 One of the questions that arose in our studies was what happens if you have a degenerate system namely one in which the linearized matrix is has a zero Eigen value. Well there are several kinds of degenerate systems but let us look at a specific example the simplest of these. So let us look at one for which x dot is perhaps A x plus B y and y dot equal to some constant times the same thing A x plus B y. So of course this immediately implies that L is A B K A K B. So delta equal to determinant L is identically equal to zero then the question is what kind of critical points you get here and the answer is very simple you have a degenerate system these two equations are not linearly independent of each other is just a multiple of that and therefore in the x y plane if you plot the locus of the points on which the right hand side is a zero you get A x plus B y equal to zero or x or y equal to minus B a over B x which perhaps is some straight line of this kind and at every point on this line critical line if you like the system is in equilibrium no time change at all then of course you could ask what is the flow like what happens if you have an initial condition which perhaps starts here this would depend on what the signs of these constants are but in general the flow would be either along this line inwards everywhere into this line or perhaps outwards all the arrows going outwards. So this is a very simple example of a degenerate system it is not of particular interest to us right now more serious would be what happens if the system is not linearizable intrinsically the example that we look at is even in a one degree of freedom system for example just a single variable if you had something like x dot equal to x squared and you ask what does this do what kind of critical points do you have in this situation well we have a phase line there is a critical point at zero and we cannot tell whether it is stable or unstable in the conventional sense because for positive x the flow is outwards but for negative x also x squared is positive and therefore the flow is inwards this direction therefore this is as far as the left hand side is concerned it is an attractor but as far as the right hand side is concerned it is a repeller it is a higher order critical point now the question is can we say something else about it and how does this higher order critical point arrive well it occurs because x squared is not a generic polynomial on the right hand side if I ask you to write down a polynomial on the right hand side or something which is expandable in powers of x about the origin you would start by writing a naught plus a 1 x plus a 2 x squared and so on and so forth and unless you have a special accident so that a naught and a 1 about 0 you never start with the x squared term on the other hand if I put a constant here just a single constant you can always absorb that constant by shifting x to that point so the constant is irrelevant but certainly the linear term is missing so the sensible way to do this would be to ask instead of looking at this system directly unfold this second order 0 if you like this double 0 by writing this as equal to some epsilon x plus x squared and examine what happens for various kinds of epsilon small positive small negative or 0 epsilon equal to 0 corresponds to the case you have but epsilon positive or negative would be a generic system and you could linearize it in the origin of these points well this instance for example there is a critical point at the origin and there is also one at minus epsilon so there is one at minus epsilon out here in the instance in which epsilon is positive then of course minus epsilon is located here and you could ask what kind of flow do you have once again near the origin near the origin here in the vicinity of the origin I could write x dot is approximately equal to epsilon x which means if epsilon is positive the flow is outwards along the direction and outwards along this direction here by continuity therefore it remains in this direction all the way up to this critical point and now you ask what kind of critical point is this for that you have to linearize about this point so the sensible thing to do is to put in this neighborhood is to put u equal to x plus epsilon so that the origin in you is the point x equal to minus epsilon and then ask what happens to this system well that system now becomes you dot equal to u multiplied by x because this is x plus epsilon times x but x is u minus epsilon which is equal to minus epsilon u plus u squared once again if you linearize in the neighborhood of this point then the linearized equation is linearize its u dot is approximately equal to minus epsilon u and since epsilon is taken to be positive it is flowing in into this attracting fixed point here and it is flowing in this way the other direction so you have an attractor at minus epsilon and a repeller at the origin and the coalescence of these two when epsilon is 0 produces for you this figure this higher order critical point and this picture is valid for epsilon greater than 0 this picture therefore corresponds to epsilon equal to 0 and one could ask what happens if I have chosen epsilon to be negative to start with that is easily taken care of because if epsilon is negative the fixed points are at the critical points are at 0 once again and at the point minus epsilon which is now on the positive side since epsilon is negative and it is easy to see that linearization about the origin is always x dot is epsilon x and since epsilon is now negative the flow is inwards into the origin on both sides in its neighborhood and it is a trivial matter to check once again that the flow is outwards from this point at the point u equal to 0 or x equal to minus epsilon because epsilon is negative this number becomes positive here minus epsilon and it is a repeller so what happens effectively is that this point becomes a repeller and this becomes an attractor so now we get a much better picture it says that the system x dot is equal to x square which is intrinsically non-linear really arises by an accident it arises by the coalescence of two critical points one at the origin and one at minus epsilon as the second critical point minus epsilon crosses 0 as it crosses the origin as it crosses the value 0 you get what was initially to start with starting with positive values of epsilon and attractor at minus epsilon and a repeller at the origin becomes a higher order critical point when two of them coincide and then as you move over to negative values of epsilon you have an repeller at minus epsilon and an attractor at the origin so there's been an exchange of stability and this is called an exchange of stability bifurcation it's an honest bifurcation one of the simplest one can think of and we draw what's called a bifurcation diagram in which I would plot as a function of the parameter epsilon at plot the equilibrium or stationary value or steady value of x in this case the value at the critical point but there are two critical points one of them is always at the origin and therefore you have a line line on this axis and the other one is at the point is at the value minus epsilon and that is a straight line which is tilted at 45 degrees and of this kind and this if you like is x equilibrium on the other hand I know that for negative values of epsilon this critical point is unstable and therefore I denote that in standard notation by a dotted line as a dotted line here for negative values of epsilon it's unstable and therefore I have a dotted line of this kind and a solid line here to indicate that this is a stable critical point so this is unstable and this fixed point is stable when you cross over to positive values of epsilon what was stable becomes unstable in this fashion and what was unstable continues on as stable so this is stable and this is unstable and we have an example of a bifurcation which in this case is called an exchange of stability bifurcation we'd like to now ask a slightly more general question what kind of bifurcation can we expect in such systems this was a simple one dimensional system but it can be generalized to higher dimensions and the question is what kind of elementary bifurcation or distinct kinds of bifurcation do we have this problem too has been analyzed in great detail and bifurcations have been classified at least the elementary ones have been classified in simple dynamical systems I should mention that bifurcations which involve a single parameter which you tune in this case just epsilon they call bifurcations of co-dimension one and this is distinct this dimensionality is distinct from the actual dimensionality of the phase space that we are dealing with this is in the parameter space and there is just a single parameter here so it's a bifurcation of co-dimension one in this instance we'll see later that we have bifurcations of higher co-dimensions and classifying them is a non trivial task classifying bifurcations in general in higher dimensional dynamical systems the fairly non trivial task now I'd like to put this in put this in the framework of a slightly more general setting and that's as follows so I'd like to look at it as two dimensional phase space but just to get a physical feel for what is meant by tuning this epsilon let's cast it in the language of a mechanical example so what I intend to do is to consider a particle of unit mass moving along the x direction and look at these dynamical equations as equations of motion for this particle in some potential v of x so the equations of motion are x dot equal to p if I consider unit mass this is momentum here and p dot equal to minus dv over dx which is the force on the particle and I'm going to tune this by writing different kinds of functional forms for this v of x and asking if I can examine bifurcations in this framework here now what would you say is the simplest of these forms that you could write down here typically I'd write down various kinds of polynomials for this force here and the simplest of these that one could write down is perhaps to say that this is a constant plus a linear term in x but it's quite clear that if you have an a plus bx here the only critical point is at p equal to zero and x equal to minus a over b and that's it there's just a single critical point and there's no possibility of any bifurcation no coalescence of singularities the next non-trivial case would correspond to putting in here some parameter and let's various symbols could be used let's use a and let's put a bx squared what kind of potential does this involve what is the shape of v of x in this case if I plot here x versus v of x remember this is the force it's minus dv over dx so this would imply that v of x itself equal to minus ax plus bx cube over 3 minus if I integrate this and change the sign I end up with this what kind of shape is that it's a cubic curve clearly we'd like to plot it and we should like to know whether a is positive or negative or what let's fix the sign of b let's take b to be positive for instance you could simply redo the whole thing for b negative there will be no essential change in what I'm about to say so let's suppose that b is always greater than 0 and a could be negative or 0 or positive let's look at all possible cases now what happens if a is negative what's the shape of this curve so I plot this for a negative and the shape of this graph is approximately linear at the origin with a positive slope and therefore the potential looks like this here but eventually this term takes over that's a large negative term so it's evident that this is going to go down and fall down on the other hand when x is large negative this term dominates and becomes positive in sign and therefore the potential has a shape of this kind so one knows immediately that the critical points of the system occur at p equal to 0 and this quantity equal to 0 and what does that look like what would this do well it's clear that there is an equilibrium point here at this point and that's a saddle point because it's a maximum of the potential and this point here is a minimum of the potential and therefore this point is a center since there's no friction in this problem you would have small oscillations about that point about the minimum of the potential whereas the maximum of a potential in the absence of any dissipation is always a saddle point this is all that a Hamiltonian system could have we have here a Hamiltonian system in which the Hamiltonian is p squared over 2 for unit mass plus v of x and these are Hamilton's equations that I have written down and this is the picture that we have so we clearly have critical points at p equal to 0 and the values here correspond to x equal to minus a over b square root of this with a plus or minus and this is the picture for a less than 0 remember the plus corresponds to the minus this point here corresponds to a center and the plus corresponds to a saddle you can write down the 2 by 2 linearized matrix and check out at each point that this would correspond this center would correspond to pure imaginary pair of eigenvalues and this would correspond to one positive and one negative eigenvalue if you linearize about these values of x we know how to do that now what happens if a is exactly equal to 0 once again if I plot at a equal to 0 I plot the potential v of x versus x what would this correspond to this term is gone and you just have a minus b x cubed and that at the origin is extremely flat it has an inflection point and it falls off in this fashion again for b positive so this point here has arisen because this maximum and this minimum of the potential have come together at the origin and it's become an inflection point where the slope is 0 in the first second derivative the curvature is also 0 at this point and now finally what happens when a is bigger than 0 so we look at the picture a greater than 0 that's the third case and if I plot x versus v of x then when a is positive you have a negative slope here and therefore this curve looks like this approximately linear and then of course as x becomes larger it falls off like a cube and it increases here like a cube there's no possibility of any equilibrium point at all in this potential there's no maximum or minimum at all there are no critical points in this dynamical system because if a and b both have the same sign there's no way this quantity can vanish so that immediately tells us that you don't have any critical points in that system you have a degenerate critical point in this system and you have here two critical points which are separated out therefore if I now plot the equilibrium values let's plot x equilibrium here versus a parameter and the critical the parameter in which you have this variation is in fact a to plot this I need one more direction I need P as well coming out of the plane of the board but since P is always 0 at the critical point I ignore this P it's always 0 so let's just plot it in a two dimensional diagram here x equilibrium versus a and what's the picture one has for a positive nothing for a negative you end up with a saddle point at x equal to plus square root of minus a over b and a center which is stable at minus the same value and this is changing or increasing in magnitude like the square root of minus a therefore it's a parabola which goes up in this fashion and falls down in this fashion however we know that this root here which corresponds to plus square root of minus b over a is unstable and this is stable so this branch which is minus square root of minus a over b is stable this branch here which corresponds to plus square root of minus a over b is unstable we could have had this parabola looking the other way had I put a plus ax here then the role of minus a and a plus a would just get interchanged but what's happening here clearly is that if you imagine changing a in parameter space from positive values to negative values no critical points at all in this region and all of a sudden at this point a pair of critical points gets created and what's happening is that if you start with this potential and start flattening it out then at the value a equal to 0 you have a cubic you have a second order 0 here you have an inflection point here execute and then below that the inflection point unfolds into a maximum and a minimum and you have this shape here and of course you go on changing a making it more and more negative these points will move out further exactly like a square root of minus a and this is what happens here this bifurcation where out of nowhere a stable and an unstable critical point emerge and move off is called a saddle node bifurcation in this case at the value a equal to 0 you see immediately that this is different from the bifurcation we looked at the exchange of stability bifurcation altogether different so a saddle node bifurcation is one where as you go across a critical value of the bifurcation parameter a pair of critical points is created typically one of which is stable and the other is unstable so much for this simple form we could make this a little more complicated let's do that in the next step is to take this potential and ask what happens if it's a x plus b x square this would correspond of course to v of x equal to incidentally the signs of these quantities a and b have taken them to be arbitrary here it doesn't matter which way these bifurcation diagrams look the physics is essentially the same thing in all cases now what happens to the potential it's minus a x squared over 2 minus b x cube over 3 if I integrate this once again we can predict what's going to happen we start again with a less than 0 a equal to 0 and finally a greater than 0 and plot in all cases the potential v of x as a function of x out here for a negative minus a x squared over 2 is an upward looking parabola because it dominates for sufficiently small x therefore this curve is going to look like this but then once x becomes sufficiently large this negative term is going to dominate and bring this potential down on the negative side this always remains negative and for x negative this number is also is positive so what happens there he seemed to have made a mistake here so let's keep let's keep be positive here this term is going to dominate so it's positive yeah there's no problem this term becomes positive yeah so that's fine this goes up here because eventually for large negative x this term dominates over this minus x cubed is negative and minus x cubed is positive so it goes up in this fashion this is fine this is fine when a is 0 exactly 0 then it's just a cubic curve exactly as I drew earlier minus x cubed which has an inflection point here goes down in this fashion and for a positive you have a downward parabola here and of course for large negative positive x this is going to become a large negative quantity go down there but then it has to eventually turn back and go off in this fashion now it's easy to see what's going to happen at a equal to 0 you have a higher order critical point here for a negative you have a maximum of the potential which corresponds to a saddle point you have a minimum which corresponds to a center at the origin on the other hand for a positive the origin corresponds to a saddle point and the minimum which occurs here corresponds to a center so what has happened what kind of bifurcation is this and where is this point this point mind you is not at x equal to 0 but it's equal to at minus a over b this point also that minus a over b it's as if the diagram has moved but what's happened is that the critical point at p equal to 0 and x equal to 0 which was initially a center has now become a saddle and the critical point at p equal to 0 and x equal to minus a over b which was a saddle point has collided with the center at the origin and has now become a center they have therefore exchanged roles and what's this bifurcation this is an exchange of stability bifurcation so it's evident immediately in the bifurcation diagram if I plot the parameter is a if I plot this versus x equilibrium as long as a is negative this center at the origin is stable so we have this picture this is stable and once a becomes positive that point becomes unstable and therefore you have a dotted line here on the a axis but the critical point at minus a over b this thing here which is just a straight line with slope minus 1 over b and we have taken b to be positive something of this kind is unstable there and is stable at this point that's the location of this critical point and for a negative it was clearly unstable and for a positive it's a center and is therefore stable and we have an exchange of stability bifurcation at a equal to 0 this bifurcation has another name it's also called exchange of stability or transcritical the saddle note bifurcation incidentally is also called a tangent bifurcation this is a matter of terminology but there are these alternative names we've looked at two distinct bifurcations and one could go on and ask are there any other bifurcations of co-dimension one because the critical parameter here is a the one that you're tuning well the next step would be the following one could play this game could continue the next step would be to say what if the shape of the force of the potential corresponding potential what if this was a x squared and this was b x cubed what then this would of course imply that the potential v of x let's make this a little stable so it looks physical let me make this minus make this also a minus sign you'll see in a minute why because I want to draw convenient pictures this case v of x then becomes equal to a a x squared over 2 plus v x 4 over 4 so the three cases that we've looked at were constant and quadratic function x squared then we had the next situation was an x and an x squared the next case I'm looking at is an x and an x cubed here the potential corresponds to minus the integral of this function the primitive of this function which is a x squared over 2 plus b x 4 over 4 I differentiate it and take a minus sign I get precisely this what kind of picture do I have now and what would you expect once again the simplest way to do this is to plot the potential as a function of x you plot v of x in all three cases so v of x versus x and let's do the same thing v of x versus x for respectively a less than 0 a equal to 0 and a greater than 0 I'm actually going the other way but it doesn't matter it's easiest to plot this potential because it's always positive we've taken b to be positive always in which case for sufficiently large x this is going to dominate and that's just an x 4 curve moving up steeply and near the origin this is a parabola turned upwards concave upwards for a positive and then after that it increases more sharply so it's parabolic at this point and increases very sharp this fashion so one immediately knows what the critical points are there at the origin there's only one critical point and that's at p equal to 0 as well as x equal to 0 and it must be a center because this is a minimum of a potential at a equal to 0 this term is absent and you have a very flat potential at that point not only is the derivative 0 the second derivative is also 0 as is the third derivative it's a minimum but it's not a simple minimum because you can see it goes like this point b x cubed equal to 0 at this point and what happens here when a is negative for small x this term dominates and since a it's negative is negative it's an inverted parabola so it's a curve like this but eventually the x 4 term will dominate and take over you have a symmetric graph of this kind unlike these cases where you just had a minimum at the origin now you have a maximum at the origin but you have two minima at these points and what are these points these points are given by the vanishing of this quantity other than x equal to 0 and those roots are at x squared equal to minus a it a over b therefore x equal to plus square root of minus a over b and this is at minus square root of minus a over b those are the locations of these points and it's evident that these are centers and this point is a saddle point in between we can therefore draw a bifurcation diagram now without further ado and this as a function of a I plot x equilibrium and what does it look like for a positive there's just a single critical point at p equal to 0 x equal to 0 and here since p is always 0 we haven't drawn it so x equilibrium is 0 and it's this and it's stable so this point this critical point is stable when a becomes negative that critical point becomes unstable so we have to replace this with a dotted line on the other side and this is unstable however two new critical points emerge and move away from the origin and they are at plus or minus square root of minus a over b so as a function of a they go like square root of minus a which would correspond to a parabolic shape there and a parabolic shape here this is stable this is unstable this is also stable sorry this is also stable with an unstable critical point in the middle and that's obvious because you can't have two minima of the potential without a maximum in between so automatically you realize that stable and unstable critical points would tend to alternate now what does this figure remind you of it's a different kind of bifurcation altogether from either the saddle note or the exchange of stability bifurcations here we have a stable critical point coming along and bifurcating continuing as an unstable one and a pair of stable ones is born what is this figure remind you of it's a combination of both in some sense but this figure because it resembles a pitch fork is called a pitch fork bifurcation a saddle note bifurcation or a tangent bifurcation had no critical point at all and then the creation of a stable unstable pair an exchange of stability bifurcation was when a stable and unstable one collided and exchange stability's and a pitch fork bifurcation is when a stable bifurcation bifurcates into a pair of stable ones and an unstable one in the middle well one could go on and ask suppose I go on increasing the powers here what would happen while it's true that you could in principle get what looks like new kinds of bifurcations the fact is these are the only three generic elementary bifurcations of co-dimension one in such systems in continuous time systems so the three elementary bifurcations of co-dimension they are a the saddle note or tangent the transcritical or exchange of stability bifurcation and see the pitch fork we've illustrated these bifurcations in the framework of a simple mechanical system drawing pictures using these potentials but as we saw right in the beginning when I gave the example of the transcritical bifurcation they could occur in dissipative systems as well so there's nothing which says these are exclusively restricted to Hamiltonian systems or potential problems or anything like that it's the phenomenon that's important and this is what basically happens these are elementary because you could have more complicated coalescences of bifurcations so in exactly the same way as saying for instance the elementary functions which I would have of a variable x expandable in power series would be one x x squared x cubed and so on and from these I can construct polynomials I can construct more complicated combinations in exactly the same way in some sense these are the basic things that happen no the pitch work is not a combination of the others no it's not so very different shape all together you can see from the potential example that I gave is very different thing all together if you go back to the potential example you had a cubic potential in that cubic curve could have a minimum and a maximum it need not a general cubic curve need not have a cubic and a specific cubic curve need not have a minimum and a maximum but it could depending on what the parameters are and that's exactly what led in the to the saddle node bifurcation similarly if you took a cubic curve then the position of the minimum and maximum could get exchanged this is what led to the exchange of stability bifurcation a fourth order curve of this kind could have three extrema but it could also have one just one right here and as it transits from the situation where it has one extremum to where it has three which is the largest number it can have you end up with a pitch fork bifurcation so clearly what has happened is depending on what kind of polynomial you have what kind of unfolding of these singularities you have you have different kinds of bifurcations and of course one could ask what would happen if I didn't have an x cube here and an x here but I had an x cube here and x to the five here for example etc those would not be the most elementary ones they'd be more complicated versions of what we already have so these are the only three elementary bifurcations for continuous time systems as I said earlier the number of bifurcations possible the classes of bifurcations possible in general dynamical systems is not known in general especially if the core dimension increases beyond two or three or four then it's hard to classify these bifurcations although a lot of work has been done in this along these lines but for the elementary cases we are looking at these are the basic ones the bifurcations we'll come across a few more we're going to study in this course a few more bifurcations which are also elementary bifurcations but which are not of these types and they occur in slightly different dynamical systems as we will see there's one more which is very very important goes along with these and that's called a hope bifurcation and to lead to that I need to introduce yet another concept namely that of a limit cycle so let me do that now yes the question is what's the correct way of unfolding a given degenerate form and there is an elaborate mathematical machinery to do this all these bifurcations are cast in what's called the normal form namely the minimal form or the minimal expression which leads to that phenomenon the bifurcation these are already the ones I've written down here are already the normal forms apart from some constant multiples or scaled months which could be scaled out some constant factors these are already the minimal forms you could make them a little more complex there is a branch of mathematics called catastrophe theory which deals with this problem of unfolding these singularities and writing things in their normal form the minimal form and this is an elaborate there's an elaborate theory to this effect and I've just given a flavor of it a little glimpse of it in this in writing these forms now so there's a systematic way of doing this of unfolding these singularities and let me go on now to the idea of a limit cycle and let me introduce this as follows in our study of two dimensional systems we looked at various kinds of critical points these are points for the right hand sides of the two dynamical equations vanished one could ask are there sets of points continuous sets of points where you have some kind of equilibrium so you don't have point attractors or point repellers point point singularities in the vector field on the right hand side but can you have lines or can you have whole sets of continuous sets of points where you could have such behavior equilibrium kind of kind of equilibrium behavior but the answer is yes because let me do this again with the help of examples and then these are called limit cycles we're going to say a lot more about limit cycles as we go along because this is a typical feature of nonlinear systems and unlike the case of critical points these are much harder to detect and study so it makes them interesting and let's look once again at an example