 We saw some properties of the Bernoulli map in the Bernoulli shift last time let me introduce you to another map which is similar to the Bernoulli map which has similar properties and is closely related to it and is perhaps a little easier to understand because it is not a discontinuous map and this is the so-called tent map which looks like this it's given by the function xn plus 1 is twice xn doubling map as before provided 0 less than equal to s xn less than equal to half and it's 2 minus 2 xn for a half less than equal to xn less than equal to 1 so again a map of the unit interval specified by this piecewise linear function we can draw this map very easily so here we have xn and then xn plus 1 and it's 0 to 1 on both sides and this goes up there and comes down here at a point half the map changes slope from plus 2 to minus 2 again it's an on to map of the interval no point is left uncovered it's not 1 to 1 because it's clear the map is not invertible for every value of xn there's a unique xn plus 1 but the converse is not true for every xn plus 1 there are two values of x sub n and therefore if you give me a final point x sub n there are 2 to the n possibilities for x 0 from which this final point could have arisen so it's not invertible and it leads to chaos for a very obvious reason because the phase space is bounded this is the bisector there's a fixed point here at the point two thirds so the coordinate of this is two thirds as you can easily check and it's unstable because the slope has magnitude 2 at this point the fixed point at the origin is also unstable again the slope is 2 what do the iterates of this map look like well the first iterate of this map if I plot x versus f 2 of x this would be something like slope 4 goes up and comes down and now you have once again you have the fixed point of the map and then you have a period to cycle between these two points and then an unstable fixed point here as before and as you take further and further iterates of this map you have many more spikes up and down and the slope at every one of those points is greater than 1 in magnitude and therefore not only the fixed points of this map but also all the periodic orbits of this map are unstable periodic orbits it's easy to check once again that the points which lie on periodic orbits are a set of measure 0 they're dense everywhere on the unit interval and they are a set of measure 0 once again and the map is fully chaotic it's completely chaotic in the sense that any typical initial condition from the set of irrational values of x would lead to an orbit that doesn't settle down to any final point and it wanders uniformly over the entire interval we still have to find out how ergodic is this system on the unit interval it's an ergodic system that's clear because typical initial points sooner or later go to the neighborhood of every point on the interval so by our earlier definition of ergodicity this is an ergodic system it's in fact a system which is exponentially unstable in the sense that it has a positive Lyapunov exponent in this case and the Lyapunov exponent is again locked to we'll verify that we'll prove that rigorously in a little while but it's exactly the same in magnitude in value as the Bernoulli shift in fact in physical terms in the Bernoulli shift what we did was to take the unit interval 0 to 1 we stretch this interval by a factor of 2 so it really got doubled when you do it 2 xn so it went from 0 to 2 and what did the Bernoulli shift amount to it amounted to taking this part of it cutting this if you like and putting it right back on top of this so you ended up with something that looked like this just pictorially between 0 and 1 so this thing was snipped off here cut and put back onto this in this fashion and this is what led to the non-invertibility of the map because you really have two pre-images for every point and then four pre-images if you go to it rates backwards and so on that this non-invertibility is what led to chaos eventually led to the complicated properties of this map so you mustn't get deceived by the piecewise linearity of the map it's only piecewise linear but it's a non-linear function all the same because we saw that a linear map was very trivial it had a single fixed point in general and that was it on the other hand a non-linear map has these very strange properties the fixed point is unstable in these current these conditions all the periodic orbits are unstable and the entire unit interval becomes the attractor in this case a completely chaotic attractor because A the phase space is bounded B there is exponential sensitivity to initial conditions and C there is a dense set of unstable periodic orbit buried in this phase space so all the conditions we laid down for the existence of chaos are met and these are fully chaotic maps both the Bernoulli shift as well as the tent map completely chaotic maps we could try to trace the origin of this chaotic behavior in the following way we could say well instead of looking at this particular map the tent map which becomes an onto map suppose we did the following yeah it's simply the same thing because what he's pointing out is that in the Bernoulli map you started with this and then you so to speak stretched it like a rubber band you stretched it all the way from 0 to 2 1 and then you cut and put it on top in this fashion this is what you did so this portion was cut and placed on top right on top of the unit interval between 0 and 1 in the tent map on the other hand you started with 0 to 1 and then you stretched it by a factor of 2 all the way and then instead of cutting and pasting on top you bent it for backwards so in that sense what you did was to go here and then bend it backwards so this was flipped over and bent backwards and again you produce this 2 to 1 effect and once again all the periodic orbits were unstable and so on so you can do it in many ways in phase space you do the stretching and then you cutting pasting cutting and putting it back or you stretch it and bend it back in this fashion so these are typical mechanisms by which chaotic attractors are produced we will see some more examples of this in higher dimensions. So coming back to the tent map suppose we look at it as a member of a whole family of maps of the following kind so here is the map function here is 0 to 1 there is the bisector and suppose the map function is of this kind it is a line with some slope r up to a half and then it folds back to 1 on this side here let's write this map down so we have xn plus 1 or f of x in this case the map function is equal to depends on what you would like to call r so let's call this 2r x its slope 2r at the origin for x 0 less than equal to x less than equal to half and what's the rest of this equal to you would like it to vanish when x equal to 1 right so it's 2r times 1 minus x for a half less than equal to x less than equal to 1 and it's continuous because when x is equal to a half it's just r the maximum value is just r and that's true here too so this point here corresponds to r the slope at the origin which is the only fixed point in this case is 2r so when is this fixed point stable when 2r is less than 1 so the fixed point at x equal to 0 is stable for 2r less than 1 or r less than a half what happens at r equal to a half at r equal to a half exactly equal to a half it's evident that this map goes right up to half and comes back in this fashion so it falls on the bisector and then comes back here so this corresponds to r less than a half this map corresponds to r equal to a half and where are the fixed points of the map at r equal to a half everywhere everywhere this map is degenerate everywhere all these points remain exactly where they are what if I started with a trial value greater than a half for this map at r equal to a half what would happen well let's start here by this staircase construction and in the next step I go here and that's the end I stay there so it's clear that this map is degenerate the entire set of points from 0 to half is a fixed point so to speak what happens as soon as r exceeds a half well here's a typical value r exceeds greater than a half and it looks like this so in this map half is less than r is less than 1 remember the peak is at r what sort of fixed points do you have now you have a fixed point here but you have another fixed point there and the slope in each of these cases is 2r in magnitude and that's greater than 1 therefore these are unstable fixed points definitely and it's easy to see that the iterates of this map would all lead to unstable fixed points and therefore all periodic orbits are also unstable immediately what happens at r equal to 1 it becomes the original tent map it's called the symmetric tent map at fully developed chaos because we'll see why it's fully developed chaos in a second so this is the map for r equal to 1 this is the original tent map and the entire unit interval is now covered it's an on to map and you have a chaotic attractor which runs all the way from 0 to 1 but now let's try to draw a bifurcation diagram for this for all the equilibrium points in x as a function of the parameter r so if you did that let's call these fixed points let me just call them equilibrium points just to have our idea straight as a function of r here is 0 and here is r equal to 1 what would this figure look like till r is less than as long as r is less than a half there's only one fixed point and that's at 0 and it's a stable fixed point so by our normal ways of drawing the bifurcation diagram this is stable here nothing else happens what happens as soon as r exceeds a half well at r equal to half it's clear this entire set of points if you like is fixed points degenerate map and notice if r takes on a value between 0 a half and 1 there's no way you're going to reach any values greater than r because the function never takes you beyond the point r itself so the unit interval is not covered wherever you start eventually you're going to fall into a window between 0 and r and in fact what you do is fall into a window and at r equal to 1 the entire unit interval is an attractor it's a chaotic attractor but in between in between till a half this is one in between after this there are no stable fixed points there are no stable periodic points either the system starts becoming chaotic and the region into which the iterates fall gradually expands till eventually it sort of falls into the unit interval here but what happens here is very very interesting a little band emerges numerically one can explore this and the system falls into this region here there's a little window in x which is never covered asymptotically you never reach that eventually so the attractor is in two bands there's a band here and there's a band here and beyond a certain value of r which you can discover numerically these two bands merge once again and the entire unit interval here there's a single band which continues goes on all the way till this point and in between you don't have any periodic cycles of any kind which are stable at all we will explore this numerically I'll bring a figure which or demonstrate this on a computer which will show you how this attractor gets built up so chaos actually starts beyond a half but it's not fully developed chaos because you don't have complete the entire interval is not care part of the chaotic attractor this portion and this portion those values of x alone correspond to the attractor and as you go along that gets bigger and bigger till eventually there's one continuous set all the way till one these curves are not as smooth as I have indicated here it turns out this curve is a fractal curve in itself one of these is a fractal curve and there's a fair amount of intricate numerical complexity that goes on here even though the map looks extremely simple so even in this extremely simple map this bifurcation diagram is fairly intricate we're going to see many more complicated examples but this itself already tells you that very simple one dimensional dynamics all you're doing is a piecewise linear map of this kind in graphical terms all you're doing is to iterate this function over and over again for r greater than a half and you end up with this very intricate kind of dynamics automatically what do you think is the Lyapunov exponent for this map we saw it's log 2 when borrows when r was equal to 1 but what's the Lyapunov exponent for this map for an arbitrary value of r exactly it's just 2r because the slope is uniform everywhere and in these one dimensional maps remember that this quantity f' of x modulus is the local stretch factor it's the equilibrium of a transformation as you can see if you change from x to f of x this is in fact the equilibrium of the transformation you take its modulus and you take the log of this this gives you the local stretch factor the local Lyapunov exponent if you like and of course once you have a constant piecewise linear map with a constant value of mod f' of x everywhere that is the Lyapunov exponent for the map we're going to shortly come across a map where this is not a uniform it's not piecewise linear there's curvature in the problem and then this won't any longer be true but right now we see that this is in fact the Lyapunov exponent everywhere so let me write that down for this pardon me the log yes Lyapunov exponent is log so my statement was mod f' of x gives you the local stretch factor and its logarithm gives you the Lyapunov exponent so this map has lambda the Lyapunov exponent lambda equal to log 2r what happens if r is less than half if r is less than half you have this figure yes the Lyapunov exponent is in fact negative what does that suggest to you that is going towards the entire phase space is shrinking towards a fixed point so even that works out as you can check and when does chaos happen when does chaos start off at what's what's the onset of chaos in this problem at exactly a half slightly infinitesimally to the right of a half you have a positively Lyapunov exponent because log 2r when r becomes greater than half becomes a log of a number greater than 1 and becomes positive so chaos is characterized by a positively Lyapunov exponent that's what I meant by exponential sensitivity to initial conditions if the largest Lyapunov exponent in a system happens to be 0 you have no chaos in this problem it has to be at least one positive Lyapunov exponent now we talked about one dimensional phase space x is a single scalar variable so there is only one Lyapunov exponent but in a d dimensional phase space or an n dimensional phase space there are n directions and therefore in principle you could have n Lyapunov exponents but you need to have at least one of them positive in order to have chaotic behavior you could have more than one positive and this can happen even in cases where the phase space is bounded even in cases where the volumes are preserved because there could be some directions in which you have stretch and some directions in which you have contraction and as long as you have a direction in which you have a stretch you have chaotic behavior under these conditions are specified you could even have a system in which there is an attractor and you could have chaos in the sense that you could have a three dimensional system in which the attractor falls into maybe a two dimensional manifold or even some fractal manifold dimensionality less than three but they could always be a stretching direction so in sort of heuristic terms think of it in this fashion if I start with points which are kind of close together in a circle like this and suppose this space space area shrinks to a line but suppose it shrinks in this fashion and becomes a line finally so in this direction things have shrunk they have gone to a line but in this direction they have expanded to system size in this fashion therefore initially neighboring points could have expanded arbitrarily far to system size itself that itself implies loss of information and possible chaotic behavior so this is typically what happens even though the phase space is bounded even though the whole thing is compact even though the system could be dissipative so that phase space volumes actually shrink with time they could still be chaos in the problem because you still lose information initially arbitrarily close points could diverge exponentially fast with a positive Lyapunov exponent at least one positive Lyapunov exponent that's sufficient to produce chaos in this figure in this particular map we see that the system proceeds very rapidly to chaos it goes straight away from a stable fixed point to some kind of degenerate map followed by chaos at once no periodic orbits this is not very generic this happened because of the particular shape of this map that we happen to take we can take other maps where this won't happen and you might expect a slightly more gradual approach to chaos and there are several routes to chaos and dissipative systems is this system dissipative or conservative how do you classify this what would you say I would classify it as a dissipative system we'll make we'll come back to this and I will point out why this is really a dissipative system in that sense in a certain specific sense we look at conservative systems conservative maps which still have chaotic behavior as far as the system itself it has to be dissipated absolutely this kind of thing has to be dissipated but we will look at a map an artificial map no doubt we'll look at an example of slightly higher higher dimensional Bernoulli shift in which you don't have dissipation in the sense that the map is invertible the area is preserved and yet you have chaotic behavior so we will get back to this okay. So now the next thing I want to do is take another prototypical map where things become a little more complicated and this has to do with the logistic map this was one of the first maps perhaps the first map where many of these features were elucidated to start with it's a very simple looking map but at the same time it can become very intricate indeed let me show you what happens here this map is parabolic it's just a parabola and it looks like this so the map function is given by xn plus 1 is equal to xn times 1 minus xn multiplied by a certain constant here and many names for this constant let me call it µ µ is a real number a positive number and when are the fixed points of this map where 0 is obviously fixed point and there's perhaps one more fixed point we have to draw this thing here so let's look at it one is not a fixed point 1 minus 1 over µ is a fixed point clearly one is not because at one this vanishes but this side doesn't so the map looks like this in fact we should draw the full map and then we'll see why I'm going to restrict myself to the unit interval so let's put x here and f of x which is µ times x times 1 minus x µ is positive the map vanishes this quantity vanishes at both 0 and as well as 1 so here's 1 and this map perhaps looks like this it's a parabola goes up and comes down in this fashion the largest value is at a half that's obvious because x times 1 minus x has the maximum value at a half now I start with µ a small positive number and if this is the 45 degree line here it's evident that this is a fixed point and it's stable because the slope near the origin is just µ and as long as µ is less than 1 this slope this magnet this fixed point is stable every point is going to get attracted to it by the way this map it goes off like this and then quadratically so it's quite clear that is going to intersect at some other point here and this fixed point is going to be unstable and this is the figure from µ less than 1 we're not really going to be interested in points outside this unit interval because if you start with some point here it's going to get flow into this and if you start with points out here beyond this fixed point they're actually going to disappear to infinity and similarly on the other side things are going to escape to infinity what happens when µ becomes equal to 1 at µ exactly equal to 1 this this fixed point becomes tangential in this fashion I've drawn this badly this value is actually ¼ because that's the maximum value of x times 1-x between 0 and ½ so it's not drawn to scale but this fixed point becomes marginally stable at µ equal to 1 what happens beyond µ equal to 1 it crosses so let's draw that separately and now let's start focusing on points in the unit interval so here we are 0 to 1 that's the bisector and I'm plotting f of x versus x and f of x is µ x times 1-x so originally from µ less than 1 I had something like this perhaps at µ equal to 1 I have this this is a marginal fixed point and it's immediately clear that as soon as µ exceeds 1 it does this the slope here has exceeded 1 but the slope at this fixed point is less than 1 so it becomes stable and this fixed point at 1-1 over µ that becomes stable we can easily compute what the value of that slope at that point is what's the value of the slope at this point let's compute that so f prime of x is equal to µ – 2 µ x so what's the value of the slope at this fixed point the slope at the origin is µ of course as we know very well what's the value at that point it's 2 – µ so it's evident that the fixed point at µ equal to 0 is stable for µ less than 1 at µ equal to 1 it becomes unstable and a µ fixed point gets in which is given by this as soon as µ exceeds 1 for µ less than 1 remember this fellow was on the negative side as soon as µ exceeds 1 you get a µ fixed point in this unit interval and its slope the fixed point at sorry x equal to 0 at x equal to 1 – 1 over µ f prime at this point 1 – 1 over µ is equal to 2 – µ so the mod of this is this guy so where is this thing stable till what value of µ is this stable till 3 because we want the mod of this so from µ equal to 1 to µ equal to 3 this fixed point is stable right at µ equal to 3 this map is out here till 3 quarters so you have this and the slope here becomes equal to 1 at µ equal to 3 in fact we can start writing down the Lyapunov exponents now because if you have a stable fixed point then the Lyapunov exponent is simply the log of mod f prime at that fixed point because if you wait long enough the iterates all the iterates fall into this point and therefore if you look at the definition of the Lyapunov exponent which remember was equal to lambda by definition was equal to limit n tends to infinity 1 over n summation j equal to 0 to n – 1 log mod f prime of the iterates xj and now it is clear that if there is a fixed point then all these logs are going to this xj is going to be dominated by the value at the fixed point and that is going to come out and that is going to divide this n is going to divide that and give you that as a fixed as the Lyapunov exponent itself. So the Lyapunov exponent is log µ as long as µ is less than 1 between 1 and 3 the Lyapunov exponent is log mod 2 – µ and it is again negative showing that there is a stable fixed point. So let us write this down lambda for the logistic map equal to log µ for µ less than 1 0 less than µ less than 1 it is equal to log mod 2 – µ again for 1 less than µ is less than 3. So you can see what is going to happen when µ becomes equal to 1 then this Lyapunov exponent vanishes and you would think the fixed point has become unstable and therefore the system is going to go chaotic like the earlier tent map but that does not happen this fixed point takes over and its log is less than 1 here and therefore you again have stability and now let us draw the bifurcation diagram and see what happens. So I need a long graph here in µ eventually I will run out of space but here is 0 till 1 which is not very interesting µ itself is the fixed point is at 0 and it is got a Lyapunov exponent which is negative so you have this as a function of x equilibrium and then this fixed point becomes unstable so you should draw the bifurcation diagram the dotted line here and you have the other fixed point which is at 1 – 1 over µ which turned out to have negative values at µ equal to 1 is 0 it crosses this and then takes over from here and this fellow becomes unstable earlier that was unstable what kind of bifurcation do we had at µ equal to 1 exchange of stability bifurcation right and then you have a fixed point which goes along and this goes along till you have till you hit the value 3 at 3 this guy here becomes 0 and you would expect okay maybe now we are going to have a chaotic behavior but what happens at 3 is that the map looks like this but what is the iterate of this map look like what would the first iterate look like if you trade this map this map here at µ equal to 3 I am not going to be able to draw it to accurately but it starts looking like this okay and let us draw it a little better than this starts looking like this and you have this behavior this was the fixed point at 1 – 1 over µ but then you have two other fixed points for the iterate this is F2 the fixed point at the origin continues unstable of course and this fixed point also becomes unstable µ equal to 3 but then you have this and that and that is what sort of point is the orbit is that it is a period 2 cycle it is a period 2 cycle so the system bifurcates to a stable period 2 cycle which you discover by solving the equation F of F of x equal to x and this is a fourth order equation because F of x is quadratic and therefore F of F of x is a quartic and you have a fourth order equation but you can solve it easily because x equal to 0 is a root which you do not want which corresponds to this and x equal to 1 – 1 over µ is this root and you do not want that so factor out x times x – 1 plus 1 over µ and the rest is a quadratic which will tell you what these two roots are simple exercise and these two points form a period 2 cycle which is stable and the system falls into that or into this flip flop between these two so the bifurcation diagram now has a new kind of bifurcation this becomes unstable but then it bifurcates into a period 2 cycle this is not a pitch fork bifurcation looks like a pitch fork but it is not remember in a pitch fork bifurcation a stable fixed point became unstable and created a pair of stable fixed points or critical points here a stable fixed point bifurcated by exchange of stability to another stable fixed point and now it bifurcates into an unstable fixed point and a stable period 2 orbit cycle so this guy here this and this is a stable period 2 cycle and the system asymptotically flips between this value and that and this kind of thing is called a flip bifurcation so at µ equal to 3 you have a flip or a period doubling bifurcation because a period has doubled what would the lambda be here suppose these two points which are some functions of µ let me call this alpha of µ and this point here beta of µ the functions of µ of course what would the Lyapunov exponent be in this region it would start at 0 but then it would become negative because you have now found a period 2 cycle which is stable and what happens to the Lyapunov exponent by this definition you have two values here in this summation isn't it so what would the Lyapunov exponent be in this case so it would just be log modulus f prime of alpha f prime of beta it will just be that right pardon me it's gone because eventually wait long enough this thing becomes a constant at this value it's a period 2 cycle and then the n here divides this and in the limited to just be this product of logs and that's less than 1 so it's stable the mod of this product is less than 1 because the period 2 cycle is stable once again the Lyapunov exponent drops to negative values and this happens for 3 less than equal to µ less than equal to 1 plus root 6 and that's easy to verify all you have to do is to find out when this number here hits 1 and it hits 1 at root 1 plus root 6 so that happens somewhere here this is no longer to scale this is 1 plus root 6 then matters begin to happen very fast as you change µ at this point this period 2 cycle becomes unstable and you have a period 4 cycle coming out that becomes stable so it's not this iterate but you have the iterate with period 4 cycle as well as this these points and these are unstable now that becomes stable so this is a period 4 cycle a little more change in µ and it becomes a period 8 cycle so the next bifurcation happens here and this period doubling cascade of bifurcation starts happening so you have cascade as µ increases and it happens for smaller and smaller intervals in µ so eventually what happens is that you have a 2 to the n period cycle where n becomes unbounded and that happens at a finite value of µ this is called µinfinity and it's equal to 0.566 or 3.566 this is known to 15 decimal places for this map one can compute the value of µinfinity numerically there are lots of scaling properties that happen here which I won't go into right now but you have the end of the period doubling bifurcation cycle this one at any rate etc. so you have a whole lot of points which form part of this attractor not a full interval it's not a continuous interval at all the limit points of all these period 2 to the n cycle out here is a set of points which is got a fractal dimensionality between 0 and 1 it's of the order of 0.57 or something like that so it's a dust it's a set of particles a set of points on the unit interval yeah I will come back and explain what fractals are so when I do that remind me to go back and tell you this it's not a continuous interval at that point at that point the Lyapunov exponent again hits the value 0 and then after that it has nowhere to go it is actually exhausted stable fixed point period 2 cycle period 4 cycle period 8 cycle etc all these are exhausted and the system becomes chaotic you might ask it never went through period 6 it never went through period 9 and so on it went through period 1 2 2 square 2 cube 2 to the power 4 and so on up to 2 to the power infinity so it still has surprises in store at this point the system becomes chaotic and after that you have a whole band in which the system rest exactly as in the case of the tent map beyond r equal to 1 but it's not the full interval as yet however in this case the system has further surprises in store and now let's draw the Lyapunov exponent and see what happens so I plot as a function of mu I plot the Lyapunov exponent lambda there's no chaos as long as mu is less than 1 there's a fixed point here and it's some negative value it's at log mu and as mu equal to 0 of course is at minus infinity but then it comes up and does something like this it's negative at mu equal to 1 it's hitting the value 0 but then it starts becoming negative once again because it's now given by log mod 2 minus mu and this goes on till the point 3 where again it hits 0 and incidentally in between 1 and 3 at mu equal to 2 something interesting happens let's draw the map for mu equal to 2 so I'd like to draw mu equal to 2 the map function f of x is twice x into 1 minus x and the fixed point is at a half because it's at 1 minus 1 over mu and what's the slope at that point it's 2 minus mu right so the slope is 0 the log of that is minus infinity so what has happened is that exactly at mu equal to 2 the map looks like this and the slope here is 0 remember that the stability of a fixed point was determined by the modulus of the slope at that point if that slope was less than 1 in magnitude then we said it was stable and if it was greater than 1 it was unstable equal to 1 it was marginally stable but at mu equal to 2 the slope here is 0 that's the least value the mod of the slope can take and it's evident wherever you start that in this case how would you get to this point I mean it's it's it's quite clear that this is the smallest magnitude you could possibly have and in fact the map becomes super stable at that point this fixed point becomes super stable the log of that plummets to minus infinity so this goes off to minus infinity and that's because at mu equal to 2 the fixed point is super stable so the Lyapunov exponent goes all the way to minus infinity pardon me the significance is that it's the it's the best stability you can have can't have anything beyond that because in this map if I start with any point here I hit this and then I hit this the approach to this is extremely rapid because it doesn't really wander around at all that was simply degenerate that was simply degenerate yeah that was simply degenerate I mean the entire map function lay on the bisector itself so nothing moved in that case yeah I agree that's the sort of degenerate case is very unusual but here when the slope is no I'm not saying this is more stable or less stable I'm simply saying it's a matter of terminology that when the slope becomes 0 then the map is the fixed point is said to be super stable it's stable if the slope is less than 1 in magnitude and when it's 0 which is the least value it's super stable the significance is that the corresponding Lyapunov exponent goes to minus infinity tends to minus infinity so the local stretch factor or contraction factor is the biggest it can have yes I'm not saying lambda is an indicator of stability I'm saying lambda if it's positive is an indicator of instability but when you have isolated fixed points then lambda tells you in some sense what the contraction rate is and the contraction rate here is the largest and that's because it's directly measured by the log of this modulus of this slope here there's another measure which says more precisely how contraction occurs in play space as you move forwards in time or expansion occurs you move backwards in time and that's measured by something called the Kolmogorov entropy I haven't introduced that as yet and when the system is super stable then the Kolmogorov entropy also behaves has a very specific kind of behavior that's the reason I called it super stable at the moment but for the moment let's leave this as just a matter of terminology we'll come back and see what it's significant signifies but it's easy to see that in this case the Lyapunov exponent to actually go to minus infinity then it comes back crosses this at free and then because now at this point a period two cycle takes over it falls back once again and then there could be possibilities that these the periodic cycles themselves become super stable if any one of them if the point half if this point becomes a fixed point or part of a periodic cycle then you can see immediately since the slope there is zero if the peak becomes always any of the peaks becomes of the extreme of this map or its iterates becomes part of a periodic cycle you have super stability there immediately so this could happen over and over again and then again at the point one plus root six it climbs up to zero but once again a period four cycle takes over and it keeps doing this till it hits mu infinity and mu infinity it's exhausted this and the Lyapunov exponent crosses over finally to positive values and the system becomes chaotic so this here is a start of this chaos here at mu equal to four that's the largest you can have here so let's jump straight to mu equal to four which is here so the maximum occurs exactly at one it becomes an on to map then it displays properties very similar to that of the tent map because now at this stage you have zero to one and zero to one here this fixed point is a three quarters and it's unstable all the iterates of the map also lead to unstable fixed points there are no more stable periodic cycles possible and this map becomes fully chaotic and the entire unit interval becomes part of so at mu equal to four the entire unit interval becomes part of the attractor but in between this end of this first period doubling cycle to the reaching of this chaotic attractor you have many intricate phenomena that go on here between mu infinity and four because what happens is although all the two to the n period cycles became unstable there are many other integers and we haven't exhausted them so it turns out that this map exhibits cycles of all integer periods 1 2 3 4 5 etc etc not necessarily all of them stable but eventually what happens is various complicated things happen here which will describe by and by including a phenomenon called intermittency so there are long regular bursts of the iterates followed by chaotic intervals followed by regular bursts and so on and eventually at the point 1 plus root 8 a period 3 window takes over so for a little bit of time there is a period 3 cycle and the period 3 cycle happens at 1 plus square root of 8 which is still less than 4 a set of tangent bifurcations occurs so you have a stable fixed point and unstable stable unstable stable unstable and the system flips between these and that remains stable for a little while so you have a stable period 3 window the chaos disappears and then once that disappears that periodic window disappears again chaos takes over and you have a chaotic behavior finally till at mu equal to 4 you have what's called fully developed chaos and we will have more to say about this and the question is what's the Lyapunov exponent at mu equal to 4 when the entire unit interval is completely chaotic it turns out that the Lyapunov exponent is can you guess what it would be because it now has properties very similar to the tent map in some sense it's locked to it again becomes locked to even though in this case the slope is not uniform so you really have a complicated behavior the system is ergodic on the unit interval and ends up with a limiting value of the Lyapunov exponent which is locked to once again and that stage so it hits something here a limiting value at mu equal to 4 this value is locked in between in this chaotic region the Lyapunov exponent has only to be found numerically there are very few analytic expressions what would be the value of the Lyapunov exponent in this periodic window it would become negative in general if there is a stable periodic window it would simply become negative so it's not as if this stays and goes up monotonically to lock to there are still complications here it still goes up and down and eventually hits the value locked yeah this is a numerical result here as to where the period 3 window emerges so the statement I made was after the 2 to the n cycles ended after that set of period doubling bifurcations you had the onset of chaos at this point exactly at mu infinity the Lyapunov exponent is 0 and right above for any infinitesimal value of mu beyond mu infinity for an infinitesimally larger value you have a positively Lyapunov exponent the system becomes chaotic but in between in this chaotic region it's interspersed with periodic motion and the last of these periodic windows happens at 1 plus root 8 as a value of mu when the chaotic attractor disappears and the system falls into a stable period 3 cycle so there are 3 points and it flips flops between these 3 points that continues for a while a small range of mu and eventually that periodic cycle becomes unstable all the other integer periods also become unstable and the system has no recourse but to become fully chaotic and this continues till mu equal to 4 okay now the question is where does this come from because we only looked at the iterates of this map we didn't ask what are the other possible periodic points it turns out that in this map for good reasons the root to chaos is via period doubling so you start with period 1 that's double to 2 that's double to 4 which is double to 8 and so on but this is only one set of possible periodic orbits you could still have period 3 5 7 or any other number which is not of the form 2 to the n all those periodic orbits appear in this region most of them are unstable when they appear but occasionally you could have stability once again simply because of the dynamics which is not trivial at all it's extremely complicated and finally the last window that appears where you have a stable periodic orbit is a period period 3 window right here and once that 2 becomes unstable you have full chaos complete chaos the attractor at this point is not an interval it's called the Feigenbaum attractor at mu infinity lambda is just about to cross over from 0 and then the attractor is called a Feigenbaum it's not chaotic because chaos hasn't set in yet the lambda is still equal to 0 it's not taken off to positive values this is the limit set of this set of bifurcation points and that's a fractal object it's a set of points with a certain dimensionality called a fractal dimensionality which is between 0 and 1 and it's non chaotic because it's not an interval but a set of points which has a certain structure it's called a strange attractor but it's a strange non chaotic attractor I earlier introduced the idea of strange attractors in three dimensional or high dimensional flows which are chaotic with at least a positive single positive one positive Lyapunov exponent but here's the case where the Lyapunov exponent is dead 0 and yet you have an attractor which is not a periodic attractor of anything like that but it's a strange non chaotic attractor in this stage but immediately after that you end up with a chaotic attractor we'll come back to some of these points in particular we want to study this we want to study fully develop chaos and see what happens at this point but first one quick question what happens if I have a map in which mu becomes greater than 4 what would that look like well it's evident that this would go like this then if you took this map seriously what would happen it's quite clear that what could happen is the following remember this graph goes on on both sides so if I start with a point here it goes here I go there and I'm out of the unit interval and it leaves the unit interval till mu equal to 4 points which started inside the unit interval remained inside there but now things have started escaping out of it all points don't escape all the pre-images of this interval would escape the pre-images of this interval would escape and so on so you have a very complicated set of points which would escape and another set of complimentary set of points which would remain inside and these would form what are called canter sets so we will talk about this in the context of the tent map so what you have here is something called a chaotic repeller because things are moving out of this interval they're getting out here and this too has its uses but when interested when you're studying chaos it's up to 4 that you would like to look at yeah yeah yeah then it's not going to be densely bounded between the 0 to 1 interval but what's going to be the question is if points leave the interval yeah then those points will not go are not going to be densely moving exactly so now what will happen once things he's got a point he says if you have a situation like this and some sets of points leave the interval then what remains as an attractor is no longer the unit interval and that's absolutely true so what remains here is what would I would call a canter set we will talk about this it's still chaotic because they still could be exponential sensitivity to initial conditions between what remains that could still happen doesn't matter doesn't matter no but if I start with the point in whatever is left as an attractor you still have you still have a Lyapunov exponent you still have unstable periodic orbits you have all those points a certain set of points has left so your attractor and so being the unit interval has split up into many disjoint pieces yeah yeah in some trivial sense I mean once I get out up there everything has gone off to infinity it's no longer stability everything has just moved off to infinity could be chaotic while it's doing that it could certainly be chaotic whatever is left inside okay whatever is left inside and yeah yeah things have now become unbounded right this motion is now going off to infinity so I know I understand that fully completely things are just escaping to infinity that's no longer of direct interest what is of interest is the following which is to say that it doesn't really go off to infinity is to take this map and make this put this not on a square but on a square lattice repeat this map over and over again so I say when something goes out it will go goes into the next square and that's how I study scattering and it is in fact a model for studying chaotic scattering for real physical systems again as a toy model we'll get back to this the next