Lecture 58 : Lorenz equations and introduction to chaos

Hello and welcome to another lecture of Classical Motion of a Single Particle. In this lecture we will discuss very briefly and also very superficially a very fundamental concept of non-linear dynamics that is the concept of chaos. A chaotic motion is a motion which is first of all unpredictable ok, but this unpredictability actually is not coming due to some noisy input or some noisy initial conditions ok. Even if you give initial conditions precisely ok, then due to the intrinsic non-linearity of the phase space dynamics ok, two very closely related or closely placed initial conditions actually in finite time would lead to the values of phase space variables very very far apart that is also possible ok. And this actually creates an apparent unpredictability ok, but if you see the phase space equations of dynamics for those type of equations, you will see that these equations are 100 percent deterministic ok. And this type of system then and the corresponding chaotic motion then is called the deterministic chaos, but how to formally define chaos? So, first of all there is no hard and fast definition for chaos ok. In general the operational definition is concerned for chaos, we say a chaotic motion is long is an aperiodic motion ok. And this aperiodicity is perceived even if we study the long time behavior. So, for some systems it may happen that if you just watch the system or study the system for a very short interval of time ok, it may or may not be apparent that the system can have some periodicity. Let us say the time period is very very large, but in a chaotic system even if you wait for very very long time the periodicity is not perceived. So, this is an aperiodic motion ok. And this aperiodic motion is actually caused by strong sensitivity on initial conditions due to non-linearity. So, the first understanding is that unlike the dynamical systems which we have already discussed ok, discussed like let us say in previous week ok, where we were talking about fixed points, the point attractors and nature of phase space trajectory in the neighborhood of fixed points. Most of the cases I mean we actually talked about the linearized system. Only in this week when we introduce the concept of limit cycles, we for the first time we took the non-linearity into account. So, for chaotic behavior non-linearity is a must. If you do not include non-linearity then you cannot explain chaos. There is a very important theorem I am not going to the exact statement or the proof of the theorem, the theorems is called Poincare-Bendixson theorem ok, but this theorem has a very important conclusion. It says that in 2D autonomous phase space flow I mean there is a subtle thing in Euclidean space ok, no chaos you just take this as an information. So, most of the element of this lecture I am just giving you an exposure or an introduction or an initiation to the concept of chaos, I will not be really digging into the physics of chaos here. So, the point is that you have to the message is you have to consider a phase space of dimension of at least 3 to obtain chaos. And Lorentz equations which were for the first time formulated by Edward Lorentz ok, Lorentz equations give us an instance for studying such type of dynamical system which we can write z and z dot equal to x y minus b z. You see these are the 3 evolution equations ok. So, the phase space is composed of x y and z and their evolution equations are given by these 3 equations. Now you see that this is not corresponding to any familiar classical mechanical system related to one single particle only because there we are actually always constructing phase space of even dimensions. For 1D motion we construct a 2D phase space, for 2D motion we construct a 4 dimensional phase space and for 3 dimensional motion we construct a 6 dimensional phase space ok. But here we are actually constructing a phase space whose the total number of phase space coordinates is equal to 3. So, it is not exactly usual mechanical systems evolution equations ok, but somehow this is very important for our understanding of chaos ok. So, you see that there are important features of these equations first is there are non-linear quantities involving x and z and x and y although you cannot see any non-linear quantity coupling y and z ok. Another thing is this sigma r and b sigma r and b they are the parameters of the equations and all of them are assumed to be positive greater than 0 ok. And also you see that this type of equations very popular because this equations were actually not derived originally by Edward Lorentz it was originally derived by. So, originally derived by Salzman in the year 1962 ok as the minimalist model of thermal convection in atmosphere in atmosphere and then what will be x y and z ok. So, x is equal to the rate of convection, y is equal to the horizontal temperature variation and z is equal to the vertical temperature variation in atmosphere in convective atmosphere. And the 3 parameters sigma is the Prandtl number, r is equal to Rayleigh number these two are dimensionless quantities and b is the physical dimension of the layer. So, this is not dimensionless that you can also see from this expression ok. Now, you see that this Prandtl number and Rayleigh number they are very very important in the case of thermal convection and I invite you to search on those dimensionless quantities and you understand that how they are actually representing the physics. You will see that both Prandtl number and Rayleigh number they are related to the convection efficiency and also the diffuse the thermal diffusivity, the kinetic diffusivity those type of things ok. I am just now going to understand this set of Lorentz equation a little bit more in detail ok. So, although this is a set of non-linear equations as I just said. So, some properties I can write some properties of I can call this Lorentz equation L e. So, why it is called still Lorentz equation because Lorentz made the analysis and somehow understood the non-linear features of those set of equations ok. So, the first point is non-linearity non-linear coupling is only between x z and x y. Number 2 you can check that that is very important if x y z are solutions of L e then minus x minus y z is also a solution of same Lorentz equation you see. So, if you just make x to minus x then you will have minus x dot is equal to sigma y to minus y and x to minus x. So, you will have minus y minus of minus x that is plus x ok which finally, gives you minus x dot is equal to minus sigma y minus x. So, you will have actually this original equation to be intact ok. For y dot it will be minus y dot then is equal to minus R x minus of minus y that will be plus y and minus x z that will simply be plus x z. So, again a minus will go away and you will have a y dot is equal to R x minus y minus x z ok. So, again this is intact and finally, z dot becomes z dot and x y remains x y because both of them are getting minus and B z remains B z because z did not change its sign. So, this one also gets impact I mean intact ok. So, that is why if this is a solution this So, Lorentz equation permits a number of fixed points. Now you can just from the first site can you check which one is a very evident fixed point of the system that is the origin itself x 0 y 0 and z 0 ok. So, that you can write. So, one fixed point is obviously, 0 0 0 ok. Now to find the other fixed points what will be the other fixed points? So, if you remember this the definition of fixed point. So, at fixed point x dot y dot z dot all should be equal to 0. So, first of all you will have at fixed points x dot equal to 0 y dot equal to 0 and z dot equal to 0 that simply said that sigma y minus x is equal to 0 comma R x minus y minus x z is equal to 0 and finally, x y minus B z equal to 0 this we can say. So, from this one of course, the first one I have y is equal to x if I have y is equal to x then I have also from here x z is equal to x R minus 1 and from that I can also write z is equal to R minus 1 if x is not equal to 0 that much I can write and also from this equation. So, x y then will be x square will be equal to B z and then z will be simply I can also write like. So, z I have already found it this. So, x will be equal to ok plus minus root B z which is B times R minus 1 ok. So, the other fixed points of the system of equations of course, are given by root B R minus 1 since x is equal to y when x is equal is also equal to that. So, I can just write like this ok comma root B R minus 1 comma R minus 1 ok minus when I am taking the minus sign from here B R minus 1 minus root B R minus 1 comma R minus 1 ok. So, you see the whole equation set of equations actually permit 3 fixed points 1 is 0 0 0 and the 2 other we have just found ok. Now you see also that when we are talking about the fixed point 0 0 0 there is of course, no restriction on the parameters. Now if you think that x y z they are real quantities for example, as I just said one is the rate of convection, the other is the variation of temperature in the horizontal direction, another is the variation of temperature in the vertical direction there is no meaning if we call them if we assign a complex value to them. So, they must be real x y z ok and if they are real then of course, you see for a realistic value fixed point ok for all these cases this R minus 1 since B is positive according to our assumption all the 3 parameters are positive R minus 1 should be also positive ok and that means, that R should be greater than 1. Of course, when R is equal to 1 then you are actually falling to the origin as a fixed point 0 0 0 you can also see that ok. Now we have seen the fixed points actually you will understand that is why I can now present the whole situation. So, 0 0 0 is a fixed point for all I can write R B sigma ok, but the other 2 fixed points are meaningful I am just writing meaningful that means, the real value ok if R greater than 1. So, I can say the whole situation as for R less than 1 I just have only one fixed point and for R greater than 1 I have 3 fixed points. So, you see that if you change the value of the parameter one of the parameters at least ok and you obtain the phase space behavior of the system changes ok. For example, for one it has one fixed point the other it has 3 fixed points in such case we call that there is a bifurcation ok. Once again this word so just to initiate you it is a bifurcation and there are several type of bifurcation this type of specific bifurcation is known as pitchfork bifurcation. So, if you just changing the parameters change the nature of the phase space behavior of the system phase space dynamics of the system then you are actually causing a bifurcation ok. So, that was just the point now of course, I understand that for different values. So, whether R is less than 1 or R is greater than 1 the number of fixed points will change. Now, can we also understand a little bit about the linear stability analysis of this Lorentz equations ok the answer is yes we will do now. So, linear stability analysis. So, first we need to first linearize Lorentz equations and if you do this. So, just once again I am writing the whole equations x dot equal to sigma y minus x y dot equal to R x minus y minus x z and z dot is equal to x y minus b z if I now linearize that linearization that means, I just neglect all the quadratic terms and higher order. So, here I will simply have x dot is equal to sigma y minus x y dot is equal to R x minus y and z dot is equal to minus b z. So, there is no non-linear term any longer ok. Now, you see what is the interesting point of this. So, in the linearized version this z and it is evolution becomes decoupled from the other 2 variables. So, you can now think the 3 dimensional phase space as a 2 dimensional phase space coupled with 2 phase space coordinates x and y and another 1 dimensional phase space with the only phase space coordinate z because you see that z is totally decoupled from here and even in the evolution equations of x and y you do not have any z variable ok. So, first thing you can do that you can actually immediately calculate the evolution of z and you can solve for z which is completely very easy and you can find this is nothing, but z will be proportional to e to the power minus b t there should be some constant of proportionality as amplitude ok. Then how should the system of equation would look like can I write the matricial form of course, I should write. So, the matricial form ok the matricial form of the linearized Lorentz equation they are actually given by x dot y dot z dot equal to. So, I can write minus sigma plus sigma 0 r minus 1 0 0 0 minus b. You see from this nature you can nature of this matrix you can actually find a block 2 by 2 over here and a 1 by 1 block over here that says that z is decoupled from x y and you have of course, x y and z ok. Now I can actually then find just for x and y x dot y dot is equal to minus sigma sigma r minus 1 x and y ok. I have already solved for z. So, I am no longer interested in finding z z I know. So, what about x and y ok. So, you see that. So, another point is that since b is greater than 0 ok z is actually decreasing with time ok that is very simple to understand. Now what about this thing ok. So, if I call that A, A has 2 eigenvalues lambda 1 and lambda 2 ok. Then of course, lambda 1 plus lambda 2 will be equal to minus 1 plus sigma and lambda 1 lambda 2 will be equal to sigma 1 minus r just from this trace and determinant property. So, the trace is minus 1 plus sigma this one and the determinant is sigma minus r sigma ok. So, that is exactly what we have written over here. If you solve you will see. So, of course, you will see that there will be 2 values for lambda 1 and lambda 2, but before even solving what we are trying to understand that if again I am going to the parameter r. Now we investigate what happens for r greater than 1 and r less than 1 ok. For r greater than 1 you see that 1 minus r is negative. So, lambda 1 lambda 2 is negative and of course, lambda 1 plus lambda 2 is also negative ok and that means, that one of lambda 1 lambda 2 is positive and the other is negative. So, which type of the situation does it correspond to it is like a saddle point right the corresponding fixed point when r is less than 1 then of course, lambda 1 lambda 2 is greater than 0 ok. So, I just write. So, that I do not forget this is a saddle point ok. Now for r less than 1 ok where you actually have only the accessible fixed point is 0 0 0 ok. So, if you remember that just again in case you are just forgetting that yeah. So, r greater than 1 you have 3 fixed points and r less than 1 you have only 1 fixed point ok. So, you will see that actually what happens for r less than 1 ok here in this case you will find lambda 1 lambda 2 is greater than 0 and lambda 1 plus lambda 2 is always less than 0 that means, both of lambdas ok are negative ok and this simply tells us that the corresponding fixed point is a stable node ok. Now again I am saying that what is the conclusion from this that is for r less than 1 every trajectory approaches because it has now only 1 fixed point the origin as t goes to infinity ok. So, if there is a point attractor I can easily say there is no possibility of a limit cycle or chaos ok. Now for r greater than 1 ok the situation becomes interesting as the L E has 3 fixed points and the origin actually acts as a saddle point. So, now the question is how the 2 other fixed points would behave ok if you do how would the other 2 fixed points behave. So, if you do some straightforward algebra calculation you can show for r greater than 1 the 2 fixed points 2 additional fixed points are stable if r is satisfying this relation this this you can do as well ok. Now since r is greater than 0 now when I am saying r is less than something that should also be positive this is positive this is positive. So, this should also be positive that means that sigma should be greater than 1 plus b. Now traditionally when Lorentz he is used this equation in the year like 1972. So, Lorentz used in 1972 ok to model the 2 dimensional fluid layers mostly for atmosphere ok and their corresponding convection ok he actually chose a specific set of values for this 3 parameters. He said sigma is equal to 10 b is equal to 8 by 3. So, you can actually check that sigma is greater than 1 plus 8 by 3 ok that is 10 is greater than 1 plus 8 by 3 ok and r is equal to 28 and if the initial conditions are given as x 0 is equal to 0 y 0 is equal to 1 and z 0 is equal to 0 then Lorentz equation has chaotic solutions. Now here of course, take the total non-linear Lorentz equation and actually that is the marvel of this. So, you see that for the choice of parameter r greater than 1 ok. So, where you can see that actually you have like this. So, if you just think of this problem that when you have r greater than 1. So, here of course, you have this condition. So, you have in the linear regime 3 fixed points then if you take the non-linear set of equation the solutions are chaotic. That means, that if you slightly change the initial conditions let us say from this one to let us say x 0 is equal to 0.005 y 0 is equal to 1.002 and z 0 let us say 0.0001 you will see that after some time the solutions are really far apart and that is actually due to the non-linearity of the whole set of Lorentz equations ok. Now why exactly for the regime of this 3 fixed points we have a chaotic type of solution that is something you have to also learn and that is not I am going to probe in this lecture and actually the important thing is that you have seen that this is the minimal setup with which one can find a chaotic solution because already I said that Poincare Bendixson theorem says that for a 2D flow you do not have chaos in Euclidean or Euclidean space ok. Now what is the meaning of chaos here there is another way of quantifying or how to say like characterizing chaos in this framework that is here also you can see that a system is chaotic when already we have talked about attractors. So, for linearized system we have found that there are point attractors ok. Now also for some linearized system is you can also find some discrete points ok not only a single point, but some discrete points as attractors ok. We have talked about non-linear phase space dynamics where we have found the stable limit cycles which work as attractors, but it is a special type of attractor which is a continuous collection of points and for limit cycle you can see that this is a closed loop. So, it is actually has a dimension it is actually having a dimension 1 ok for points as a dimension 0. Now let us say can we with this framework of attractors can we also define chaotic motion. So, for chaotic motion it is actually very interesting something which we have already observed that whenever there is a chaotic motion and somehow we can also define chaotic motion in terms of that then there exist also an attractor, but the dimension of that attractor is not just 0 1 2 3 like this, but it is a fractal dimension and this is called a strange. So, the corresponding solutions corresponding I should write the corresponding phase space trajectories trajectories in large times are converging to an attractor of fractal dimension. I invite all of you to have a look at what fractal dimension is ok. So, an attractor of fractal dimension and which is called a strange attractor. For Lorentz system we call the corresponding strange attractor as Lorentz attractor ok and one can show the picture of such attractor. So, if this is x direction this is y direction and this is z direction ok. So, I can actually draw. So, I can simply it is very very complicated ok. So, it is topologically very complicated and they look like a butterfly and that is why it is called a butterfly diagram. So, it is look like a butterfly wings ok. So, butterfly wings, but this is actually a strange attractor and actually this is not void this part is also you can see things are going ok. So, exactly how it looks like for that you can go to internet and find a 3D picture. So, this is a strange attractor for our case this is Lorentz attractor ok. Now, with this I am just trying to say that this is a characteristic feature of a chaotic motion that it will always have a strange attractor which will have a fractal dimension ok. Now, once again just before wrapping up this lecture I want to say you something that is what will be now from a measurement point of view what will be the what will be the outcome or what will be the property ok or what will be the signature of chaos ok. So, that will be the fact that and this is very important to understand for a specific choice of parameters the solution at large times tends to infinity settles into an irregular oscillation which is non repetitive and. So, the resultant motion is a periodic motion and this is very sensitive to the initial conditions and this is formally known as chaos ok or deterministic chaos. So, although all our although all our analysis like fixed points the phase space trajectories the stability where based on linearized system basically when the chaos is observed of course, this is due to in a in a deterministic system of course, this is due to the non-linear terms in the Lorentz equations or any other systems non-linearity must be there ok. Finally, from a measurement point of view a chaotic motion causes very rapid divergence of two very close phase space points in time. So, for example, if in a chaotic system this is let us say the phase space and two points where they started from there and if this is chaotic after sometime it can go there it can go there or even like this it can go here it can go there ok we do not know. Now, for the Lorentz attractor specifically for the case of a Lorentz attractor two trajectories starting very close initially say at some x 0 and x 0 plus delta 0 point just for like for one phase space coordinate x ok same thing will also happen for the other phase space coordinates as well ok. So, if they are just starting from these two values where delta 0 is very very small with respect to I mean is very very small. So, you can simply say that it is of the order of 0 actually very small then this delta t ok will be actually is equal to delta 0 e to the power lambda t. So, that is the main point for which the divergence becomes a very small initial divergence become very very important. So, now, let us say if you take delta 0 as 10 to the power minus 15 ok. So, I mean after some. So, if you follow this result and this one is so small like 10 to the power minus 15 ok. So, you will see that after some finite time and that you can calculate this this delta t becomes no longer negligible and one for one typical case which is the observation in numerics ok lambda is find to be found to be 0.9 ok. So, this lambda is known as the Lyapunov exponent. Now, if you have such type of thing for delta. So, delta basically is their separation. So, you can actually write in log log delta is equal to lambda. So, e to the power lambda t yeah. So, you will have just a minute you will have log delta 0 plus lambda t yeah yeah. So, you can now write log delta if you now just draw the picture of t and log delta you will see that it is a very realistic picture actually that this slope is given by this Lyapunov exponent lambda ok, but after sometimes finally, a very large times this separation becomes almost saturated ok. And this is the case ok when the separation is of the same order as that of the attractor or the strange attractor ok. The size of the strange attractor then it actually saturates otherwise is always increases ok and it increases exponentially that is why the log value has a straight linear increment ok. So, that is all about the introduction to chaotic motion ok. You see all this unpredictability in the weather forecast is actually due to this chaotic motion ok. You see that it is not that the initial values or the initial conditions they are noisy. Even if you know 100 with the 100 percent precision very very high precision 100 percent precision is a too much very high precision the initial conditions just because of the nonlinearity you will see that the just if you if you change a little bit initial conditions then the final solutions will be very very far apart and that a quantitative measure of that is called this Lyapunov exponent ok. So, with that I am wrapping up this lecture I hope that you have somehow understood the basics of chaotic motion. So, the basic message at this level of this course is that for having a deterministic chaotic motion first of all you need at least 3 dimensional phase space and also you must need nonlinearity ok. Thank you very much.