# Lecture – 27 Dynamics on a Torus

Let us briefly recall where we were in the last class We said that when there is a periodic orbit like so in continuous time and suppose you have placed a Poincare plane here, you would normally see a point but supposing this orbit becomes unstable which means the fixed point here becomes unstable and not just any type of instability It is that instability in the map where the Eigen values are complex conjugate and their magnitude is just equal to one When that happens on this, if you blow it up you will find that in the neighborhood of that you will get an outward spiraling orbit while outside there will still be an incoming spiraling orbit as a result of which there would be a closed loop develops A closed loop on the Poincare plane would mean that there is a torus in the whole continuous time That is the picture that we have seen We forget about what is happening elsewhere, we keep it in the back of our mind but let us concentrate on the Poincare section Since on the Poincare section we will see these are rotating orbit, since the fixed point that is sitting inside has become unstable while the Eigen values are complex conjugate, there should be a rotational motion and the rotational motion will give rise to a turning this way We said that suppose we consider the rotation on a topological circle which is no different topologically from any closed loop and then we defined the angle and said that let us define thetan +1 is equal to a function of thetan Let that be the equation We define a 1 D map that represents the angular rotation or angle in any particular iterates measured from any data That is where we were and we had used for that purpose a specific map given by thetan +1 is equal to thetan plus omega minus k by twice pi sin twice pi thetan and then you have to take mod 1 This is the sine circle map that we have taken We had taken a sine circle map and then we had gone ahead with this argument but if this is the representation of the dynamics, dynamics as it turns then we can identify very easily The situation when there would be a 0 to 1 mode locking, we can easily identify the 1 to 1 mode locking What would represent the 0 to 1 mode locking? The condition where it does not rotate at all in the small circle It goes around the big circle and comes back to the same point which means nothing but

the period one fixed point To find out the condition we had set this equal to theta star, this equal to theta star, this equal to theta star Solved it and we obtained the condition Likewise what would be the condition for 1 to 1? 1 to 1 mode locking would mean physically that while it goes around the big circle once, it also goes on the small circle exactly once Going around the small circle exactly once means addition of one in this case because it is normalized to one In order to obtain that condition we had simply put theta plus one We have solved it and we got the condition But that may be convinced ourselves that there would be two parameter ranges There are two parameters, the omega is parameter and k is another parameter There would be two parameter ranges in which you would have a 0 to 1 mode locking and one to one mode locking We had more or less come up to this stage in the last class Do you exactly recall what the ranges were? The 0 to 1 mode locking condition was 0 less than or equal to omega less than or equal to k by 2 pi The 1 to 1 mode locking condition was Notice that as k goes to 0 because k is a parameter and we have already understood that k is a nonlinearity parameter If k is set to zero it is a linear map and then omega becomes the frequency ratio parameter and k is a nonlinearity It brings in the nonlinearity If k is 0 then you would notice that there is no range But as k is increasing you get some range and if k is equal to 0, you get a lot of range Now k is equal to 1 you get some range for this one and some range for this one and it is customary to visualize this as a graph in the omega verses k plane in which how would this condition look This is 0, this is 1 and here it is 0 to say here it is 1 For 1 I will draw a dash line This would be nothing but a straight line condition that goes up For k is equal to 0 the range of omega for which this happens is 0 For k is equal to 1 it is 1 by 2 pi, so you would get a range like this No, we are not limiting We are only now studying what happens between 0 to 1 It is not really limited to one We will see what happens if it goes beyond one Then here also this is a range that is limited to one but its starts from 1 minus k by 2 pi and so here is also a linear range that goes like this This is the range in which you will see 0 to 1 locking and this is the range where you see 1 to 1 locking Now the question is that in between these two, will there be other mode locked conditions? Now other mode locks conditions means what? It means as you can see that in case of this kind of a map, the mode locked condition is

represented by periodic orbits If it is periodic then it comes back to the same state after some time which is nothing but the mode locked condition By the same procedure that we have already learnt, we can find out the range of parameters for which it becomes period two for example If you try to do that yourself this will give some trouble because you will land up in a transcendental equation That needs to be solved numerically If you do so then you will find that here also there is a range that goes like this It is 1 to 2 Let’s try to figure out how the 1 to 2 situation looks in the graph Here is your thetan here is your thetan+1 graph of the map Here is a 45 degree line so it would be something like this and then from here it will come back here and it will start like this You can visualize an orbit something like this The drawing on the left side of the above slide is wrong I will the draw again Supposing it goes here, it comes here so it has to come back somewhere here An orbit can be something like this If it is like this that means one iterate falls in this chunk and another iterate falls in this chunk; falling in this chunk means it is actually gone up like this, not here Here it is continuation of that line and it has come down here because it is mod 1 Actually the graph is going up like this but since you are taking mod 1, it starts all over again here As a result if there is a iterate falling here, it means that it goes around in one circle This orbit would mean that it goes around the small circle once and period two means it goes around the big circle twice, so it is a 1 to 2 mode locked orbit Why it is 1 to 2? Similarly you will be able to visualize a 1 2 3 mode lock orbit something like this Drawing this is difficult, so I will just show you the graph from the book Can you see this graph? This is the period three orbit fine and in the period three orbit I mean they have shown it starting from some initial condition with some iteration that means the Kobe wave diagram Finally it converges on to the period three orbit What would be the winding number for this one? Looking at this, can you figure out? Not difficult that all, because there are two iterates in this chunk which means that it goes around the small circle twice While in total there are three points therefore it goes around the big circle thrice It is 2 by 3 mode locking window Now for these two situations, one was this situation and the other was that situation For these both you can plot thetan+2 verses thetan Can you see or is it too faint I will draw again on the paper In one case I will have to draw thetan+2 verses thetan The way we did for the logistic map in order to understand the character of the period two orbit and the other case we will need to do three plus In this case there would be this kind of behavior so as a result of which there would be few fixed points In this case there would be more number of this things and it is not difficult to see

that the more you increase this number that means which iterate I am talking about, the more would be these oscillations Is it difficult to visualize that? No, because we know in case of the logistic map we have seen that in case of period two there would be two of these, in case of period three there would be many more of these As you go on increasing there would be more number of oscillations, same thing will happen here The point that I am trying to make is that therefore the periodicity I am investigating will have to take n plus that many The more there would be those numbers of oscillations Now look at the character of this What does omega do? Imagine I am drawing the graph of this and what is the role of omega? It is just moving in vertically As a result if you move this vertically, the intersections will exist only for a certain range I am trying to give you a sort of intuitive argument that you can appreciate Mathematically everything can be worked out algebraically but very long calculations will be necessary I am trying to give you an intuitive argument There would be oscillations in this graph Now the more this periodicity, the more will be the oscillations As a result, as the omega is increased, this will be moved and naturally the higher the oscillations, the smaller will be the range of omega for which this graph will intersect the 45 degree line Let’s put it this way First let us understand the argument for k We find that this broaden as you go for higher values of k What actually happens for higher values of k? If k is zero there is no oscillation at all, they are straight lines If you introduce k, it becomes slightly like this The more it increases, it becomes larger oscillations As a result the intersection will exist for a larger range of omega as this goes up and down That is exactly why for larger values of k, we have a larger range of omega for which these are stable Is that argument convincing? Now we will get into the interesting part of it, if these arguments are understood The larger the value of k, the larger will be the range of omega for which we will get the periodic mode locking windows Second point is that higher the periodicity, the narrower will be the range in which they will occur Now you would notice that if the periodicity is p by q, if the winding number is p by q then periodicity is this q, the denominator As a result you immediately conclude that larger the denominator, the smaller will be the range of the parameter for which it will occur At this stage number theory comes in because here we are talking about rational ratios Rational ratios starting from 0 to 1 or I will write it this way, 0 to 1 basically 0 and ending it 1 to 1 In between these two, how many rational ratios are there? Infinite As a result it is not difficult to see that within these two ranges, here is a range of occurrence of 0 to 1 mode locked window Here is a range of occurrence for 1 to 1, in between there should be all infinite period mode locking windows We also conclude that higher the periodicity that means higher the denominator, the narrower

will be the range A nice way of visualizing that is if we plot omega versus the winding, it must start with 0 It must end with 1 It goes from 0 to 1 Now you are using omega as a parameter say you are setting some value and you are slicing through this You will see all these ranges in which there would be mode lock windows As a result this graph looks like this It is extremely difficult to draw There will be a range here 1 to 2 mode lock, there will be a range here, there will be a range here, so I will not draw it to continuous line It will be a graph like this, so you will get a graph like this What is the specialty of this graph? Each of this horizontal lines represent the parameter range for which a periodic orbit occurs, a mode locked window Since if I ask you how many of these mode locked windows are there? Your answer would be infinite because there are infinite number of rational ratios between 0 and 1 If you imagine that this is a staircase and then n starts from here and tries to climb up to 1 How many steps does he has to climb? Infinite No I don’t want to get up to that level I only want to get up to that level How many steps? Still infinite No I don’t want to get up to that level, I only want to get here How many steps? Still infinite You see in order to go from any place to any place, he has to climb infinite number of steps He actually cannot climb That is why this typical structure is called Devil’s staircase structure This is called Devil’s Staircase structure because you never can climb such a staircase but it is a staircase That follows from number theory How will you actually calculate this? Omega is a parameter, for every value of omega you will actually have to calculate the value of omega Every value of omega you will have to calculate w How will you calculate w? The average number of turning so you will have to take fn of theta minus theta, you have to take limit n tends to infinity This way you have to calculate w, the winding number and then you have to plot this graph It’s possible to plot, I mean the ant cannot climb it doesn’t mean that you cannot plot It is possible to plot this graph Of course you will be able to calculate these to a finite precision of the computer but nevertheless it is possible to plot these graphs Another very interesting thing follows from number theory See there is a range with zero winding number, there is a range for one winding number Suppose there are two rational ratios I am talking about Now forget about dynamics just consider the number, two rational ratios p by q and p dash