circle theta decreases at angular velocity equal to 1 in the clockwise direction because theta is decreasing so this means it does this on the circle therefore a point starting in the neighborhood of r equal to square root of mod mu this circle had a radius r equal to square root of minus mu what happens to a point in here it's clear that it's going to spiral in and fall into this fixed point as in tautically similarly it's evident that if I start a little bit outside I'm going to spiral away towards infinity outwards that isolated closed orbit is called a limit cycle so let's define it this limit cycle and isolated a limit cycle is an isolated closed trajectory and it's important to notice that it should be isolated in other words in its neighborhood there are no other close trajectories there exists a neighborhood of this circle in which there are no other close trajectories in this problem there isn't one in any case anywhere else except here notice on contrast that near a center if this were a center you'd have a whole family of close trajectories of this kind arbitrarily close to this trajectory closed orbit there are other closed orbits and this is not a limit cycle on the other hand in this situation it's the clear that this is a very special trajectory everything inside falls away from it everything outside falls outwards away from it therefore you'd intuitively see that this is an unstable limit cycle you could have a situation where everything falls into it then it would be a stable limit cycle you could have a situation where things fall into it from one side but fall away from it on the other side of it and then it would be a semi stable limit cycle but the general situation is that the limit cycle is defined as an isolated closed trajectory it should be obvious immediately that you can't have a limit cycle in a conservative dynamical system because this means that whole sets of points are falling into this point here the whole sets of points are moving away there so there's no question of conservation of a space volume here at all a limit cycle therefore a stable limit cycle is an attractor which occurs only in dissipative systems no possibility of limit cycles in a Hamiltonian system in particular and in conservative dynamical systems in general we've defined conservative dynamical systems as those for which the phase space volume is preserved under the flow limit cycles therefore are a feature of dissipative systems in this instance the limit cycle was actually unstable but let's look at a slight modification we tweak this model a little bit till we get a stable limit cycle what would you need for that we'd like to have things go into it from both directions what would you require for that what should we do to this model we change things a little bit here so that we end up with stability there what would you suggest well had the situation been reversed namely had this become unstable and this becomes stable then the job would be done right all we need to do is to ensure that this becomes stable and this becomes unstable what should I do therefore pardon me minus mu okay we put a minus mu x then what put a minus here put a minus here this do the trick well let's try what do you think happens well the linearized matrix near the origin is now minus mu a 1 here a minus 1 here and a minus mu here and the trace is equal to minus 2 mu determinant is still mu squared plus 1 so lambda 1, 2 is equal to minus 2 mu plus or minus square root of 4 mu squared minus 4 mu squared minus 4 so this becomes a 2 I so it's just minus mu plus or minus I what next and what happens to the equation for RR dot what happens here for RR dot pardon me where should I put a minus in the non-linear term I should put a minus sign here this is not going to work we'll try that's exactly I'm just replacing mu by minus mu exactly simply replacing mu by minus mu so do you think anything is going to happen maybe not okay so what's his suggestion what do you suggest what do you think I should do all of them should have negative signs this should remain as it is the non-linear term should have a negative sign okay we put a negative sign here and we put a minus sign there the linear the eigenvalues are not changed so let's draw this picture all possibilities can occur so let's draw what see what happens now it's evident here that these are the eigenvalues the way I've written it here because this is not affected doesn't affect the linear terms and it says that when mu is positive you end up with a spiral which is stable when mu is negative you end up with a spare spiral which is unstable right yes okay so let's go back to our original way of looking at it we had this and we change the sign here in which case this is mu and that's mu so we go back we did not change anything now it's this and it's this so this doesn't change it's an asymptotically stable spiral for mu negative and an unstable spiral for mu positive but the equation for RR dot that definitely changes because this immediately says that RR dot is equal to multiply this by X multiply this by Y and add the two and you get mu times R squared minus R to the power 4 which implies that R dot is equal to I take out an R squared and calculate one of the R so you get R times mu minus R squared so once again we see there's a limit cycle that R is equal to square root of mu but this time for positive values of mu so what does the picture look like this one negative values of mu and this for positive values of mu now I have X as well as Y so I really need to write down a proper bifurcation diagram but let's first draw what happens in the phase plane here's X here's Y here's X and here's Y and this corresponds to mu positive this corresponds to mu negative and for negative values of mu there is no root here R dot does not vanish anywhere other than R equal to 0 and this is an asymptotically stable spiral and we should also check what theta dot does so let's do that theta dot is equal to X Y dot minus Y X dot divided by X squared plus Y squared and I think it's not going to change so I multiply this by X I get minus X squared plus XY they multiply this by Y I get a mu XY and I put a minus sign so it's minus R squared minus one the nonlinear terms cancel out in any case so once again when our mu is positive there exists a circle of radius square root of R equal to square root of mu this time on which the flow is in the counter clockwise direction and inside you have an unstable spiral point so it's evident that wherever you start other than R equal to 0 you're going to flow into this limit cycle if you start with a value of R bigger than square root of mu R dot is negative and therefore R shrinks in words and therefore if you start here you're going to flow in into this point from outside on the other hand when mu is negative you just have an unstable spiral so this point is unstable here sorry you have a new negative you have an asymptotically stable spiral and therefore wherever you start you're going to flow in this direction into the asymptotically stable spiral point it's quite clear that what has happened is that if you start from negative values of mu and move towards positive values what started off as a asymptotically stable critical point has bifurcated it's become unstable but it's also given birth to a stable limit cycle something which is asymptotically stable so the bifurcation diagram in this case looks like this I need to plot x and y equilibrium values so let's do that here and let's put this to be the mu axis and this is x and this is y I mean the steady state values values on which things don't change at all then the only equilibrium point for mu less than 0 is an asymptotically stable spiral point at x equal to 0 y equal to 0 so it's on this axis as soon as mu crosses over to positive values this point becomes unstable therefore in keeping with our usual notation we should really draw this with the dotted line like this and in its place you have a limit cycle which is stable of radius r equal to square root of mu but on the limit cycle x and y change without changing r and this radius of this limit cycle increases like the square root of mu therefore the picture would be something like this this really is a kind of parabolic bowl which comes out whose size this radius here increases like the square root of mu and it's in both x and y you can see so it's not a critical point it's not an equilibrium point there is dynamics going on there but the moment you hit r equal to square root of mu it remains at that square root of mu and this is a stable limit cycle this is an unstable critical point and this was a stable asymptotically stable critical point in the bifurcation which happens at mu equal to 0 of a stable critical point into an unstable critical point and a stable limit cycle is called a hope bifurcation it's one of the most important bifurcations in non-linear dynamics happens all the time we will see examples of this when we study chemical and biological systems and oscillations in these systems this is a standard mechanism by which periodic behavior suddenly emerges from nowhere as you change a bifurcation parameter so on this thing you can see that it's a periodic orbit it's a periodic it's a close trajectory but an isolated close trajectory because I've tailored this dynamical system to illustrate this point these were very simple functions here very specific functions and cancellations occurred therefore the limit cycle had the shape of a circle that's not necessarily true it could have a very complicated shape it could change they could be families of limit cycles but the basic definition of a limit cycle is that it's an isolated periodic trajectory in a dissipative system and discovering limit cycles is not anywhere near as easy as finding the fixed points or the critical points of the system for which all you had to do was to equate the right hand sides to 0 in some sense but this is not so for limit cycles especially if you don't if you're not able to reduce things to very simple things simple algebra using say polar coordinates or anything like that so the shape of the limit cycle is complicated you can only discover this numerically but there are theorems and criteria which will tell you when these limit cycles could exist and whether such limit cycles could exist incidentally from this picture it's also clear that inside this stable limit cycle they must exist at least one unstable critical point otherwise there's no way these things would get thrown out and move into this limit cycle here and that's measured by something called the winding number of the Poncare index which I'll come to a little later yes can and I oh yes open trajectories we're not considering here at all because they're not of much interest to us at the moment but close trajectories would mean periodic motion and we'd like to start with things which are periodic and maybe apply perturbations and go to things which become quasi periodic or a periodic this is going to be the thrust of what we're doing open trajectories unbounded motion in general is not of direct interest there are of course many physical situations where it's important but they're not directly of concern to us here specifically we're always looking for periodic motion or things close to periodic motion in some fashion the earlier example what happened in the earlier example if I put a plus sign here what happens now what kind of behavior do you have once again since we've chosen this mu to be exactly as it is this is certainly true you have an asymptotically stable spiral for mu less than 0 and an unstable spiral for mu greater than 0 but because we change the sign of the non-linear term this becomes a plus here it becomes a plus here in this case and then the picture gets reversed this is for mu less than 0 and this is for mu greater than 0 for mu greater than 0 this was an unstable spiral here so this is unstable and things are going to go flow away from it for mu less than 0 this is a stable spiral point asymptotically stable spiral point things are going to flow into it but this trajectory here at r squared equal to square root of minus mu is certainly going to be an isolated trajectory here but I should really draw it as a dotted line because it's unstable it is evident that here things are flowing away in this fashion on the other hand here things are going to flow into this origin this is stable so it's going to flow in here and things are going to flow starting here flow outwards that's right this is therefore an unstable limit cycle and it contains within it a stable critical point so here what's happening is as you move from positive mu towards negative mu what was an unstable critical point has bifurcated into a stable critical point and an unstable limit cycle if you like it's the image it's the complement of what happened earlier and let's draw the bifurcation diagram in this case and you have a picture where as a function of mu so here's x here's y and here's mu greater than 0 you had an unstable critical point therefore it was like this and it becomes stable for mu less than 0 so this is unstable and this is a stable critical point asymptotically stable critical however you have now a limit cycle that's unstable and therefore I should really draw it with a dotted line like this and these are the trajectories on it and this is an unstable limit cycle compare this with the earlier case where you had it doesn't matter which way this parabola looks the way we've written it drawn it this is moved to the right here but it's not necessary this was stable and inside it you had an unstable critical point and this was a stable critical point so stable critical point came along bifurcated into an unstable critical point and a stable limit cycle and this was a stable critical point and this was an unstable. This is what we call a hope bifurcation this to is a hope bifurcation this is called a subcritical and this is called a supercritical of bifurcation. In either case it is the bifurcation by which a critical point bifurcates into a critical point and a limit cycle and the stability is in the two cases or as I have shown here. The fact that this parabola looks to the right and that looks to the left is an artifact of the way we have drawn this, the way we have chosen coefficients but the essential point is that a stable one, a stable critical point bifurcates to a stable limit cycle here and loses its stability. In a subcritical bifurcation on the other hand an unstable critical point gains stability but also gives rise to an unstable limit cycle in the process. Both these processes happen very often in nature but as I said earlier they happen only in dissipative systems. You have limit cycles only in dissipative systems. Here is a simple example of a system at limit cycles. So let us consider r dot equal to sin pi r and let us say theta dot is equal to 1 in polar coordinates. So I have already gone to polar coordinates and done this. What would you say is the behavior of this system? What does it do? In polar coordinates if it does this and we could go back and write the equations in x and x dot if you like but we can analyze it directly as it stands here. What would you say is the behavior of this system on the right hand side? When does the right hand side vanish in the first equation? So it is evident that limit cycles at r is equal to 1, 2, 3 and so on. We do not yet know what is happening at r equal to 0 which is the origin and you have presumably a critical point at that place. We do not know the nature of this critical point. What kind of limit cycles do you have at various places? Would it be stable or unstable or what? How would you decide this? Yes, you are quite right. They would alternate in stability. One would be stable, the next would be unstable and so on. What would be the one at r equal to 1? For instance sin pi r near r equal to 1 you have to do a Taylor expansion about r equal to 1 and of course sin pi is 0. So the first term is 0 plus the next term is r minus 1 times the derivative of sin pi r evaluated at r equal to 1. What is that? It is pi cos pi r. So there is a pi cos pi r at r equal to 1 so that is a cos pi and this is equal to 1 minus r in the vicinity of r equal to 1 plus higher order terms. So it is really telling you that r dot is of the order is of form 1 minus r. Is this therefore a stable or an unstable limit cycle at r equal to 1? If r is less than 1 it is growing and if r is greater than 1 it is shrinking and therefore it is a stable limit cycle. So we immediately realize that at r equal to 1 you have a stable limit cycle and it is not hard to see at r equal to 2 you get a cos 2 pi here and that would become plus 1 and therefore it would become unstable. Therefore the picture in this case would look like this at r equal to 1 you have a stable limit cycle at r equal to 2 you have an unstable limit cycle at r equal to 3 you have a stable limit cycle once again this corresponds to what would you therefore expect is the behavior at the origin which is a critical point it should be unstable so that things would flow into it and I would therefore expect that in this case everything goes around counter clockwise direction therefore I would expect trajectories to do this. So all points between r equal to 2 and r equal to 1 1 would eventually fall into this trajectory they would fall into this stable limit cycle what is the basin of attraction of the limit cycle at r equal to 1 so we have the concept of the basin of attraction of the stable limit cycle r equal to 1 is all points which lie inside they are going to move outwards except the origin which is a critical point by itself so you really have an unstable critical point there so this is 0 less than r less than 1 and what happens to points between 2 and this way to fall in so in fact less than 2 points on r equal to 1 are already there and similarly for the basin of attraction for this is everything between 2 and less than 4 would fall into this and so on. So this is the second kind of attractor other than a critical point point attractors we now have closed orbits as attractors and you can easily see that one could generalize this and have in higher dimensions you could have a torus attractor you could have an n dimensional torus as an attractor but the interesting thing and we will see this little later is that for high dimensions dimensions greater than 2 you have another kind of attractor possible called a strange attractor and those could be very complicated fractal objects and they would be the ones of central interest to us as we go along in real non-linear systems. So I stop here today and then we take it up from this point we have seen some of the elementary bifurcations here and we will see what other possibilities can exist for dynamical systems of this kind no they need the question is can limit cycles do limit cycles have to be circles the answer is no of course not it happened to be they happen to be circles in the examples I showed because I have contrived it in that fashion I have written this already in polar coordinates in a simple way and in the earlier example I had very simple functions of r squared there is no reason why this should be so at all the very first example of a limit cycle which occurred was in triode oscillations and it is called the van der pol oscillator and it has a term which goes like it is again a two dimensional dynamical system there is an x double dot term which would be the inertia term plus a term which is proportional to x so some natural frequency squared times x plus a damping term and then if it is a simple harmonic damped simple harmonic oscillator linear oscillator this would be a constant times x dot but you have here a term which looks like 1-x squared times x dot times some coefficient and this equal to 0 is a nonlinear equation because of the presence of this term x squared x dot and you can see that this damping for x squared less than 1 acts like normal damping but x squared greater than 1 it feeds back into the system or if you like you put a minus sign here this would reverse the situation and say for sufficiently small x things would tend to grow outwards and for larger x they would tend to come back so the damping acts in a state dependent manner it is dependent on x so either you have for positive feedback or you have negative feedback depending on what x itself is and the limit cycles of this system for the first limit cycles to be discovered and this is the van der pol oscillator I am not too sure about the signs here but this is essentially what it is you could change this constant it does not matter but this is the way this system behaves and as I said earlier this occurred for the first time in the consideration of triode oscillations in a triode in the old vacuum tube we will see further examples of limit cycles.