So, good morning everyone and welcome to today's lecture. So, today we will be looking at a short introduction to the stability of dynamical systems. So, in the previous lectures so far in this course we have seen how to formulate problems in fluid mechanics including two phase flows and how to obtain a solution for the case of a steady state in most of the problems. Ultimately we are building up to a position where we want to be able to study some complex two phase flows which have time dynamics. So, how it fits in with the previous part of the course is that so far we have seen how to find one particular solution under certain assumptions to the Navier-Stokes equations. But the problem is that that solution is not always stable which is what we discussed at the end of the previous lecture that often there will be multiple solutions to the Navier-Stokes equations because they are highly non-linear. And what will happen physically is that for some set of parameter and Reynolds numbers and other parameters you will have one solution stable and that is what you will see in experiment. But on the other set of conditions when you increase the Reynolds number or in terms of an experimental parameter if you turn up the flow rate at some stage you will have a transition to a new type of behavior and spatially you will see a new kind of pattern. So, what has happened is a solution has become unstable and a new solution has become stable. So, to understand these transitions we can draw on the theory of dynamical systems and the stability of these different steady states. So, in this class I will take a very simple second order two dimensional dynamical system and we will track go through how we can understand the stability of a steady state and how that transition happens. So, before I start how many of you all are familiar with the idea of linearization and then stability of systems in ODE's you can have a show of hands. So, I can gauge where to pitch the lecture. So, in for the audience watching this lecture also this is the basics of what we would do in a course like process control as well and I will just focus on the simple ideas here. For those of you all who already know some of this what I will do is try to give it a spin that allows me to generalize what I am going to do today to the partial differential equation setting which is what we are really interested in this course. So, let us begin. So, to let us consider very simple system. So, I have two variables x1 and x2 which are dependent on time and system is governed by two equations that tell me about the time dynamics are going to be and in general this first derivative will be some function f1 of the two dependent variables and since the right hand side does not have time in it it tells us that the system is basically the derivatives are time independent or how the system evolves depends only on x1 and x2 it is current state. I can rewrite this these two ODE's in a more condensed form where the dot represents the derivative f is a vector function and this capital X is a vector of the two quantities x1 and x2 and capital F is. So, this is a standard representation of the system in a condensed notation. So, whatever try to do today is lay out the steps that one would need to follow to analyze a system for steady states and stability. So, the very first step is to identify the steady states and generally it will be plural for non-linear systems. So, in order to identify the steady state what we need to do is we are looking for x1 and x2 values of the system such that the left hand side derivatives go to 0 or there is no evolution in time. So, these states are simply given by setting x1 so x1 and x1 star x2 star rather steady states to the system which must satisfy these two equations because only then my evolution in time will be 0 or have a time independent steady state. Again in vectorial form I can write this as so this first step is what we have already covered in the course so far. So, if these were if you are looking at the Navier-Stokes equations we have already seen how to calculate steady states for some condition and generally the idea will be that this x star this steady state is something that is quite simple I mean it is the it is an obvious steady state solution of the problem for some parameter values. So, for example, it could be a situation where there is no flow or for example, Poiseuille flow in a pipe just a simple Poiseuille for solution at steady state and then we want to understand how that solution undergoes a transition. So, in the ultimate case transition to turbulence in some simpler cases that we are looking at this course maybe some convection cells will set in and different features may arise. So, this will be a simple base state that we are going to start with and now we want to see whether that x star is stable or not. So, that brings us to step 2. So, step 2 is called linearization and the idea here is that because we want to see whether the system is stable around x 1 star and x 2 star we give small perturbations to the current position and see whether the system returns back to the original steady state or whether it deviates off and this ties in very well with the perturbation theory we have just completed where we have seen how to derive equations when you have small changes epsilon order changes. So, that is what we will do here. So, the variable x 1 which is my dependent variable I am going to give it a small perturbation about x 1 star. So, I will have x 1 is equal to x 1 star plus epsilon which has the amplitude of my perturbation into x 1 tilde. So, here tilde represents the perturbation that I am giving but this x 1 tilde is bounded by 1. So, that is magnitude is an epsilon. So, this entire quantity is very small compared to x 1 star. So, this quantity is a small quantity similarly I can do that for x 2. So, x 1 tilde x 2 tilde tell me in this 2D case the direction in which I am pushing the system or the what kind of deviation I am going to give to the system and epsilon just reminds me that that has to be a small deviation because I am looking at small perturbations. So, again I can write this in vectorial form as x which was the vector of my 2 dependent variables as the steady state plus epsilon times x tilde over x tilde as the deviation variables. So, this is the expansion and now like any perturbations calculation that we have done we need to substitute this into the original equations and then obtain simplified equations for the evolution of x 1 tilde and x 2 tilde because that is what we want to identify with stability. We are giving perturbations of this form and we want to see whether these how these grow in time. So, to be explicit I have x 1 of t and x 2 of t they are both time dependent and for small deviations about the steady state this time dependence is simply going to be of this form. So, you will have the steady state part and to the steady state there is some time dependent deviation and the idea is to find out whether this time dependent deviation is growing in time or whether it is going to decay back down to 0 and give me my steady state. So, to do that we come back here and substitute it in these equations. So, if you put it into the left hand side we will get simply the derivatives of the deviation variables to order epsilon naturally because these are of course time independent. Now, coming to the left hand right hand sides here f 1 and f 2 is any general function. So, while I cannot make the substitution mechanically what I can do is use Taylor series and that is what we have been doing in the course when we do not have simple functions. So, we can expand f 1 as a Taylor series. So, let me write it down here explicitly. So, x 1 is simply x 1 star which is some constant plus epsilon x 1 tilde. So, that is f 1 and now if I expand it in a Taylor series about x 1 tilde and x 2 tilde because it is a 2 dimensional function. Then I will have the function f 1 at x 1 star the value of the function at the steady state plus the deviation into the derivative of the function with respect to x 1. Of course evaluated at x 1 star x 2 star same thing for the variable x 2 and I can write the same over here. So, that is pretty clear I just use the Taylor series up to first order epsilon. So, to be exact I should add the fact that there will be order of epsilon square terms which I have neglected to write down at the moment. So, now the first thing you can see is that the function value f 1 x 1 star x 2 star is naturally going to be 0 because that was how we got x 1 star x 2 star in the first place. In other words these two are steady states. So, naturally they will evaluate to 0 on the functions f 1 and f 2. So, these guys are just knocked out and I mean that is by construction it is not a miracle that is going to happen every time naturally. So, what we are left with if you see is just a linearized system that is why it is called linearization. So, the derivative of x 1 tilde which is now the growth of the deviation is going to be related linearly to x 1 because this is simply a constant evaluated at the steady state. So, this is a function the derivative but we have to evaluated at the steady state. So, it is a constant factor multiplying x 1 tilde. Now of course, you will ask me that there are order of epsilon square terms as well. So, if I go to epsilon square I will get high order corrections which will involve x 1 tilde square and so on. And then we know from the perturbation theory that we have already looked at that since the left hand side is order epsilon. So, naturally we will equate it to the right hand side terms of order epsilon itself because these order epsilon square terms will be much smaller. So, if we look at it as a perturbation problem we just equate the coefficients of epsilon on both sides and what will be left with well again the dots represent derivatives is simply I just put the star here to indicate I am evaluating at steady state. So, that is what I get after I equate terms to order epsilon. And now I can write this in my vectorized notation. So, then I will get the growth of x tilde is simply equal to a matrix which I will call j and use the double under bar to signify a matrix times x tilde itself which is a classic first order linear algebraic system. And this j is called the Jacobian matrix and it is simply given by the partial derivatives. So, the matrix j has got the 4 partial derivatives all evaluated at the steady state and note that it is f 1 in the row and f 2 in the second row. Of course, if you ever forget that you just have to come back and re derive this which is quite straight forward. So, ultimately what we have seen here is I mean we should not get too caught up in the formal notation. I want you to remember from the beginning that we have some dynamical system this could for all you know be a partial differential equation also on the right hand side. But the point is we found a steady state and then we linearized about the steady state by the basic idea is giving small perturbations. Now, in the case of this simple 2D system dot small perturbation procedure ultimately led to this very simple x dot equal to j x form. This same form may not arise in an immediately obvious fashion when we are doing PDEs. But the idea will be the same we will have to give small perturbations and we know how to deal with those perturbations using perturbation theory. So, ultimately we will land up in some linearized description of the system. So, that is the point about step 2. So, moving on to step 3 and step 3 is where we now want to ask the question given my linearized equation how is x tilde going to grow in time. So, imagining that we have absolutely no clue how to solve this system by any formal rules what we would do is assume a solution for the time dependence. So, if you look back there you can see the full matrix form and since it is linear I can assume that x tilde which is the function of time is going to be sorry x 1 tilde is going to be some constant u 1 which is a number like 2, 3, minus 5 some number which I have yet to determine and the time dependence can grow as e to the power sigma t or exponential growth. The reason I would do e to the power sigma t is quick note that suppose I have a system d y by d t equal to say lambda say sigma y. This is your classic first order single equation O d right and what is the solution to this problem? This is e to the power sigma t into the value at t equal to 0 that is just an exponential growth. So, you can see that the time dependence has e to the power sigma t. So, in analogy to this we are looking for a similar solution when we have more than one equation and I am postulating beforehand that this growth rate for all the variables that is your x 1 tilde and x 2 tilde have the same sigma. So, right now just look at it like a hypothesis and let us proceed and then if things work out in the end we can look back and think about why it happened. So, right now I am saying that this has e to the power sigma t and so does x 2 all right. So, what I have done now is made a statement about how the time dependence is going to be and what this tells me now is that suppose my sigma is positive then I will have this exponential term growing unboundedly which means that x 1 tilde and x 2 tilde will both grow. So, if sigma is positive I will naturally have an unstable system right because my initial perturbations here x 1 tilde x 2 tilde are going. So, I give some small initial perturbation that initial perturbation you will see here is just u 1 and u 2 correct in the form that I have shown if time t is equal to 0 the initial condition I will just have u 1 and u 2. So, u 1 and u 2 represents that small perturbation I am giving now the question is whether this factor will grow. So, if sigma is positive that initial u 1 and u 2 is going to grow without bound if sigma is positive and this x 1 and x 2 of t will very rapidly move away from the steady state because of these terms. On the other hand if sigma is negative then e to the power sigma t dies off because sigma is negative. So, then the system will be stable because this portion will rapidly decay to 0 with time and I will return back to simply x 1 star and x 2 star. So, in this form of my perturbation sigma is crucial and that is why sigma is often called the growth So, typically again in these problems at step 3 we look at the linearized equations and these equations as you will notice will be homogeneous which means that if I say x tilde 0 it will satisfy the equation and that is it has to happen because saying x tilde 0 basically is saying that I am on the steady state and I know that the steady state satisfies the original equation. So, I will have a linearized homogeneous system for the evolution of the disturbance variables and the next step is always to assume a form e to the power sigma t for the growth where sigma is the growth rate. So, now for this particular system we will see where this leads us. So, if I substitute this again into the equation that I have here. So, let us look at the expanded notation and let us see what we are going to get. So, what will the right hand side be x 1 tilde derivative is simply d by dt of this and e to the power sigma t gives me back e to the power sigma t. So, I will have u 1 but with a sigma and my right hand sides this is of course a constant I have x 1 tilde which I will substitute from here. So, I will get u 1 e to the power sigma t back right do the same thing for the second equation. And now you see that because the equations were linear I have this e to the power sigma t term common everywhere right and so naturally I can knock that off. So, what am I left with I am left with an equation for u 1 and u 2 which of course are unknown at this point I mean that is one way of looking at it sigma is also on that is what we are trying to find. So, this is what I have. So, now if we again look at this in terms of the matrix notation remembering that those 4 partial derivatives are j I will just have sigma times u capital U where capital U is I will write that at the end.  So, this process of assuming this form e to the power sigma t has led us to this equation in this particular system. And now the question is how do we use this to evaluate sigma what does this tell us about sigma what does it tell us about u. So, suppose sigma has some random value maybe 5 then I need to find a u such that 5 u is the same as Jacobian times u 1 sure solution is if I take u equal to 0. So, u equal to 0 will always solve the system that is because it is a homogeneous set of equations and saying u is equal to 0 means I have not given any disturbance. If u is non-zero this equation may not be solved there is no guarantee that for any non-zero values of u I will have this equation satisfied. But there could be some values of sigma such that a non-zero value of u will satisfy this equation. So, my objective really is to find those values of sigma and those non-zero disturbances such that this equation is satisfied. And this is the classic Eigen value problem that we have solved in general dynamical systems theory this is what an Eigen value problem looks like where you have a homogeneous system of equations which always has the 0 solution. But there will be some parameter like sigma and in this case the growth rate it is. So, there will be some values of the growth rate such that you will have non-zero disturbances propagating to the system and that is what we want to solve. Now in this particular 2D system it takes this simple matrix Eigen value form which we can now solve with the standard theory we already know. But in the PDE system we will also reach the stage we will have an Eigen value problem to solve and then we need to proceed with that solution either analytically or numerically. So, that brings us to so the end of step 3 is the Eigen value problem. So, in this case sigma is the Eigen value and the disturbance is the Eigen vector. So, that brings us to step 4. So, step 4 is to solve the Eigen value problem and determine the growth rate and determine the disturbance. And how do we solve classic matrix Eigen value problem? We realize that this is equivalent to saying. So, this itself is a matrix and for this to have non-zero solutions u this matrix has to be singular which means that the determinant should be equal to 0. So, for a second order system like the one we are looking at this will lead me to a quadratic equation and I will get 2 values of sigma. In general this equation can be some polynomial nth order if my system is nth order if you have a partial differential equation then this can be some extremely complicated function more complex than we can imagine sometimes. But it is always solvable if you cannot do it analytically we can do it numerically. But this is the equation is called the characteristic equation that leads us I do not know where lambda pop into my head but it should be the identity matrix naturally. So, this is the characteristic equation that gives me the characteristic values or Eigen values which in this case is simply the growth rate and you will have many values of sigma in general. For a 2D system you will get 2 growth rates for a partial differential equation you potentially have an infinite number of sigma or infinite number of growth rates. So, what that means is that there are various key directions in which you can disturb the system and if sigma is positive along even one of those directions or if even one value of sigma from here is greater than 0 then the system is unstable and the Eigen vector corresponding to that Eigen value sigma the positive guy will be the direction along which the system will grow and that Eigen vector will contain information of the new pattern or the new state. So, this equation can usually be solved and we can obtain the values of sigma and once we know sigma we can come back here and calculate the Eigen vectors knowing sigma we then calculate u which is the Eigen vector and that u will contain information u contains the information of my new state if sigma is greater than 0. So, if sigma is greater than 0 the previous state has become unstable and I am the system has to go somewhere. So, where it is going to go what will be the new pattern that I see will that if some of that information at least is there in you and we can get a lot of information if you do a slightly non-linear calculation. But from linear calculations we can tell if it is stable or not to very small perturbations because that was the initial hypothesis it has to be infinitely small. So, only to those types of perturbations we can tell whether the system is stable or unstable. So, that was step 4 where we solve the Eigen value problem and that will immediately tell us and answer all our questions about stability. Are there any questions at this point? Yes. So, then what you do the same thing that you do in the case of solving these linear algebraic equations because they are linear you will recognize that each of those sigmas and each of those u's correspond to linearly independent solutions. So, then the general solution will be given by the sum of them. So, because it is a second order system you will have 2 solutions and you will have 2 different directions in which the system can grow and in the general case your actual growth will be some linear combination of the 2. So, I will write that down formally. Yes, the question is that after solving this problem I have a value of sigma 1 and a value of u 1 right and I have also calculated another sigma 2 and another u 2 that is not a good idea. Let me call it sigma 1 sigma 2 u 1. So, u is Eigen vector corresponding to sigma 1 the first Eigen value. Second Eigen value has another Eigen vector phi and you will have only 2 for a second order for a 2 dimensional system and if I have a higher order high dimensional system I will get more of these. So, then I can represent x 1 of t for example as the value u 1 that comes from here e to the power sigma 1 t right and multiplying with some constant alpha some linear combination or in terms of my so all this is some linear combination of the 2 fundamental solutions I got from the Eigen value problem and you see that makes sense because it is a second order system. So, I need 2 I mean I will have 2 unknown constants because I have 2 initial conditions 1 for the initial value of x 1 1 for the initial value of x 2. So, there are 2 constants here which I can determine from my initial condition. Now, if I substitute so then this is definitely a solution because if I put it back in the equations they will each satisfy the equation on each half. So, the first part will be 0 second part will be 0 independently and there are solutions. So, a standard superposition of solutions. So, this is why if sigma 1 if either of sigma 1 or sigma 2 are positive it is unstable. So, even if sigma 1 is negative this part will go to 0, but this guy will blow up if sigma 2 is positive. So, for stability I need all the sigmas to be negative and even if one of them is positive it will grow. So, if everything else is reasonably clear at this point what we can do is look at a simple example of a 2 dimensional problem where we will get a feel for the actual mechanics and see how it plays out in a real system. So, there are a lot of subtleties that I have left out here. For example, what would happen if you get sigma equal to 0. So, that would mean that my Jacobian matrix is I mean it has a 0 eigenvalue. So, then I have some problems I cannot calculate the eigenvector and the eigenvector is can be anything. So, in such systems you actually read a you will end up at a contradiction in the mechanics because the fundamental theorem that lets me do this linearization fails if you have a steady state where you have a 0 eigenvalue. And those points are very special actually and that is usually what happens the problematic points are the ones that should be paid special attention to. So, in such situations honestly this theory completely fails. So, if the sigma is equal to 0 I cannot tell whether the what will the dynamics be whether it will be stable or unstable I have absolutely no idea. So, to find that out we need to go to the next order of epsilon. So, here we stopped at first order in epsilon we need to go to the epsilon square order to get more information. And the formal theory is called center manifold theory and we discuss more of these higher order dynamics in the course that we do in the alternate semester which is steady state and dynamical systems. So, next semester probably we will be having that course again. But for our purpose in this course we will be mainly wanting to identify the transition point. So, sigma equal to 0 is the transition point of the system. So, we will be varying some parameter like the Reynolds number and for small Reynolds numbers say the sigma will be negative system is stable. For higher Reynolds numbers suddenly one of those sigmas in a calculation will become positive. So, that point where it is 0 marks the transition and that is very important for us. Because that will tell us the critical parameter values for which my I am going to have a change in my flow. And how to investigate the dynamics over there we need to go to high order in our perturbation. So, to read more about the ODE systems theory dynamical systems theory and go through some of the things I have said you can refer to professor's book and its mathematical methods for chemical engineering. I think it is a Prentice Hall India. So, this book is available in the library and so I think in his last chapter he goes through this stability theory in some detail with a lot of practice problems and even answer some of the questions about high order dynamics. So, that is the place you can refer for what we have done today. So, we will now look at a specific example to which we can apply the past steps that we have just discussed and analyze the stability of a dynamical system. So, now I will work in x, y, z variables rather than subscripts. So, the change in x where the dot denotes the derivative is minus x, y and the change in y is minus y plus x square minus lambda. Here lambda is a parameter, write that clearly. So, if you look a bit carefully at these equations you will see that for positive values of both x and y the change in x pulls it back towards 0 as thus this term in y, but the x square term takes y further away from 0. So, there is a chance for some instability and some dynamics in this system and we will get right to the very first step of our analysis which is to find the steady states. So, to find the steady states we put the right hand sides of both these equations to 0. So, from the first equation we can get that say x is equal to 0 then from the second equation I would immediately find that y is equal to minus lambda. So, this is my first steady state and the other option from here is that y is equal to 0 in which case I am going to get x square is equal to lambda or x is equal to square root of plus minus square root of lambda. So, you see that I actually have 3 different possibilities one is 0, minus lambda and the other 2 are and you see another interesting thing here that these 2 steady states are possible only if lambda is positive because if lambda is negative naturally both of these would be imaginary and our vector field to begin with is real. So, these exist only for lambda greater than 0 whereas this steady state exists for all lambda right. So, this is the picture that we have and clearly at lambda equal to 0 I have a transition from one steady state to 3 steady states and this is something that we should keep in mind as we go on and we will get back to this at the end after we analyze stability. So, now that we have looked at the steady states we will move on to step 2 and that was of course linearization. So, in this step we write the variables as the steady state value plus a perturbation of order epsilon and substituting this into our original system of dynamic equations and linearizing we already know that we will get equations of the following form right where the Jacobian matrix contains the partial derivatives of the 2 right hand side vector fields. So, why do not we compute the Jacobian matrix now? So, here I will have derivative of the first function with respect to x moving on to the second term right and now that we have the Jacobian we want to evaluate the Jacobian at the base solution which in this case the base solution that we want to study the stability is the one that exists for all values of lambda and that is when x is equal to 0 and y is minus lambda right and we store this need result. Okay, so you might ask the question why are we picking this particular steady state to look at of course we should study the stability of these as well. But for this lecture I am going to focus on this for 2 reasons firstly this is the steady state that exists for all values of this parameter lambda and so in some sense it is the trivial steady state that is always a solution and in this sense which comes to the second point this steady state is very similar to the kind of simple steady states that we find in the fluid systems that we will be studying in this course. So for example Rayleigh-Bernard convection experiment when the fluid is at rest without any convection that is the simplest possible steady state of the system which exists for all values of the parameter like the Rayleigh number or the amount of temperature difference of the plates and then after a certain amount of heat input you get new steady states arising which is the convection cells which correspond in this analogy to these new states. So that is why right now we are going to focus on this trivial simple steady state which exists for all lambda and we can see that the Jacobian at that point at 0,-lambda is simply this diagonal matrix. So having completed the first 2 steps we now move on to the crucial step 3 which involves the Eigen value problem. So if you remember at this stage we make a substitution for our variables some constant which is governed by its initial condition or at this point itself I can say x0 e to the power sigma t. So we assume a exponential growth having the same growth parameter sigma for both variables and we can make this substitution into a linearized system where the Jacobian about a base state is lambda and minus 1 the diagonal matrix. So after doing this as we know we are going to get sigma times after simply substituting this in this expression we will get sigma times the vector is equal to Jacobian times the vector and this has the classic Eigen value form matrix times vector gives me the Eigen value which in this case is sigma times the vector. So directly as we had already said sigma is going to take the values of the Eigen values of the matrix j of course evaluated at 0,-lambda at the steady state of interest. So in this case sigma can take 2 values plus lambda and minus 1. So in a 2 dimensional system the 2 Eigen values or growth rates are lambda and minus 1. So we can see here directly that this minus 1 term this Eigen value is only going to lead to a decay and exponential decay of the solution. Lambda on the other hand can make the system unstable and lead to exponential deviation away from the base steady state 0,-lambda if the values of lambda are positive. So at this stage now we can conclude about the stability of the steady state of interest 0,-lambda and say that it is stable when lambda is less than 0 and it is unstable when lambda is greater than 0. And this is now very interesting result because you see it fits in very well with the multiplicity of steady states that we had observed right at the beginning. So if you come here you will see that for negative values of lambda only one solution exists because these 2 solutions will become complex. So when lambda is less than 0 I have only my base steady state and in that case the base steady state is stable. However when lambda is greater than 0 my base steady state becomes unstable and that is exactly when the 2 new steady states arise with plus minus square root of lambda,0 and now these come in so to speak to take the place of the original steady state. So I can represent all of this in terms of a diagram of the steady states plotted against the parameter lambda which in the literature has been called a bifurcation diagram. So in the bifurcation diagram now I will plot so on this axis I am going to plot values of lambda and on this axis I am going to plot x steady state and here lambda is 0. So on this side lambda is negative and here lambda is positive and if I plot the steady states now I will get a situation where I have only one solution all the way up to lambda equal to 0 and then exactly at this point this original solution which has simply x equal to 0 becomes unstable which I have denoted by the dashed line and instead of this I will get the 2 new solutions which grow as the square root of lambda and you can see that they will be symmetric because they are plus minus square root of lambda this is 0 and positive and negative. Now I will leave it up as a homework exercise to do the linear stability analysis about these 2 steady states which are plus minus square root of lambda, 0. So you can linearize the system about these 2 steady states calculate the Jacobian it is Eigen values and find whether these are stable or unstable. So that is the homework problem but I can tell you now that in fact these will turn out to be stable and this is a classic exchange of stability concept where one system one state becomes unstable and 2 new steady state solutions emerge which are actually stable. So for lambda less than 0 the system will be at this steady state globally and as soon as lambda crosses 0 it is going to leave this steady state now which is become unstable and move to either the one above or the one below and this bifurcation diagram is called a pitchfork bifurcation and is in fact a classic one of the classic bifurcations in one dimensional systems and is in fact seen in high dimensions as well as in pattern forming systems involving partial differential equations and there is special reason for that and that is the inherent symmetry of this bifurcation diagram and the reason for the symmetry you can trace back to the vector field which if you see is symmetric for values of plus or minus x. So what that means is if I substitute x equal to minus x I will get a minus sign out from here as well as from here which will leave the equation unchanged and you can see the same thing happens here because of x square. So this equation also remains unchanged if I make the substitution x goes to minus x and that is the reason why along the x axis I have this symmetry about 0. So you will find that in our physical systems this is quite common where there is the inversion symmetry about positive and negative values whole and that is why the pitchfork bifurcation is quite common in many of our systems. So we will be seeing this later on in the course again in a more complex context but it is good to remember this right here. So with this we have analyzed today the key stages in a stability analysis and I have shown it to you in ODEs but the same ideas propagate forward when we look at partial differential equations and therefore very often even though we may not be able to find out these steady states because in this case we have a simple vector field. So we could directly obtain the solutions analytically but in complex partial differential equations which have many solutions it is very difficult to find out to solve those nonlinear PDEs and get all the steady states in the system. However we will usually be able to find a simple steady state which exists for all values which in this case is this base steady state x equal to 0. So we would know this and by doing a stability analysis we can find when this becomes unstable and possibly leads to new solutions. And so during the next the latter part of this course we will be focusing on analyzing these systems and studying their part information.


This is the CBS radio workshop There's a woman And a woman is the blues cause blues is a woman and a woman is the blues And two is two life is rough stuff if you ain't got any Love is a hunt that few of us can feel heartache That you take and you chew CBS radio a division of the Columbia broadcasting system and its 217 affiliated stations present the CBS radio workshop Radio's distinguished series dedicated to man's imagination the theater of the mind Tonight transcribed an original story ballad the legend of Jimmy blue eyes by Edmund Brophy With original music by Ray Noble Starring William Conrad as the narrator and Jimmy Dodd as Jimmy blue eyes Horned by Manny Klein in Storyville where blues were born There's a legend of the golden horn and our hot lip kid blue-eyed and fair Who tried for a note that wasn't there?   Oh, he went first class. He was the best Now Jimmy blue eyes loved to dolly at a card room down Long Pig alley He could riffle the deck with gambler's ease and clean a coach game like a breeze one night a player met his raise with a Silver horn from better days I guess that's it boys. I don't see no more money on the table. I'm folding up hold on Jimmy blue eyes I got me some grouch bag dough Delight one hand a stud one hand and I quit win or lose. No, no good chippy You blowed plenty of horn on the riverboat for that. I don't want no grouch bag money. Who says you're gonna win me? deal the cards King Trey King Betz 20 Queen 6 King Queen Betz 20 10 8 Kings high 20 again Jack Deuce possible straight bets 20 more I raise you 20. I'm 20 shy I'll put up my horn. Oh, that's a pretty beat-up piece man.  King high You Deuces Jim showed his wind to his loving Jill and her fancy flat in the morning still Why Jimmy honey? You couldn't call home the cows at sundown with that old thing.  You can hear the echo down the riverside But you never blew horn before I was top horn in the reform school band I don't make for playing Tom Anderson saloon, and you're not filling in alongside King Oliver in the funeral band. That's a cinch Sweet Jill. I'm sure no fancy Dan, but I'm a dead bang natural music man I'll take this battered silver horn and make it talk come Marty Graham on Jimmy my love since I was born. I've loved the music of a horn. You learn to make it weep and shout I love you to those stars While me and the blues are kid and candy for St. Louis and Memphis and Mr. Handy if you stuck a pin in my heart, it's true a drop of my blood had come out pure blue But summers came and winters went and Jimmy's loving heart was bent As for he hit that master blow his sweet Jill blew with hard lips Joe Hardlips Joe he had no peer he could shave the head from a glass of beer with a win from his educated horn Just as shows you were born just as shows you were born The kiss off gave poor Jim Jolt He loaded up his blue steel Colton headed for the circus house to croak that double-cross and loss Hardlips Joe was holding the floor when a cold spit lead type 44 The bullets sang around his head Jim killed a tourist guy instead In sure discuss a gun he slammed upon the floor and any lamb Along the streets of jasmine night well whistles blew to all this flight and back at many circus house Hotlips Joe that heart thief lost case of sucker on the floor and smiled and whispered never more He did put a mirror to his mouth hotlips. Don't need no looking glass Man got no pulse. He's a goner.  I say amen to that amen No, nevermore will Jimmy blue eyes catch me with a lead surprise This hog wild kid is through for fair the cinch to dance upon the air Before he ever got a mile poor Jim was nailed and brought to trial The old judge burned him with a look And up and hit him with a book And the cell where Jimmy locked steel secured and granite blocked He played a music rich a part a gift God gives a contrite heart Long summers came long winters win And all of Jimmy's time was spent to reach a cord to cut the air and Blow that note that wasn't there The son went down in store of ill when love went out with Jim and Jill First war came in prohibition and a district went to quick petition a die-hards died on bathtub Jim But the music lived like hidden sin to tickle toes in northern lands for the advent of the Dixie bands But no all-eens was much alive in the year in 1935 Jim took the long road home at last the dark and bitter days were passed Now when Jimmy blue eyes hit the bricks He was master a hot licks his trumpet clawed and tore the air in search of a note that wasn't there He played a hearty spots in town Nearly blew the ceilings down When his uncles all were done. They said oh that man is Gabriel's son Well, it's Jimmy's feet and full-blown gale the silver sleet he smiled and played right on Until his mind ran back to his heartbeat Jill Now you show me an artist's fine affair who seeks a note that isn't there And I'll show you a guy that most men ain't He's alone in the clouds an uncrowned saint For he scatters joy to his fellow man Though he might wind up and also ran To drive past glory Fortune fame it's nirvana sure But a heartbreak game And Jimmy's heart and soul sent out the soft sweet tones of his trumpet shout He blew it hot and low and high. He hit the fringe a heaven sky The multiplying strains made not he couldn't reach a peeky sort He blew until the notes were pain elastic stretches of his brain I tried a horn gone and mumbo pills, but they wouldn't bring a trumpet trills He killed a quarter-eye each day and it didn't help for his high note play Jimmy boy your case is tragic. You'd best resort to Mammy magic. She cooks a pot like jungle stew There's conjure in her devil's brew Where's this Mammy magic live? Where tell me where two miles north of nowhere One mile south of someplace Look over your shoulder man the old hot in the wood the old hot in the wood Now Mammy magic was her name a voodoo witch of power and fame whose spells were famous as a blues from New Orleans to Newport news Mr. Jimmy blue eyes Audrey You know me Mammy magic, you know my name few things are secret to one who has a third eye You are troubled speak Some walls and windows have ears. I Read your thoughts Where? Your desire is beyond my power. You lie witch. I got the price woman of darkness I'm desperate. Do you hear me desperate? Level with me. I tell you Jimmy on the level you got to see my boss Devil You're asking one thing. I can't do Despite the magic I can boom. I'll have no truck with devils I am selling you are here to buy What's he asking for price? I Don't make out his bills. He keeps them plenty private Get your devil man.  Yeah Speak up sir. There is no occasion for fear. I pride myself on an understanding heart Well Here's the situation mr. Just call me red.  Okay, mr. Red. Here's the setup So the red king made a deal with him And a secret locked in silent Jim Then with the evil art of old red turned that silver horn to gold Jimmy blue eyes walked on feet which never touched upon the street He wore a broad smile upon his face for that never never note was his at long elusive note was there The most immortal anywhere, but when he blew it come what made He had an awful price to pay All hotlips Joe had just come down from a long run in Chicago town He'd been the world's top trumpet king for 20 years come one more spring The jazz folk down a new Orleans dug folding money from their jeans To bet on Jim or hotlips Joe to contest for the master Put your money where your mouth is I got a century says Jimmy takes it. Well, you get it up man It's like shooting fish in a rain bird You can't just sit in bets on the job and Memphis takes your bet. Chicago goes for hotlips That boy can really blow off a storm. Here's money marbles talking beans on Jimmy the pride of new Orleans Don't cover five g's with cash not talk. It's Jimmy blue eyes for all New York Most ever parish up and down bet scads on Jimmy's horn renown The high and low of fortune's birth came on from ever end of earth The joint was jammed And the 88 was under the dukes of a solid gate The SRO sign hung outside Now was hotlips Joe and his beauty bright sweet jill the night and gave us song Dead wrong braw dead wrong dead wrong Playing the puff at a ringside table sep and iamda draped in sable And Joe took a stage with a master's pride And cut his trumpet open wide He blew hot notes heard round about he turned that trumpet inside out He blew till all the glass was broken. He blew so hot the joint was smoking His horn turned inside out and curled the last note traveled round the world Joe sat on the churran sounds bust tombstones in the burial grounds His look told jim with unfeigned joy.   I'm so afraid His horn sang smooth and educated and blue and true and dedicated The music of that Dixie man was greater than the pipes of pan The highest soft sigh of a trumpets cry can tell what magic words can't try Well, the horn sings true as it only can unmatch the nature bird a man Man sings his heart with tongue a pen words give and live through time again, but it's very hard and so Ring clear when a true horn speaks and the crowd sat frozen round to geth Jim split the ceiling right in half He blew the walls down in the doors. He raised the carpets off the floors And a cotton this disinherited miss begotten son of a slum and sin and gin blew that scatter outside in And yet He seemed like a tired life going home from a weary earth and a heartbreak room To that promised land of a fairer climb out there On the other side of time Jimmy blue eyes hit some bars that blew out half a million stars and then That never never note Went clear 10 trillion miles to heavens here blue-eyed kid Went up and smoked now some rounders claim they're in the know that jimmy blue eyes Fries below But anew all leans they'll lay your odds He's playing trumpet with the gods For a deacon man was there who preached it before the last note cut and reached the edges of eternity and died Father forgive me His trumpet cried No matter where when hot music blows if you're not hairpass dad. He knows Jim's golden horn the love of faithless jill When blues and we were young in story film Now jimmy blue eyes came on earth a child of hunger from his birth They spiked his milk with old-time gin and he played all day round the streets of sin So come along for dito street where the hotlicks tickle dance and feet that shuffle in patent leather shoes Where jimmy blue eyes blew the blues In storyville where jazz was born There's a legend of the golden horn and a hotlip kid blue-eyed and fair Who hit that note it wasn't there Tonight transcribed from hollywood the cbs radio workshop has presented the legend of jimmy blue eyes An original story ballad by edmund brophy with original music by ray nobel and starring william conrad as narrator With trumpet by mani kline adapted and directed by san pierce Jimmy dodd in the part of jimmy blue eyes appeared by arrangement with walled disney Producer of the full length motion picture in technicolor song of the south starring uncle rimas and the critters Song of the south will be released nationally easter week Also featured in the cast were roi glenn georgia ellis sam edwards lou meryl nan bordman jack moiles and tony barrett Featured in the all-star band were tom peterson on trombone maddie matlock on clarinet sammy white drums nat barber piano and larry brine bass The workshop is produced in hollywood by william froome Who's there's a woman who's there's a woman Is the blue half is rough stuff If you ain't got love is a hunger take And you choose sometimes you win


Hey everyone! Merry Christmas! I contemplated not putting my tree in this video, but I have decided that it is necessary because it is still Thursday and we don't know the results of the election, so we need some Christmas cheer, okay? We just need some Christmas cheer. Hi everyone, my name is Katie and I am on a gap year in 2020, like many students across the globe. So even if you are in college this year, you're probably not having the typical experience. So this video is really just for anyone sort of feeling bogged down by 2020 and like looking for just some things to do. If you're interested to see more of what I'm getting up to this year, I post new videos every Monday and Thursday, so if you want to subscribe, that'd be cool. If you don't, that's cool too, but I just think that it would be fun for you. So in the year of 2020, it's very easy to distract yourself, and I was doing this with Instagram for like six months, and that was bad. So I really wanted to compile a list of the things that I have done, A, since I've quit Instagram, and just since I have, you know, been living as a human in 2020, let's just jump right into it. Activity number one, drive around and scream, sing, Taylor Swift in your car, or any song of your choice, but Taylor Swift has just been my artist of choice for the last like month or two. So, but do whatever you want. I do recommend something dramatic though, because it just like really lets awesome steam. Like, even if you're one of those people who like listens to like screen music, like even to that, like just like just let awesome steam, it's great. I'm driving one bit, not at all, not one bit. You can see that there's a parked car right there. Oh, I have a neighbor going by. Two, go outside and paint the landscape, because it's really pretty and the trees are just like, you know what, we just really want to be painted. There's a like a kind of a little like reservoir, like lake thing, like right near where I live, and I saw these ladies like just like painting outside the other day, and I was like, wow, I should do that. Um, mine did not turn out the same probably, but you know what, it's an activity. Number three, go for a walk. Um, this is something I started doing in college most of the time. Well, okay, my family is like a big walking family, like we're that type of family, like when we go to New York City, like you don't take cab, you don't take the metro, like you walk, and if you don't walk fast enough, you get left behind. Oh my god. So, I'm a walker. So, I walk a lot, but so when I got to college, I like really needed an outlet to get away from my roommates. So, I would walk around the Charleston City a lot. So, if you're looking for something to do, going for a walk, even in your city, like put your mask on, go outside, go look at the trees, play some good music, um, and it's just, it's a really like peaceful experience. It's like, it's like driving, but you're outside breathing fresh air. So, that's, that's kind of good. Number four, start a YouTube channel.  So, if you're like, oh that's something I've always wanted to do, like, you should do it, it's really fun, like I don't, I don't know how to tell you. Number five, write that book that you've always wanted to. Um, it's a really good time to do that. Write it about 2020 and your experiences, and um, like here is me writing my book. Number six, call your friends. Catch up. When I call and talk to my friends, like, it's almost like we're having lunch, because like sometimes I'll sit and talk to them for like an hour and a half. So, like, make like dates with your friends, not like, you know what I mean, like make a date and like set a time and be like, okay, on like Wednesday at seven o'clock, like we're gonna talk and have a good chat, because it's just, it's really good to connect with people, like if you're at home, like other than the immediate people who live in your house. Number seven, stare at a wall for five hours. Number eight, cry. Have in-depth conversations about human existence and existentiality with your dog. Are you a good person? Are we good people? Is Kim Kardashian the pinnacle of modern society? What does that mean for us as a generation? Ten, try on all your clothes. Hi, I'm Katie. And could I be wearing more clothes? I jerked you. I jerked you. You thought that it was gonna be like me wearing different outfits, but it wasn't. It was me pulling a Joey from friends doing the opposite of not getting dressed, which is wearing all of your clothes, so. Number 11, learn a new recipe. I sometimes make cookies, which that's not a new recipe, that's just like me wanting cookies, but it's still like something to do, you know, some, try like making a cake. I did that for someone's birthday, like instead of like going out, like I was like, oh, like let me try and make like a cake from scratch. I'm better at baking. Cooking is like kind of my thing. I cooked like for myself last year in college, but like, I didn't really hit my stride because like I'm a college student and like, it was kind of a miracle, like if I made like a potato for dinner, like that was like the peak of my cuisine. Number 13, pretend to go to France. This is just a shameless plug. I made a video where I pretended to go to France. Go watch it. Number 14, read a new book every week. If you like love book clubs or like, you are like a literature person, read a book or join like an online book club. Like I know a lot of places have that, like there's a lot of small YouTubers who have like small book clubs or like things like that. So you could totally join that. That'd be so fun. Number 15, get an internship. This might be a little bit harder because A, we're in a pandemic, but B is, I don't know what B is. I'm just going to leave that there, but try to get an internship, throw your name out there. You know, it'll get you some professional experience, which is always good. I love having internships. I have one right now, which is super cool. Getting an internship might be a little bit harder right now. I know a lot of people are sort of struggling with finding them. But even if you are having trouble finding an internship or a job, try making a LinkedIn profile. Try making a resume if you haven't had one before. Taking any steps toward that goal of getting an internship, whether it's like a remote one or maybe it's in an office where there aren't a lot of people. It's just definitely something to consider because it can add, you know, another layer to your life. Number 16, put up your Christmas decorations way too early. She's so cute. All of my other Christmas decorations are packed away.  Like since I was like 15, I have like Christmas throw pillows. I have specific lights that go up. I have another Christmas tree. This one was for my dorm last year because I just, I can't physically live through the month of November or December without a Christmas tree. It's just, it's like, it's like air. I need it because it starts getting cold and then it's just like, well, what am I supposed to look at? I can't just look at a regular lamp. I have to look at one that's twinkly. Okay. Number 17, watch every movie that you have ever wanted to see. So I think this is a really fun one because like, it's fun to watch movies. Um, but start writing down like all of those movies that you were like, oh, like I'm meaning to see that. Like being a young person, like there's a long list of like iconic movies that like I have wanted to see or people have mentioned to me. And I think that's a really good way to just sort of like occupy your time. Like if you're not like, if you can't see friends, because like we're all going into like probably another lockdown. I know that people in the UK like are already maybe, I think they're already in a lockdown, which is rough. Like it's, I have a feeling that the next couple of months are going to be rough, which is not like super cool. So try, you know, compiling a list of movies and make it sort of like a nice night. So maybe you make like a nice dinner.  Like just giving yourself like those little things that maybe seem a little bit more special, um, can just like add a little bit of like something to look forward to. Okay. Number 18, challenge yourself. So I have been in school since preschool and it was really weird to think of spending a whole year not challenging myself, like not challenging myself academically or in any sort of like career way. Like I think especially in the area that I am from, like there is always this sense of like needing to better yourself, which is always, that's a good thing.  And I think that a good way to sort of occupy your time is to focus on some of the things that maybe you've put off because you were in school. This can be something very small. Like for me, I wanted to try to give up social media for a month. So I haven't checked Instagram since like September, like the first or second week of September, I forget when. And after that month, I have just decided to keep going with it. So I actually haven't checked Instagram in like a full two months, which has been really actually very nice. And it's changed my life in a very positive way. So even something small like that. Okay, so number 19. I said this video is going to be 20. I think it's 19. I can't come up with another one. I don't know. But number 19 is have a spa day. Like paint your nails. I need to paint my nails. Do a face mask.  Like just have a day. Like that just sounds nice. I don't know.   So if you haven't yet already subscribed, we're very small over here. So if you want to subscribe, that'd be cool. Let's let's hope we find out who the president is that that'd be a nice way to end this week. That's number 20. Finding out that we have a good president. That'd be nice.


So it's been a few years since U.S. Navy divers have been here in Odessa, Ukraine, in support of the Sea Breeze exercise, and in 2016 when this yacht was sunk, the planning began for coming and trying to rig and prep for the ultimate lift of spoiling the whole yacht off of the bottom. It became a navigational hazard for the ships and the movements for the Ukrainian Navy. Overall, the experiences gained here between Mutsu-2, UCT, NEDU, and the Ukrainian Navy provide unique opportunities for future partnerships.


How do you make your marriage last a lifetime I buy her anything she wants Still money from the church every day Husbands love your wife When you love your wife love is is is there certain elements of love that we need to remember first Corinthians 13 Gives to us a variety of ways to see the elements of love. It gives us a scriptural way of understanding it But one of the things Marina we're talking about that tonight. That's one of the things we spoke about we're talking about love in marriage and Love endures, you know, so Marie and I when we got married Determined that we would have a biblical understanding of what a relationship is and we put biblical principles to work and so Over the years We've become better at living for Jesus independently and we've become better at living for Jesus corporately Because we have each one of us has an individual walk with the Lord that blends into the married life And so it's very important for me as a man to guard my walk with God And it's important for my wife Marie to guard her personal walk with God and we Though we have individual Walks with the Lord. We also have roles within the home that we've learned to work together in So I've learned what my role as a husband is She has learned her role as a woman of God And so a man of God and a woman of God working together in the things of God With the love that endures that mentality of the patience and Kindness and the gentleness and the goodness and those things that pertain to love to apply those within our marriage to learn to talk as well as to hear and Learn to hear sometimes is more important than to talk and to respect one another and to pray for one another and And and all of those things That's how Marie and I have remained the way that we have for the years that we've been together You know a long time ago I knew what it was like to be without her and I discovered what it's like to be with her And I want to be with her more than I want to be without her and so because that's the truth I do everything that I can to make sure that I knew never do anything that would make it possible for me To be without her and so I love my wife You know and I love her like Jesus taught us to you know when Paul was speaking in Ephesians He spoke concerning husbands love your wife. He said even as Christ loved the church And he gave himself for her and so that element even as Christ, how did Christ love you? He loved you absolutely and totally without reservation and without any kind of stipulations he just loved you and so that's how I want to love my wife and Was I that way when I was a young man?   I'm this and I'm that, you know, and you're gonna you know I started out with that attitude and and I eventually realized that no I'm better with her than without her and I need her and she needs me and why not be allies rather than belligerent and and I learned to to shut up and I learned to listen and I learned to value the things she values And no, I'm not perfect at that, but I'm a lot better now than I've ever been and I mean, I'll even sit sometimes for a few minutes watching a lifetime movie now that's sacrifice That is sacrifice. Oh that's sacrifice, you know, but Man's greater love has no man than this than to watch lifetime movie channel It's just it just pick your Bible up Read 1st Corinthians read 1st John You know read read your Gospels and see what love is and then just say how does that work in a marriage? And it does, you know, love is patient. Love is kind, you know doesn't boast You know, it's enduring you see the different qualities of it and then you say God make me that that's what I want to be I want to be this and And you know what over time he molds you into his image and And you become that but I entered into marriage with Marie with one thing And that was this we are gonna make it no matter what I we went it We never had any reserve clause. We never had any if it doesn't happen. Well, that's okay No, it was like we burned the boats. We landed on the shore. We're gonna go forward. There's no turning back That's a fact and that's and that's how we entered into marriage Never had any Consideration of divorce there were times when we were struggling where I thought maybe we've made a mistake being together But I also said I know God brought us together. He'll keep us together We're gonna learn to work these things out. There's nothing that can separate us. We're gonna remain Consistent. She's the same identical way I had the the advantage of having having a mom and dad who were married for 50 some years and she did too So we saw stability in marriage. That was our model and with Christ I can do all things and I can die to myself and I can become a man of God and I can be a good dad I can learn to do these things. It's not impossible to I just determined that that's what I would do and And no secret every married man in here knows what I just said we all know that you just hang on You just hang on and you know, what's good. It's good For me it is for you Josh, you're not married Don't get married bro


What's going on guys got Kyle here with laden sports cards, and we are ripping 2022 23 impeccable team all eight box number five Did it randomized for the break guys? I'll show you guys the results again real quick had Corey a with the Lakers down to Matt and with the Pacers Good luck here guys. Let's see what we got. What's up, Elliott doing good sir doing good. How you doing? Peckable team all this is eight brick eight box number five Good guys still got seven left in crown team all gonna try to get that one going next If we can get it filled up here. Good luck here guys impeccable team all see some big rookies Switching off the baseball and now we're on the basketball We've got a bar in the first first pack got the number to eight nice one there Luca Donchage good start for the maps Luca number three out of eight That's a sick one there on the Luca to start us off We have base of Keldon Johnson spurs This one is to 88 we got a impeccable elegance RPA Johnny Davis Nice one there. This stuff is fire, man Get all the RPAs the autos low numbers Nice one there to 88 for the whiz We've got auto this one is numbered five out of eight gonna be BJ Armstrong canvas creations auto bolts Nice one there BJ five out of eight for the bulls Got our bar card nice It is three of six on these silver bar. It's gonna be Jaden Ivy nice one there pistons Great hit on the Ivy rookie bar I'll be Bradley R nice one there on the one fine ounce of silver One Troy ounce of fine silver. I should say three six there Ivy Nice one Bradley. There's a base red of Cam Thomas We've got base rookie Jabari Smith, Jr. That's one there Rockets It's a clean-looking car with the red very clean And nice to 88 we got J. Dub on the impeccable elegance RPA nice one there thunder That's Keith with the thunder J. Dub 36 of 88 on the RPA nice one there and We've got auto to 88 rookie autographs of Musa Diabata. That one's Clippers This one is not numbered.  We'll be entered into our national promo We're going all the way through next Tuesday with national promo breaks Definitely check out the promo as well guys. We haven't seen it yet You can see it on the header of our website. You can see what are the actual results of the promo are To eight we got Paul George nice one there Clippers Hey, what's up Ron? Hey, man, I hope you enjoyed it. I hope you enjoyed it, man. It's a fun time out there I know it can be overwhelming if it's your first time for sure Donovan Mitchell on the red Glad you got to meet Sean forest Lane rich It's awesome, man to 88 we got thunder on the RPA again elegance of Usman d'ang Thunder picking up a couple RPAs and rookie auto this one's to 88 of Blake Blake Wesley spurs And another bar card here. This one is two out of six on the silver bar. It's gonna be Anthony Edwards Nice one there. Um T wolves That's Christopher L taking home the one Troy ounce of silver on the T wolves Nice bar card there Box for this one is not numbered stainless stars red of Jason Tatum It's when they're Celtics on these stainless stars You see any crazy any crazy cards today, Ron?  This one is five out of eight Jalen Brown for the Celtics there on the red to eight Got a band out of bio base This is RPA to 88. It's gonna be Jane Hardy Mavericks there now go well with the Luca to eight Mavericks doing well Jane Hardy elegance RPA And rookie autos also to 88 it's gonna be window more T wolves box number five stainless stars This one is rookie on these stainless stars nice one there for the Pacers Ben Matherin Ben the bucket Stainless stars rookie. Yeah. Yeah, I wasn't sure on I know if the team also for less or whatever But there's none fair there's a Coloco red Raptors God Jamal Murray on the red I mean backed up that Jalen Williams RPA for you Keith there's elegance RPA to 88 tari Easton go into the Rockets Elegance RPA rookie auto to 88 this one's Pat Baldwin jr. Warriors Worry box number five Or number six. Hey going back on Sunday. Nice Ron. Nice Enjoy that man Did you uh, did you get the cards back already from PSA? Is it a is it a one-day turnaround? Are you going back on Sunday and grabbing them? There's a rookie stainless stars Johnny Davis wizards I'm sure if you're one of the first people and it's probably a one-day turnaround I feel like it's probably a little bit later Not one of the first people grabbing Can only grade so fast got a Jimmy Harden there number six of eight for Philly on the red Base red is Frans for the magic Frans Wagner Hey, well, good luck on those Ron. Good luck, sir We've got nice one here shading sharp on the RPA number to 88 Ego blazers on the shading sharp elegance RPA Now when it's Patrick P Patrick got the blazers We've got a Kennedy Chandler number to 88 rookie auto grizzlies Two boxes left still seven left guys Still seven left in crown Royale team all 20 box full case 20 box go guys box number seven impeccable surprised. We got two bars out of this already famous star This one is stainless star Steph Curry Go into the Warriors there Nice Steph on these stainless stars base So we're magging that um shading sharp as well for you there Blazer spot got a red scary Terry Hornets Got dame on the red this one is to 88 rookie auto of AJ Griffin steal the draft For the Hawks someone says that at least steal the draft AJ Griffin and RPA elegance RPA to 88 we got Christian Brown Nice one there Nuggets on the three color RPA And looking cards here Last one guys number eight Peckable team all this one is the MVP stainless stars of Joelle and bead Nice one there Philly getting a nice and be a nice Jimmy Stainless stars and be nice one here for the pistons. We got Jalen. How you doing? Number two eight on the team all exclusive nice one there pistons Bradley Going well doing well for this break Had the Ivy bar earlier We've got a John Marant rookie that one's a mag. I know that John Marant and magic We've got elegance RPA to 88 gonna be Paolo Bankero. Boom. There we go Marcus Marcus H with the magic Nice Paolo gel so this one We'll mag this for you Marcus nine of 88 on the Paolo RPA Great hit there sir for the magic And on the back we have number to 88 Musa Diabata on the rookie autos second one here for the Clippers Diabata Alrighty guys that will do it for the break there. Um impeccable team all that was eight box number five We'll do a quick hit recap here We'll get trying to go on to crown next Alrighty guys here's some autos and some bars Nice break here, man. Really nice Diabata to 88 auto RPA Christian Brown Griffin to 88 Chandler to 88. We had Tara Easton RPA Pat Baldwin window Moore Jayden Hardy on the RPA. These are all the 88s. I don't have to say that Usman Dieang RPA. We've got Blake Wesley Musa Diabata again We got BJ Armstrong number to eight on the road on the auto And to 88 Johnny Davis RPA We had two bar cards. We had two of six and three of six Had Anthony Edwards on the one one Troy ounces silver and we had Jayden Ivy on the one Troy ounces silver And then max we had RPA to 88 Paolo Um Shane Sharp to 88 and we had Jaylin Williams to 88 Some nice hits there guys really nice Great case of impeccable team all there. Um, that was number five. Get these cards right out to you guys. Thanks