Thank you very much, Miroslav for an excellent presentation and thanks to all of you for still being alive after a long day and I have a plea to you all which is to please stay awake now. I know it's been a long day but today you will see the first direct systematic evidence of profit shifting through transfer price manipulation in a developing country. Profit shifting is when you move reported profits without moving actual activity, all right, so say that you are the owner of a multinational group with a subsidiary in the Cayman Islands and one in South Africa, then you have a tax incentive to shift reported profits from South Africa to the Cayman Islands. Now if you do that without moving any actual activity then you're profit shifting and this is illegal but if you close down your factory in South Africa or close down the production and move the entire production to the Cayman Islands then you will also see the profits moving to the Cayman Islands but this will be the result of an actual activity movement and this is not profit shifting, okay, so this is a bit of an important distinction and this is as you can imagine also why it's difficult to estimate this, right? Did we observe a movement of activity or did we observe an artificial movement of reported profits? So I just want to reiterate why this topic is relevant and particularly relevant in a developing country setting so as most of you will know the corporate tax revenue is very important in a developing country setting so when you look at the contribution of corporate taxes to total taxes in developing country this is 20% compared to 10% in developed countries so as you can see profit shifting which erodes the corporate tax that hurts twice as much in a developing country setting so that in itself makes this topic very important and what we are at the same time observing is we have this very rapid expansion of M&E multinational activity in developing countries we're seeing this expansion all over the world but it's happening particularly fast in developing countries so as relevant as this issue is today it's only going to become even more relevant and as the corporate tax is such an important part of the developing countries economy you can see that a lot is at stake. The final point is that there is this hypothesis which is widespread that developing countries they lack the proper institutions to monitor and regulate and follow the multinationals when multinationals are not playing by the rules and you heard Michael also commenting on this yesterday so this is an assumption and a hypothesis which has now been somewhat backed by the data so I myself started getting into this topic really three years ago and what I have seen in that time is this rapid expansion of research on this topic so you have really seen the frontier of research on profit shifting in developing countries moving the last two years so I think the contribution today by Murislav is an important part here we have Caroline Schimanski sitting here in the room also made a very good contribution on profit shifting in developing countries I myself have written two papers with co-opers and then we also have Michael Keane sitting here I actually hadn't even realized that Keane in this paper I've read so much was Michael Keane of course so just to say there's really a lot of things happening and all of these papers have been truly amazing in understanding now the overall size of the issue full-park estimates of the losses and for example we now have support of this notion that multinationals indeed more aggressive profit shifters in developing countries however all of this research on the previous slide relies on this approach which has been framed the indirect evidence approach and that is an approach where you essentially find patterns in profitability which are consistent with profit shifting that is the identification strategy and to explain it you here have two pictures of cookies and if you can imagine the cookies as as profits as you can see I'm trying to keep it interesting you can you can see that that firm a they they doesn't have a connection to tax havens but they do have a lot of profits from be they do have a connection to a tax haven and they don't have any problem and now looking from this picture it seems as a reasonable assumption that the tax havens ate the cookies and I completely agree with the with with coming to this conclusion from this picture and that's also why I wrote two papers using this approach but of course there are some issues when you use this indirect evidence approach one is for example are we modeling the profitability correctly when we see that there's a lack of profitability are we sure that that the scales of return in in FDI are just constant for example are we observing actual movement of activity it doesn't have to be capital it could also be knowledge moving moving etc. I'm capturing some of this but in general when I when I present my my previous papers what I often meet is just a dissatisfaction that when we're not observing the movement of the profits we're just observing that they're gone there are no more cookies the profits are gone but we didn't really see how they disappeared so today we're gonna try to combat that a bit by by zooming in on what is known as direct evidence and and in particular direct evidence on transfer mispricing of of goods so so this this is done by using highly highly detailed data at the transaction firm level where we are then able because it's goods we're able to have prices unit prices so so goods I can say that this book here that's one book and I can say what's the price of this this book and when you have prices then you are able to compare the price on internal and external transactions so this is what allows allows me in this case to almost directly observe transfer mispricing which is one form of profit shifting and come with this really really credible evidence and I think the truly amazing thing here is that as I said a study like this was really not possible outside OECD and and we only have these these four countries Denmark being one of them where I'm from that has actually been able to produce a study like this and of course this is this is only possible due to the amazing work done by by the UNU wider and the National Treasury in South Africa which which made this tax administrative micro database that I can only highly recommend other people to start using and I can say coming from the country of of tax administrative data that that this database is really on par and and and top top of the line so so before we move forward I think it's important to say that that what I'm gonna present today is not something that's gonna replace this previous research I was I was citing instead it's gonna be a compliment right so what we're gonna do today is that we're gonna zoom in on a corner and and even a small corner of profit shifting and we're gonna find some really really credible evidence of this now what Miroslav was showing was global evidence of profit shifting so that is the ultimate scope right and I see at least today that we have this trade-off so I have papers now and in each of these boxes and I think we need all the evidence in order to to get a clear idea but today we're really really now trying to get this this very credible evidence which can also be applied in practice using algorithms in tax authorities so as I promised when I'm gonna zoom in on these transactions of the multinational firm and to understand Transformers pricing you first have to understand that of course any multinational firms will engage in both internal and external transactions so the internal transactions I that's the multinational trading with itself that's why they exist to begin with and the external transactions that's when the multinational for example sells the final product to a consumer but it's also buying all in inputs that it's not itself constructing etc etc so by law there's an arms length principle and that is that when the multinational is is buying from itself it has to close its eyes and and imagine that it's not buying from itself it's buying from an external party and so that is that is the law and essentially all countries where I have looked into the tax laws and and as you can see it's actually quite a requirement that you're giving the multinational sometimes it's it's even hard to imagine how you can even make this fault experiment and and of course firms on top of this have really an incentive to deviate from their arms length principle because when they're trading internally there's a tax incentive to price the products differently than they would have done when they were trading externally so say for example that I'm now importing a book from my subsidiary in the Cayman Islands well if I could choose freely I would like to price that at a million dollars because that would allow me to shift one million dollars from South Africa to the Cayman Islands when I'm trading externally I really don't want to pay a million dollars for that book because why would I do that I want to buy the cheapest inputs as possible and as you can see that that really it creates an incentive to have a wedge a tax wedge between these two type of transactions so so here I I just put a fictional example of what we're we're expecting if you imagine the company only trading in bolts buying them externally and internally then we expect that the internal price will be really high compared to the external price where they're importing from low tax destinations and this won't be the case when they're importing from high tax destinations so now we come to the to the data part where we really can exploit this this amazing data set that has been created and we can there and we can calculate the unit prices of imported goods in each transaction when we have the unit prices we can then go to a regression model and estimate what we would think the transfer price would be in each case and so we can estimate the arms length price and when we have estimated the arms length price we can calculate the deviation for that in a regression framework and we can see if we estimate if the estimated deviation from the arms length price correlates systematically with the incentive to deviate from the arms length price so this is really the the the approach or the to-do list when you want to do this and and of course it requires this this very detailed transaction level data which South Africa has created so so just looking at the descriptive data just at a first glance you can you can you can almost see here that there's something going on right so here you see the average unit price and how that that in percentage deviates from from the when you compare related to unrelated import so the red column shows you that in average just looking at average unit prices we see an almost 60 percent higher unit price on imports coming from a low-tax partner when they are related compared to unrelated right so so when I'm buying the book from from my subsidiary in the Cayman Islands I'm pricing that 60 percent higher than when I'm buying a book from an unrelated party in the Cayman Islands this is just an example of course however when you look at the high-tax partner she actually see no difference you see that related and unrelated imports they are priced the same way of course right now we're not exploiting any dimensions of the data because this is just average unit prices we are literally comparing apples oranges bolts and books so next day up is is then to use these many many dimensions of this amazing data we even have a digit product codes which which which is a bit of an abstract concept what is what how detailed isn't a digit product code well here is one of the categories patches for puncture repair of self-volcanizing rubber so I have no idea what this is but I can see that it's very very specific and and that is really the amazing thing about this data and where it actually supersedes what we have even in Denmark is that you have this amazing amount of detail which really allows you to very accurately predict the unit prices so when you have these these these product categories natural next step would be to do the same analysis I did before but now within categories so we saw that in average that's the bottom part here the all products line we see we saw the 60% right just average unit prices from no tax destinations seem to be overpriced by 60% now you can do that same analysis just within a computer park or a specific taps and cocks or the seal of rubber that was the patches for puncture repair so you can do this this this aggregate analysis I did just within each product category and as you can see in some way that the estimate actually remains within the same ballpark but you can dig even deeper than that you can actually see the same firm importing the same product so so one concern when digging that deep is that that maybe the firms will be aware that it looks a bit suspicious if they're actually importing books at widely different prices but even when you use these firm product fixed effect you still see that that imports from from low tax countries are overpriced when they're internal yeah and and so this is in general I try to limit tables in these short presentations but but if you're interested these these are the results I think the interesting part is here in the bottom where we we actually find or I find that that that the responsiveness to to these tax incentives is actually not a significantly different from what have been seen in Denmark France etc so so that is that is interesting I so a word of caution is that these estimates are a bit all over the place but at least we're not seeing any dramatic differences compared to for example Denmark so I think that is that is interesting I I also think that that's something like transform is pricing of goods is maybe what I would have expected less deviation than say set for example with with royalties but but again now we're getting some very concrete evidence on on what these differences are between countries and and this is a tool that can actually be used in the flagging of firms yeah so to conclude we have we have managed to to now get this this very credible evidence of profit shifting in the developing countries and and a bit surprisingly so far it doesn't seem that that there are it's an order of magnitude different than than what we observe in in the OCD countries but but but I would say that the work has only begun there are there are so many I was speaking with Elizabeth there are so many things that would be nice to investigate further and and I can only if there are any listeners recommend that that other developing countries also start allowing researchers to use these databases because as I said you can you can actually use this in the day-to-day work


I wanted to show my workflow for baking in captions or subtitles into videos. And there's various ways to do this. The reason to do it is for accessibility for people, obviously, who can't hear the audio track in your files, and for video players that don't have closed captioning. This works really well in a 3. It might not be the fastest method out there, but it involves the least amount of manual work. So what I do is upload my videos to YouTube. And so in YouTube Studio here, I've got a list of videos. I go to Details. If I click on Subtitles, I can see here's my automatic subtitles. If I click these three dots, I can click Download. I'm just going to choose SRT. This method will work with multiple formats. Here's my caption file. And I'm just going to put that in here. And now what I use is a free program called Handbrake. It's available for Windows, Mac, and Linux. I have Mac. Now I'm going to choose Open Source. Go to that Wayfinding video. And I'm in this, you might start in the Summary tab. I'm in this Subtitles section here. And what I want to do is under Tracks, click Add External Subtitles Track and Wayfinding. There's my captions file from YouTube. And I want to choose Burned In. You may not want to use Burned In. You may want to use a layer that can toggle those captions, then you don't need to do this. And then, of course, with your video, you can set all the options you want for frame rate, quality, et cetera. I'll just leave them at the default for now. I'm going to call it Wayfinding CC. Put it in the right folder and hit Start. You can see it moves pretty quickly. And so now when I open this video, we've got these Subtitles Tracks in here. And they're baked in, meaning they're part of the video.


So let me share the screen so you can see my slides, right? Yes. Perfect. So hi, everyone. Thank you, Jacopo, for the introduction. So today and tomorrow, we will be talking about nonlinear dynamics. So let me first show you how I have organized all the material for today's and tomorrow's tutorial, and then we can get started right away. So I will start with a very simple and brief introduction just to remind ourselves what are nonlinear systems, why are they interesting, and I will also give you a few examples with a particular focus, of course, in ecology. Then I will show you how, for example, by using stream plots, we can actually understand the general behavior of the solutions of a nonlinear system without actually solving the equations' analytic help. Then we will be talking about stability in nonlinear systems. In particular, I will show you the two main tools that can be used to study the stability of equilibria in nonlinear differential equations. Of course, throughout all these topics, I will show you plenty of examples. We will also do a couple of very simple exercises. So we will get a practical sense of, we will get our hands, let's say, on the topic. So we can better understand what we're doing.  So please, at any time, if what I'm saying is not clear or if you have questions, please ask me. So if everything is clear for now, we can get started. So what we would like to do in general is, let's say, solve or understand something about nonlinear differential equations. Now, by nonlinear differential equations, we basically mean an equation or a system of equations that look like this. So with this function, f here is any nonlinear function. So this, for example, will all be simple cases. They can be in one dimension or in two dimensions here, where this function is nonlinear. So these are all simple examples of a nonlinear system. And they, I mean, they are interesting, and we want to be able to understand something about them, basically, because any interesting phenomenon in nature is described by a nonlinear differential equation. Let's see some example. One of the, I think, most simple, but also overlooked physical system that is actually described by a nonlinear differential equation is the pendulum. Now, you may be used from introductory physics courses to see this equation with sine of theta approximated by just theta. But that is only an approximation of the true, let's say, pendulum equation, which is actually a nonlinear one. Then another important case where of nonlinear system are fluids, because fluid dynamics is regulated by Navier-Stokes equations, which generally are nonlinear equations. For example, this is the equation for an incompressible fluid, where u here is the velocity field of the fluid. And this term, for example, we can see that is not linear. So again, fluid that an important example of nonlinear system. Now, I want to introduce two examples that we will consider over and over again in these tutorials, and that are actually relevant for ecology. The first one is the logistic growth equation. You may already have seen it somewhere at some point in your life. But basically, this equation simply describes the growth of a population x in a system with, let's say, a limited amount of resources. Now, of course, I'm going to show you this in more detail in a few slides. But basically, you see that if we didn't have this term here, this equation would basically be a simple linear differential equation. And so we would have that the population x here is growing exponentially without limit. But this term, as I will show you later, basically makes it impossible, sorry, for the population to be larger than k. So this system has a maximum population, let's say, that can be sustained. The other system that is relevant for ecology is log-cavoltaire equations. Again, this is a very famous system, which basically describes the dynamics of the population of a prey, which is x here and a predator, which is y here. And these parameters basically measure the interactions between these two populations. You see that in both cases here and here, the functions that define our differential equations are nonlinear. So these are indeed nonlinear systems. OK, so what we would like to do generally with, let's say, an ideal world with nonlinear differential equations would be to solve them analytically. But unfortunately, this is almost never possible because we don't have a theorem or a general recipe, let's say, that can give us a direction into how to solve any given nonlinear system. So it is reasonable to ask ourselves if and how we can understand something about nonlinear systems without actually solving the equations. Now what people generally want to do with nonlinear systems is studying their equilibria. Now I'm going to introduce definitions that probably are very familiar to you, but I just want to first refresh their meaning. And then I also want to build a small vocabulary on a nonlinear system that can be useful throughout this course. Now in general, if we have a general nonlinear system like this, a given point x star is said to be an equilibrium of the system if the function that defines our equations here is equal to 0. You see that this means basically that the equilibria of a system are the points where the variable doesn't change because if x is equal to x star, then f is 0, x dot is 0. And so the variable will not change once it is equal to this equilibrium here. Now what people generally want to do with equilibria is understand if they are stable or unstable. So we need some notion of stability and instability to study them. So let me introduce some very informal definitions of stability and instability. I will make these definitions a little bit more formal in a few slides, but I just want you to get an intuitive idea of what they mean. So in general, an equilibrium is said to be stable if any solution of the system that starts with that initial condition that is close enough to the equilibrium will always remain close to x star. Now of course, this notion of closeness, close enough is anything but rigorous, but still I just want you to get the intuitive idea from now. On the other hand, an equilibrium is said to be unstable if it is not stable. So if we can find solutions of the equations that start close to the equilibrium and eventually go away from it. Now in light of these concepts, we can basically reformulate our initial question as follows. So can we understand something about the stability of the equilibrium of a nonlinear system without actually solving its equation? The answer, of course, is yes. There are several ways, the several tools that we can use. And one of the simplest one, but also quite effective that we can use in this direction, is drawing stream plots, which basically means drawing the trajectories of the system in the state space. Now I think the easiest way to understand how stream plots work is to see how they are done practically. So let's start with a simple example. So assume we are given this differential equation, which is, of course, nonlinear, because this function here is a simple cubic function. So let's draw it. This function basically looks like this. And so we can see very easily, we can also factorize the expression in this way. So we can see very easily that the system has three equilibria in this case, which are the three points where this function is equal to 0. In particular, in this case, are minus 1, 0, and 2. Now the basic principle behind drawing stream plots is the following. Now wherever this function f here is positive, x dot will be positive. So the solutions that start from point where f is positive will be characterized by the fact that x is increasing. So for example, if we consider this interval here between minus 1 and 0, in this interval the function f is positive. So any solution that starts from these points here will be characterized by the fact that x is increasing. And the same will happen, for example, in this interval here. On the other hand, wherever f is negative, x dot will be negative. And so the solutions will be characterized by the fact that x is decreasing. So if we start from any point in this interval here, we will have that the solutions actually go towards the left and the same here. So in the end, what we can draw are these trajectories. So this is the stream plot of our system. And so you can see that by doing this very, very simple drawing, we can already guess which equilibria are stable and which are unstable. In particular, we can guess that this equilibrium here is unstable because you see that if we take any initial condition that is slightly larger than 2, we will have that this solution basically increases without limit. And something similar happens here if we take an initial condition that is likely lower than minus 1. On the contrary, we see that solutions here are actually going towards the equilibrium in 0. So we can guess that this equilibrium is stable. Let's see another. Is everything clear here? OK. I think it's a good time if you have a question. Exactly. Or ask a question, too. Please remember that this is a tutorial. So this is really meant to have everyone being on the same page on the top. So if something isn't clear, don't hesitate to ask a question. OK? OK. So let's see basically the same thing applied into a different case. So let's consider this equation. x dot equal cosine of x. So in this case, we know that our function here looks like this. So we will have basically an equilibrium in every positive and negative on the multiple of by halves, because these are the zeros of the cosine function. Then if you apply the same principle as before, we will have, for example, that in this interval here, in this interval here, and in this interval here, the function is positive. So the solutions will go in towards the right. On the other hand, in this interval here and in this interval here, the function is negative. And so the solutions will be going towards the left. So in the end, the stream plot that we can draw of this system looks like this. So you see that in this case, we basically have an alternation of stable and unstable equilibria in our state space, which, of course, is the x-axis here, because this system is a unidimensional system. We only have x as a variable. Let me now show you basically how we can use these stream plots in cases that are ecologically relevant. The first thing I want to show you is the stream plot of the logistic equation. Now, actually, this is one of the few lucky cases where we can actually solve an equation analytically. So I hope that with this example, it will be clear that, I mean, comparing what we see with the stream plots with what we see with the analytical solution, the stream plots can actually help us understand the general behavior of the solution of a nonlinear system. So first thing, let's try to, I mean, let me show you how these equations can be solved analytically. So let me write here in the whiteboard the equation. OK. So to solve this equation, I'm going to use some physicist's tricks. So if you are a mathematician, I apologize because you will probably be horrified about what I'm going to do. But basically, what we can do in this case is we can separate the variables. So we bring everything that depends on x on one side and everything that depends on t on the other one. I'm sorry, this whiteboard is a little bit slow, but I hope you can see everything. So once we have done this, now this fraction here, we can actually decompose it into two terms. So we can write this as dx over x plus dx over k over 1 minus x over k. So you see that if we simply add these two fractions, we get exactly this. So we have this equation. But these are very simple terms to integrate because, for example, this one will be the logarithm of x plus a constant. This one will be minus the logarithm of 1 minus x over k plus a constant. And this here will be simply RT plus a constant. So in the end, we can rewrite everything as logarithm of x over 1 minus x over k equal to RT plus c, which basically means that x over 1 minus x over k is the exponential of RT plus c. Now, of course, we have to determine the value of this constant, so we can evaluate this expression for t equal to 0. And by simply rearranging basically the expression that we get in this way, the final analytical solution of the logistic equation looks like this. This is the analytical solution of this equation, where x0 is, of course, the initial condition on x. Now, you see that here we have an exponential with a negative exponent because remember that both r and k are positive parameters. So as time passes, so as t becomes larger and larger, we can neglect basically this term here. Then x0 and x0 cancel out. And so we can see that the solutions of this equation tend towards the value k. The only case in which this doesn't happen is when the initial condition 0 because you see that in this case, if x0 is equal to the numerator, is there a question? Yes? Yes, yes. So could I explain what is r and x and k? Yes, of course. Basically, the biological meaning of r is the growth rate of the species. You see that if we didn't have, let me write it down, if we didn't have the quadratic term, so if we had only this equation here, the solution of this equation would simply be constant times, would simply be an exponential. So in this case, we would have a population growing exponential. The biological meaning, on the other hand, of this other parameter, it's going to be clear in a few seconds. But basically, this is the maximum population that the system can sustain. Because let me draw basically how the solutions are. So for example, if we start from x0 equal to 0, as I just told you, basically, the solution is constantly equal to 0. So this is a trivial case because it basically describes a system with no population. Something similar happens also when x0 is equal to k because you see that here we would have k minus k. So this is always equal to 0. And again, this and this cancel out. But then if we take any initial condition between these two values, this function basically behaves like this. So we have a population that grows. And the growth, the initial growth here is actually an exponential growth. Because you see that when x is very small, we could neglect it. So at the very beginning, the growth is well approximated by an exponential. But eventually, this population saturates towards k. On the other hand, if the initial condition is larger than k, what happens is that the population quickly decays towards k. So basically, the meaning of this parameter is the maximum population that the system can sustain. Because basically, the idea behind the logistic growth equation is that the population here, this could be microorganism, could be animals, could be anything. But it is in an environment with limited resources. So at a certain point, the system will reach a population that is not sustainable anymore. And so x cannot grow larger than k. And even if x is driven to a value larger than k, then it quickly decays back to k. Does that answer your question? I mean, was I clear enough? Yeah, yeah. Please request to all. Please don't put the message regarding the witnesses. OK. Sorry? OK. OK, it's correct. It's OK. OK. So the fact that k here is basically the maximum population of the system is also why this parameter is called the carrying capacity of the system. So now we know how these solutions work. In particular, if we look at what is happening in the state space along these axes, you see that these solutions here basically are going towards k. And in the same way, these solutions here are going down towards k. So let's see if we could understand this general behavior already from the stream plot. So again, this is our question. The function, the non-linear function that describes the system now is simply a parabola. And so we can draw it. Now, of course, here I am not considering the negative part of the state space simply because this variable here represents a population. So it makes sense only when it is non-negative. So this is the aspect of this function. So we see immediately that we have two equilibria, in the int, 0 and k. So now let's apply what I've shown you before for stream plots. In this interval here, the function f is positive. And so the solutions, the trajectories of the solutions will go from 0 to k. On the other hand, in this other part of the state space, the function here is negative. And so the solutions will move again towards k. So in the end, the stream plot of the system is like this, from which we can also guess that this equilibrium here is unstable because the solutions are moving away from it, while this equilibrium here is stable. So you see that if we look at what is happening in the state space, we are actually recovering the same behavior. So of course, by using the stream plots, we can't say, for example, how quickly these trajectories are moving or the exact way in which this is happening. We couldn't guess, for example, that there is an exponential, like I've shown you before, that regulates the speed of this motion. But again, without even trying to find the analytical solutions, we can see how the solutions are moving in general. Now, if everything is clear here, so if there are no further questions, there is a question. Yes, of course, of course. What does it mean of unstable and also stable? By unstable and unstable, in this case, I'm meaning the very informal definition that I have given a few slides ago. So unstable, in this case, means that the solution are moving away from the equilibrium. Because you see that in this case, if you pick any initial condition that is slightly larger than zero, the solutions will eventually go away from this equilibrium. While, on the other hand, stable equilibrium means that the solutions are remaining close to the equilibrium. So in this particular case, for K, for example, we have that the solutions are actually going towards the equilibrium. But this is not necessary for the equilibrium to be stable. So this is a particular case of stable equilibrium. In general, for an equilibrium to be stable, we just need that the solutions remain close to it. Does that answer your question? Yeah, thank you, thank you. OK, no problem. So I hope it is clear with this example that what we are seeing with the stream plot makes actually sense if we compare it to the analytical solution. Can I ask? Yes, of course. So ecologically, it means that the K is the point where it achieves equilibrium, right? So there's no resource limitation, like we can say, for the growth. No, the resource limitation is in the fact that this population is not growing exponentially. If we had a system where we had an unlimited amount of resources, basically here we would have only an exponential growth with a fixed growth rate. OK, OK. What happens in this case is that K is the maximum population that can be sustained by the system, meaning that if for any reason the population grows larger than K, there are no enough resources to sustain that population. So you see that eventually the population will go back to K. OK. So in this case, in this sense, there is a resource limitation. OK. OK. So what we can do now is basically apply the same thing so we can try to draw the stream plots in the case of the log-cable data system. So our equations in this case are this and this is our state space. Again, here I am considering only positive values for x and y because they represent populations. So it doesn't really make sense to look at the other quadrants of the state space here. So if we look at these equations here, we can find very easily two equilibria, which are the origin and this non-trivial equilibrium. Now the origin is very easy to see why this is an equilibrium because if x and y are both equal to 0, then both x dot and y dot here are 0. The other non-trivial equilibrium is very easy. Let me show you. For example, we have that x dot is equal to alpha x minus beta xy. So if we want this to be equal to 0, we can rewrite this as x times alpha minus beta y equal to 0. So if x is not 0, we simply have that y is alpha over beta. If we do this exact thing with the other equation, we get that the equilibrium in the end is this point here. Now let's try to draw the stream plot. The first thing that we can do is see how the system behaves on this axis here. So for example, if we take an initial condition where x0 is equal to 0, so we start from the y-axis here, you see that there is a question. Sorry. Am I having a problem with the audios? No. So yeah, I think I can hear you. Yes. Can you hear me? Can you hear me? OK. OK. What if the k is not constant? So what do we do? I'm sorry, I'm not hearing you well. There is a little bit of background noise. I think the question was what if the carrying capacity is not constant? What if the carrying capacity here? OK, you mean here. Well, in this case, we would have basically a different system. k could be a function of x if you want or a function of any other variable. But I mean, it would simply be a different system. It would require a different stream plot. So we can use the same tools, but it would be simply a different system. OK. So where were we? OK, so if we take an initial condition, for example, that is on the y-axis here for the log table data system, you see that our system reduced to this. So the solutions basically will be decaying exponentially towards the origin. Remember that all these parameters here are positive, so minus gamma y is negative. On the other hand, if we take any solution that starts on the x-axis, so if y0 is equal to 0, we can write our system like this. And so the solutions will be growing exponentially on the x-axis. So the first thing that we can draw about the streamplots are these trajectories here. And notice that it would be enough for us to guess that the origin here is an unstable equilibrium. Because you see that along this direction, we have solutions that actually go away from the equilibrium. So by definition, this equilibrium here would be unstable. So let's try to see what happens in the rest of the state space. What we can do is see exactly when the components of this function here are positive and when they are not. So for example, if we look at when x-dot is positive, we have, again, as I've shown you before, that this must be positive. Now, we are not on the y-axis, so x is different than 0. And in the end, we get that y must be lower than alpha or beta. So we can divide the state space into two regions, one where x-dot is negative and one where x-dot is positive. Similarly, if we look at where y-dot is positive, we get that this is true when x is larger than gamma over delta. So in the end, we can basically divide our state space in four regions. And for each of these one, we know that x-dot and y-dot is other positive or negative. This basically means that we are able to see the general direction, let's say, towards which the solutions are pointing. For example, here, if x-dot and y-dot are both positive, it means that the solution is growing both in x and in y. So in general, the solution is pointing in this direction. And this is true also for these other areas. So you see that we haven't even tried to solve the equations, but we can guess already that the solutions of the Lord Cavalterra system oscillate around this equilibrium. Of course, we don't know exactly how this happens, but we know that this is happening. Is that clear? Are there questions? OK. So I hope it is. Sorry, I was muted. So I hope it is clear now that stream plots are actually useful to understand the general behavior of the solutions of an linear system. But of course, the powerfulness of this approach is limited because we can't always understand something about the stability of the equilibrium. For example, if we consider this case here, we know that the solutions are oscillating, but we can't say anything on the stability of this equilibrium because we could have the solutions here are spiraling towards the equilibrium. We could have the solutions are spiraling away from the equilibrium, or we could have any kind of behavior. In this case, we can't say anything about the stability of the equilibrium. So how we can study in general the stability of equilibria in nonlinear system? There are two main tools that can be used in this direction, which are Lyapunov functions and spectral analysis, which is also known as simply linearization. Now, is anything clear up to now? OK. So before I go on and I talk about how to use the Lyapunov function, I want to introduce the formal version of the definitions that I have given you before about stable and unstable equilibrium. So we consider a generic nonlinear differential equation and assume that we know x star is an equilibrium. Now, using the language of mathematician, now I'm not a mathematician, so I won't be very formal. But I just want you to let you know how the formal definition of stability is. So this equilibrium in the language of mathematical is said to be stable if for every neighborhood A of x star, there is a neighborhood B in A such that the solution starting from points in B will always remain in A. Now, this is just the formal way to say what I told you before, which means that an equilibrium is stable if every solution starting close to it always remain close to it, where the notion of closeness in the language of mathematicians is given by the use of neighborhoods. So let me show you graphically because I think it's easier to understand this way. Now, assume we have an equilibrium here. Now, this equilibrium will be stable if for every choice that we can make of a neighborhood A, so a set on the state space that contains this point, we can always find a smaller set B. And if we pick any point inside here and we use it as the initial condition of our nonlinear system, the solution starting from here it can move around all it wants, but it will never go out of A. So it will always remains close to the equilibrium in this sense. Now, if this is true, I want to introduce a couple of other definitions of stability because I mean, this way we can build this vocabulary that will be useful throughout the school. Now, if this definition of stability is true, not only let's say in the future, so looking at how the solution behaves for positive time, but if this is true also for negative time, we say that this equilibrium here is not only stable, but stable at all times. On the other hand, if this solution here, instead of going around this set A here, at a certain point, it moves toward the equilibrium. So if the limit of the solutions is the equilibrium, we say that the equilibrium is asymptotically stable. So the difference between a simply stable equilibrium and an asymptotically stable equilibrium is that in the asymptotically stable equilibrium, we know that the solutions are moving towards the equilibrium. While if an equilibrium is simply stable, this doesn't necessarily happen. Is this clear? Are there questions? Okay. So the other side of the coin is instability. So again, using the language of mathematician, an equilibrium is said to be unstable if it is not stable. So if we can find at least one neighborhood of X star such that for any choice of a smaller neighborhood, there is always at least one initial condition such that its solution goes out of A. Which again, it's just... Yes? Yes? So Shamolina, sorry for misspelling the name. You have raised the end. Yeah, I have a question. I could not understand the difference between asymptotically stable and normal stable equilibrium. Of course. The difference basically is in the fact that if an equilibrium is asymptotically stable, it means that we know that the solution is actually going towards the equilibrium. Yes. While if an equilibrium is just stable, so not as stable, but not asymptotically stable, then the solution is not going towards the equilibrium. It can be oscillating, it can be going around the equilibrium, but it's not actually going towards the equilibrium. Then why you are calling it stable? If you don't know its fate, I mean, ultimately when it will go towards that equilibrium, then we can call it stable, right? Then in what sense you are calling it stable? Yeah, I'm calling it stable in the sense that the solution is not going away from the equilibrium. So the idea of a stable equilibrium is an equilibrium where the solutions are always remaining close to it. If on top of remaining close to it, the solutions are actually moving towards the equilibrium, then we call the equilibrium asymptotically stable. Does that answer your question? No, I actually don't know that how could I differentiate? They say if I have a set of dynamical equations and then I solve that and say after a long period of time, my simulation shows that it is approaching to equilibrium. Then it's asymptotically stable. But how could I know from the graphical simulation that it is stable? Because in my system, I don't know where is the boundary of this B and A thing. This is mathematical definition. Yes, yes. In this case, I mean, of course it depends on the system you are considering, but I would say that you could see if the equilibrium is actually stable by using different initial conditions. So if I don't know, you guess, for example, from your simulations that X star here has a particular value, you start sampling some solutions around this value. So for example, if we have just to make things simpler, if we are in one dimension, let's say you guess from your simulation that X is more or less equal to one, then you could sample several points around one, so I don't know, 0.75, 0.80, 0.85, then 1.05, 1.10, and see how these solutions behave. If you see for example... Like starting from those initial conditions, exactly, you take like, I don't know, four, five, 10 points close to the equilibrium and you see how the solutions starting from these points behave. If you see for example, that all of them are going towards the equilibrium, then you have a good indication of the fact that the equilibrium is asymptotically stable. So going towards the equilibrium means like, how long time or there is no time sense here? No, in this case, there is no time scale here. I mean, the mathematical requirement is that for T going to infinity, the solutions goes to the equilibrium. So you don't have any kind of measure in this sense. Oh, then we have always played to the asymptotically stable thing. We cannot... That means we always see the asymptotically stable equilibrium, never the stable equilibrium. It depends. I'm going to show you some examples in this tutorial, but for example, if we have an oscillating system, so for example, if we have... Okay, so if we have, for example, I mean, I will show you the Laudka-Volterra system or for example, a pendulum. So assume this... Excuse me? Yes? Leonardo, can you hear me? Yes, I can hear you. At the last slide, can I go back to it? Yes. Yeah, so you have written X star is stable not only for all T greater than or equal to zero, but for all T included in R, right? But time for the sake of its definition, it is supposed to be contained in R plus, right? Yes, I mean, physically, yes, mathematically, not necessarily, meaning that... I mean, if you... Let me write it this way. If you have a system like this, so X of T, sorry, X of T equal to F of X of T, yes, you would say that I can solve this for T greater or larger than zero, but then I could define, I don't know, tau equal to minus T and solve this system here, X of tau equal to F of X of tau. This is another nonlinear differential equation which can be solved, but when tau here is larger than zero, T is lower than zero. I don't know if I'm being clear enough. So in any case... It is supposed to be a transformation on T, but T itself cannot be negative.  If you want, I can change what I'm saying here by exactly what I've written here. So if you can say that the equilibrium is stable for T larger than zero, and then it is also stable if you make this change of variables, then it is stable at all times. Okay, thank you. No problem. So about what the question I was answering before, so for example, if we have... This is our system, this is our equilibrium. So for example, if we have a solution that goes in circles like this, this is, for example, what happens for the pendulum equation when we approximate it for very small angles. So you see that in this case, the solution is going around the equilibrium, but it's never going close to it or far from it. So in this case, the equilibrium is not asymptotically stable, because the solution is not... Now I understand the periodic orbit. Yeah, because the solution is not doing something like that, so the equilibrium is not asymptotically stable, but still the solution is not going away from the equilibrium. Yes, yes, yes. Okay, is that clear? Yeah, thank you. So... Just another fact, Leonardo. Yes, yes. So actually, in the linear stability analysis, you are always constrained to near about these stable states, stable or unstable states. Yes. Never far from it. And then you have to invoke this large deviation, right? So it has to be all the way... Then you have to invoke, if you far enough, you have to invoke large deviation. Yes, I mean, it depends on the system you are studying, but I'm going to show you something about it, but please continue. Please. So again, all the time, whenever you are applying these analytical results to conclude something, you are again very close enough to your steady states. Yes. So this asymptotically stability always matters. Yes, thank you. This is also that we can say the price that we have to pay for not being able to solve the equations analytically in general. So of course, if we would be able to solve analytically on non-linear differential equations, we would be able to say everything globally, basically, about an equilibrium, but since we cannot do that, we have to restrict ourselves to the points of the state space which are close to the equilibrium. Does that answer your question? Yeah, please go ahead. Okay, thank you. So I was saying the other side of the coin is instability. So graphically, what happens is what I was saying before. So an equilibrium will be unstable if I can find some initial conditions for which the solutions will eventually go away from the equilibrium. When in this case, go away is the idea of going away is given by neighborhoods. So if there are no further question, what time is it? Okay, so we still have 10 minutes more or less. If there are no other questions, okay, I can start introducing liapunov functions. So okay, now one of the tools that I have anticipated you we can use to study the stability of equilibria in a linear system are liapunov functions. So let me show you the basic principle that is behind using liapunov functions so that things will be more clear. So let's assume we have a system like this, a nonlinear system and we know that X star is an equilibrium. Now here I am drawing this in one dimension simply because it's easier but what I'm saying is true in any number of dimension. So assume now that we are somehow able, I'm going to explain you how we can be able to do so in a few slides but assume we are able to define in a neighborhood of this point a function w that has a minimum in X star in the equilibrium. Now, if we had the solutions of the system and we computed the value of this function along the trajectories of the system, for example, we could find out that this function is decreasing along the trajectories. So for example, we would be in a situation like this. So again, let me repeat, we define this function here in the proximity of an equilibrium, then we take the trajectories of the system and we compute this function along the trajectories. I'm going to explain you, of course, how we can do that. If, for example, we find out that this function is decreasing along the trajectories, so the time derivative of this function here is negative, then we will be in a situation like this. So you see that necessarily this means that our solutions are moving towards the equilibrium. And this happens because we know that the function here has a minimum in X star. So this would mean in this case that the equilibrium of the system here is asymptotically stable because we know that the solutions are going towards it. On the other hand, if, for example, we find out that this function here is increasing along the trajectory, so if the time derivative of this function is strictly positive, then we are in a situation like this. And so you see that in this case, necessarily the solutions are going away from the equilibrium. So in this case, we could say that this equilibrium is unstable. Is that clear? Okay. So now what I've... Yes? The question by Mgaka. Hello. Are you approaching from just one side? Yeah, of course I'm showing you just from one side, but I mean, if you flip this image vertically, you see that the same is true also from the other side. If we are in this situation, sorry. If you are in this situation, so if the solutions are decreasing, sorry. If the value of the function is decreasing, you see that if I started here, let me draw that probably this will be a little bit clearer. So if this is my situation, this is my equilibrium X star. So let's say that I can define this function here. Let me draw it a little bit better. So you see that if I start here, for example, and I find out that the value of the function is decreasing along the trajectories, then I will have exactly the same situation. So the solution will be approaching the equilibrium from the other side. Does that answer your question? Okay, great. So, okay. So what I've told you now is absolutely non-Rigorus. The rig, I mean, what I've said, of course can be made mathematically rigorus. And this is done by what is called the Lyapunov's second theorem. This theorem basically states that if we are exactly in the situation that I have just described, so a general nonlinear system with an equilibrium and this function defined here with a minimum in the equilibrium, then if the time derivative of this function along the trajectories is equal to zero, the theorem says that X star, the equilibrium will be stable at all times. So if this function here is constant along the trajectories, this equilibrium is stable at all times. On the other hand, if the time derivative is non-positive, so it's either negative or equal to zero, the equilibrium is stable. If it is strictly negative, sorry, the equilibrium is asymptotically stable. And if it is strictly positive, the equilibrium is unstable. Finally, a function that I have in, I mean a function like this that I have just introduced is simply called the Lyapunov function for this equilibrium. Now, we still have five minutes, 10 minutes, more or less. It's about 10 minutes. So, I mean, perhaps we can see if there are more questions from the audience. Yeah, absolutely, absolutely. I mean, it depends on whether, because Leonardo is giving a second part of this tutorial tomorrow. So, Leonardo on whether you want to, how you want to divide it into parts. But I would say, let's see if... Yeah, absolutely. So, I can stop here, no problem. Okay. So, are there questions, clarifications, something that is unclear? So, I have one question. Yes. Yes, please. My question is that in this case, the equilibrium is the minimum. Is this property is useful for when we have an equilibrium is a maximum. Yes, I mean, you could write basically the same exact theorem when the function has a maximum in the equilibrium instead of a minimum. Of course, you would have to change the names here because of course, if here we have a function with a minimum in the equilibrium and we find, for example, that the time derivative is negative, then we are moving, let me draw this, sorry. Probably this will be more clear. So, what I've just told you in the slides is basically that if we are in this situation, so I have my equilibrium and I have this function here with a minimum. So, what I've shown you in this case is that if the function decreases along the trajectories, then we are moving towards the equilibrium. So, the equilibrium is asymptotically stable. But for example, if we decided to define the function differently, so instead of having, sorry, a minimum in the equilibrium, it had a maximum, sorry. Okay, so the situation looks a little bit like this. You see that the equilibrium in this case would be asymptotically stable only if the value of the function increases along the trajectories. Because in this case, we would have that the function is increasing like this. So, we are actually moving towards the equilibrium. So, yes, we could very well do the exact same thing with a function that has a maximum on the equilibrium, provided that we change the definitions here accordingly. Is that clear? Mohammed, is that clear the question, the answer? Yeah, it is clear, it is clear. Then there is a question from Juan Jose. Please unmute yourself. Yes, thank you. I wanted to ask you if when we are looking for an attractor of a chaotic system, is there around those trajectories that are closed in a kind of space near the equilibrium points, there are we talking about the same notion of equilibrium or it's a bit different. You know, I would say that it is similar, but the notion of attractor is actually wider, let's say than simple equilibrium because by equilibrium, we mean a point, a specific point in the state space while an attractor can be something more, I mean, it can be also a different set because for example, I don't know if you have ever heard of cycle limits, limit cycles, sorry, but there are some dynamical systems, for example, where we find out that there is, I mean, I'm just trying an example, there is, for example, this circumference and any solution that starts, for example, here, instead of tending towards a particular point, it will tend towards this trajectory. So the solution will do something like this, sorry, I am terrible at drawing, but the idea is that the solution, instead of going towards a particular point, it goes towards a trajectory. Now, the intuitive idea is the same, so if we have that the solution instead of tending towards a point, it tends towards a trajectory, I mean, the idea is the same, but mathematically these things are very different and there is a whole different set of tools that can be used to study them. Thank you very much. No problem. The second line for asking a question was Ayan, Ayan? Yeah, I can hear you. Yes, yes. So I was just wondering if we go back from dynamic systems theory to study ecological networks, the concept of equilibrium and steady state, so non-equilibrium is something when you have this exchange of energy, right? Yes. So then, which I guess is always the case in ecological networks. So the proper terminology over there maybe is to use steady states or unstable steady states, rather than using these equilibrium states. Yes, when we talk about thermodynamically open systems, let's say, so system where there's an input of matter or energy or any kind of thermodynamic quantities like ecosystems, technically we should say steady states and not really equilibrium because there is no equilibrium. Yes, so you are right. Thank you. Okay, there was another question by Pablo Liacciano. Hello. So, but when you're talking about not being able to solve the linear system and then in order for you to talk about stability, you have to talk about stability in a region near equilibrium. Why is that? Is that because if you don't know how to solve the system, would you linearize it around the equilibrium or? Yes, exactly. I mean, if you don't, I mean, if we would know the full analytical solution like in the case I've shown you a few slides ago on the logistic equation. So if we could find the analytic solution like this, like this here, we could say anything that we can say about the system. We have like the maximum information possible on the system. But when this is not possible and this is not possible almost always, basically, we have to find ways to determine the maximal information which necessarily cannot be the maximum information because we cannot solve the system. So we have to do our best and often doing our best is studying the system close to equilibria. Did that answer your question? Yes, thank you. No problem. Great. Next in line is Debas Mita. Can you hear me? Yes. Hello. Yeah. So my question is that construction of this Lyapunov function, like whenever we change the model, so there I face problem with construction of Lyapunov function. So can you suggest some techniques that, some base techniques to assume what kind of Lyapunov function we can assume for to test the stability of a participant? Sorry, I lost the audio for a moment. Can you repeat the question? Yeah, so I'm asking about construction of the Lyapunov function for any participant. Yeah, I mean, this is the great problem of this approach. I mean, as you can see, it's actually very powerful because you can tell us a lot about an equilibrium but the problem of this approach is exactly building a Lyapunov function. In general, it is not easy at all to find the Lyapunov function for any generic system. So we have to be either lucky or we have to use some intuition to build the Lyapunov function. So there is always this kind of trade-off. So we have a very powerful tool that this powerful tool is not easy to use and it's not possible to use it always. Okay, thank you. Then there was, okay. Sorry, you were saying something, Devas Mita? Yeah, yeah, it's fine. So then there was a question by Samson, what I need. All right, thank you very much for giving me opportunity to ask my question. First of all, I want to appreciate the presenter for this wonderful presentation. My question goes straight like this. Where can we have a question of global asymptotic stability? Sorry, I don't think I heard correctly because of some audio problems. What was your question? I'm talking about, my question is tended towards when are you going to have a global asymptotic stability? Well, it depends. A case when we can say globally that an equilibrium is asymptotically stable, for example, our systems with concert quantities, for example, if we have a particle in a potential, so a very simple physical system, this particle can move only on the x-axis and we know that it is subject to a given potential, energy potential V. And for example, we know that this potential, for example, is a parabola. So this is defined on all the x-axis. So we know globally how the potential work. We know that, I mean, from physics, we know that the equilibrium, the equilibria of the system are the minima of the energy potential. So in this case, since this is a parabola, we only have a minimum. And so in this case, we can say, okay, there is one asymptotically stable equilibrium and this is the only one that there is in the system. So, I mean, this is a lucky case in which we can tell anything about the system globally and not only locally. Did that answer my question? Oh, thank you. Yes? I think that we're right. I'm satisfied. Oh, okay, thank you. So there is a question which is quite popular in the chat and I think could be the last question of today's session. So the question is, is there any general principle guiding us to the construction of a Lyapunov function? No, this is the great problem of this approach. There is no general principle. We have to be lucky or smart. I mean, there is one big exception. I'm going to talk about this tomorrow, but if we know, for example, that we have a system with a conserved quantity. So if we, for any reason, know that there is, like I've shown you before, potential energy or any kind of quantity that is conserved. Generally, these conserved quantities are a good first guess for a Lyapunov function. This is not always true. We have, every time we have to check that they are a good Lyapunov functions, but generally when we have conserved quantities, these can be good Lyapunov functions. If we don't have conserved quantities, then we are completely on our own and we have to find them on our own, which can be difficult or even outright impossible in some cases. Okay, great. So I would say that this is the perfect time, the end of today's session, today's tutorial. So as a reminder, Leonardo will give the second part of this tutorial tomorrow at the same time. So you can also watch again toward, I mean, as many times as you want, the tutorial that you'll navigate today on YouTube. So if you want to, you need something and you want to watch it again, please do. So the next slot in the next lecture by Joshua Weitz is starting in about 13 minutes. So what we're gonna do now is to splitting randomly assigned breaking rooms. You are free to stay in the breaking rooms, chat with whoever you are assigned randomly to.  I guess Leonardo is staying with us, so you can also. Yes, absolutely. More informally, but also you are free, of course, to stretch your legs, to get a cup of coffee or take a break from the meeting. So with that, since I'm not host, if I'm ready. Okay, yes, I give you. Here we go. Yes. Okay, 20 breakout rooms, is it okay? Yes, okay. Cause 150 people. Yes. Okay. Can someone on the ICTP side, if you can hear me, let me know. Yes. I have answered you in the chat. So your sharing is good. Okay. You can see your sharing very good. So it's okay. I just remember you that reminded that we are now live on streaming. Yeah, no, I understand. Okay. On streaming. Hello, streaming people. Yeah, perfect. Great. Hello, you need help? Breakout rooms. I mean, there is only one breakout room that is room number 15 where I am assigned. But can I switch the rooms or can I see the available rooms? You can leave the room. And then I am going back to the main station, but I cannot see the other rooms or... If you want, I can move you. Do you have a bottom breakout room on the bottom of your bar? Bottom bar? It maybe depends on the version of your Zoom. It's after five. You have to go at least version number five. What is this Zoom version number? Do you want that they move you in another... Yes, actually I want to ask a question to the person who was giving lecture. I think he's out of the breakout room. So just leave the room. Oh, okay. Okay. Give breakout room, just on. Hey, Josh. Can you hear me? Hi. Yes, I can hear you. Oh, great. Okay, so I was... Because we have the participants paired randomly in group, randomly in meeting groups. So I was in one of those. Okay, that's fine. I'm sorry to find you. So how is it going? It's going fine. You have a big crowd. I think we're live on streaming, so... Okay. Oh. Yeah, just keep that in mind. Hello. Everyone who's watching us have this chit chat. I see. Okay. Great. Yeah. If you want to discuss anything before we start, we can go to a breakout room. But otherwise, I think I'm set now. I'll leave with you the coffee. I think you'll check that. Okay, great. Thanks.


Welcome back to my channel guys. It's your girl Coco styles and on this channel. We talk about beauty fashion and lifestyle I also run a blog called Coco styles NY so go check that out and I write a post every week for women of All races for fashion for tall women and some lifestyle stuff also because I am a vegan So I do push that a lot for women But today I have 10 tips for you guys that are 10 tips for healthy habits that every woman should have I'm gonna say Every woman, but I'm talking to my 40 plus women because I'm 40 And I'm well, I'm not 40. I'm sorry. I'm 41 I'm gonna jump right into it. These are pretty simple and this is something that I think that anybody can kind of put into their routine Even if they have children, you know Even if they have husband and you know, they're kind of busy with home stuff or even if they have a nine to five I truly believe that women of all ages should take some time for themselves Every single day or once a week or even once a month depending on your life's jump right into it So tip number one would be And I do this I've been doing this for some years now and it helps me So I want to share it. So Tip number one would be to before your feet hit the floor out of your bed Always give thanks. This is even before you actually pray or meditate Before your heat before your heat before your feet hit that floor give thanks to God give thanks to the universe or whoever and Just be grateful. Have gratitude. Look around your room. Look around your home and just be thankful for what you are blessed with. I Think that a lot of people To are not They don't have enough gratitude and they think things are just given to them just because but honestly, it's it's not So I would always recommend being grateful And you will be rewarded the universe loves gratitude and so does God So that's tip number one is to before you get out of bed Before your feet hit the floor give thanks and be grateful for what you have number two Make your bed now. I have been making my bed since I was a kid and Personally, I think that if your bed is not made Good day is kind of like all over the place. Okay for me, I work from home and If it's not made I kind of feel like things are off in my house So it doesn't take me but five minutes even less to make my bed Pull the sheets over fluffy pillows. Boom. You're ready to go and you're out the door So I think that's a tip that millionaires use and I'll see if I can find like the Source but I've heard that a lot of millionaires make their beds if they can make their beds. That's okay. We okay guys so number three would be to pray Meditate or whatever it is that you do before you actually get your day started and I mean it doesn't have to be 20-30 minutes take five minutes before you get your day started because most of us have days where we're super busy And we're dealing with people from all over between traffic between work emails phone calls take some time for yourself and meditate or pray whatever it is that you do and just take a second and just Kind of just breathe and just get in touch with Everything okay before your day gets started and I think this works because like I said Most of us do have busy busy days and it's so easy for us to get wrapped up in the day-to-day crap and Then we're home and we're tired, but we've never taken the time for ourselves So always feed your soul before you feed anybody else's soul because if you're not good Then they're not good. So that is number three and Number four would be to read books. I think reading books is a total total Healthy habit for women of any age being knowledgeable on things, you know, it's something that I am Well, I will always fight for and push because at the end of the day, you know Things are happening every day in this world and we're not knowledgeable of some of the stuff. I eat the vaccine You know, a lot of people are just jumping to take it because they're saying to take it But sometimes you have to take a step back and think about your life your health And will this be good for you in your lifestyle? It may or it may not So that's number four that is to read books right now. I'm actually reading The 48 Laws of Power and it's a really really good book and that's by Robert Green And it's really good I'm taking my time with it because there's so many things in this book that I'm just kind of taking back but I'm kind of like Twisting it into my own way where I can use it or whatever not in a bad way But you know, some of the stuff is it seems a little conniving But when you really think about it, you put yourself in situations. It's gonna help you So that's on the fourth. That's reading books number five would be clean eating Okay, guys, you know, I'm a vegan, but I am not gonna push you to Be a vegan. I would say try to start in taking more vegetables Less white rice more whole grains more water More smoothies Try to cut out as much sugar as you can And I will always say this cut the dairy out Cut it out ladies Cut it out. Cut it out. Okay. It's no good for us and You know as an African-American woman, I know for us is definitely not good I don't know. I just the cow's milk and the eggs and all that in the cheese It causes mucus in your body and this is how a lot of us Get sick. So cut the dairy out first and then kind of work on the other stuff but definitely bring in some more greens drink lots of water and Stay away from sugar guys. So that's number five is clean eating Number six would be have a skincare routine. It's a healthy habit As we get older our skin does not snap back like we would like it to snap back I'm 41 and My entire life. I've pretty much had pretty decent skin. I've never had any acne or anything like that My mom was really big on if you have a pimple. Don't touch it. Don't pick it. Don't pop it Don't even look at it.  It doesn't have to be expensive, but just put a little bit of time into Yourself again. So all of these tips are for you as a woman No matter what age you are for the young girls You can start young.  I had nail polish.  I'm very clean in here. I'm very structured Um, it's just how I was raised. So that will be number six. That is have a skincare routine All right. Number seven is workout Now I'm gonna tell you guys I think I've said this before I go to the gym, but I'm not like I don't go super hard. I go I go I'm banking on the way I eat honestly, but I do go. Um, I'll go three times a week And if I don't go through times a week is twice But that's because I you know for me the gym is not really fun I think I do well with a trainer and I do well with classes So if this is something that you struggle with also then I would recommend getting a trainer Um, they are pricey, but you can find somewhere they do like free classes sometimes Um, I know here in Atlanta. They do free boot camps all over the place Or they're like super dirt cheap and there's a trainer on every corner here So you wouldn't have a problem if you live here in Georgia So that is number seven. That's work out if you guys seem to be looking down and reading my paper Because I keep forgetting them. Um number eight is Really really important Have a clean bill of health now. We cannot Determine if we get certain illnesses and stuff like that But I would always say for the ladies because this is for you if you are 40 plus Please make sure you are getting your mammograms like you are supposed to Okay Now if you are Younger than 40 and you have a history Of breast cancer when I mean history.  That's a history. Okay So You should also be getting checked You tell your doctor. I need to get checked because I have a history of it They're not going to turn you away You tell them that and I will also recommend you guys getting the bracket test.  Make sure you are getting your annuals your physicals all that goes to going to the dentist All of that all of those things Affects how you feel and you feel good knowing your status on your your breast your status when it comes to Any stds all that good stuff So that's number eight having clean bill of health and being on top of your doctor visits. Okay number nine would be Therapy All right, a lot of people Kind of stay away from the word therapy. They don't like it Um, I am gonna say in the african-american community It's something that a lot of us do not do A lot of us will go to church and pray and all that good stuff But I also think that is really good to talk to somebody and they have if you're not comfortable with going to Someone of another race, which I really don't think Because some people say that they can't relate to them, but I don't know I feel like at the end of the day a therapist is a therapist Um, and you're just there to release and just get it off your chest Some of us are holding in things that we have buried for years From being a child So I think therapy is a good habit for us or even journaling taking time out To get the thoughts out of you and just kind of release them and let them go So that's number nine is therapy or even journaling guys and number 10 drum roll Is probably the hardest one Being determined, okay And a lot of people will say that they want to do something they're started then they'll stop Their start they'll do two three four months then they'll stop Until something happens in their life meaning Let's say a breast cancer scare, okay Now you want to go back and start eating properly Be determined know what you want at the end of it all remember that our bodies are shell, okay This is a shell. We've been given this body to pretty much Borrow, okay, and I always say take care of it the best that you can And let it work with your soul Your soul So those are my 10 healthy habits that every woman should have I would say 40 plus, but this is for all women of all ages All races. I think it's something that we all should start doing And again take time for yourself ladies and make sure you're healthy and keep on top of everything for yourself guys So that's it guys If you like videos like this, please comment below and if you stay to the end of this video, please comment Coco we see you and guys don't forget to hit that like comment and subscribe button and do not forget to go Follow me on instagram at cocoa underscore styles and why? Cocoa underscore styles underscore and why and go check out my blog at cocoa Styles and why so I do articles over there once a week and again It's for tall women But anybody can pretty much read the articles because I am Doing other stuff also on this so I will catch you guys in the next one and bye for now


The narcissist will infect everyone you know. The narcissist will try to contaminate everyone in your family, friends, social circle, workplace, everyone that you know, everyone that you talk to, everyone that you come into contact with. The narcissist will contaminate all of these people through exposure to their distorted way of thinking, their ideas and beliefs.   You're trusting your intuition and staying grounded in your own beliefs understanding of the situation. You might make it known to the narcissist that you're planning on leaving them. You might make it known to them that you no longer want anything to do with them but the narcissist cannot just let you go. They can't just allow you to move on and start a new life without them. When you start confronting them and talking about how you want to leave and move on, they feel as though they are losing control over you and the situation. They have to engage in these types of situations where they have control or an aggressive exercise of power over a person to boost their stature or feelings of self-worth. It gives them a sense of importance or reputation gained by their ability or achievement of locking you down and securing you as their source of supply. It is an activity or a way of behaving that makes them feel powerful. It is something that they do for the pleasure of using power to control you. When you are trying to leave the narcissist, in their minds, you are threatening their sense of power and control. They will begin to associate with the family members that you talk to, your friends, people in your social circle, people in your workplace, people that you know or talk to, anyone who you come into contact with. The narcissist will associate with these people and then convince them to turn against you. They will plant certain ideas, suggestions or ways of thinking into their minds. Although you may have made it clear to the narcissist that you want nothing to do with them and you wanted to move on.  In their minds, if you can move on and be happy with someone else, they have lost and they don't want to lose. The narcissist doesn't want to lose. They want to win. They cannot just allow you to leave them and then start talking about your experiences with them, with your family, friends or associates. The narcissist intentionally establishes connections with the people around you so that when you do decide to move on, you have to detach all of the connections that they have made. This is designed to isolate you. It is designed to separate you from your family, friends and associates. They don't want you to be with anyone. They want you to be completely alone. They want to leave you alone to yourself. This gives them the ability to be your only source of influence. It gives them the full capacity to have an effect on your character, development and behavior. They can make you believe whatever they want in these isolated conditions because there is no one around to tell you otherwise. There is no one around to validate or correct you. Your only source of influence is the narcissist and they are able to make you believe whatever they want and they can mold your character, development and behavior into whatever they want it to be.  This is why they want to isolate you. This is why they don't want anyone to believe you. They don't want anyone to be on your side. They put so much effort into going behind your back and telling people about the conflicts or disagreements that you have had with them. They have expressed it in a way that makes you look like the bad person. It makes you look like a person that your family, friends and associates would no longer desire to associate with. They put so much effort into doing that because they want you to be alone. They don't want to see you with anyone. This is designed to give them full control and influence over you. They are also very jealous individuals and they know this. They know that they're not going to be able to stand the thought or idea of you being with someone else, whether it's a relationship partner or even just a friend. They don't like the thought or idea of someone taking it away from them because they already know that you are going to be far happier than you ever were with them. That's why they went behind your back and started talking to people about you. This was done intentionally to separate you from anyone who could potentially help or support you. Someone who could befriend you or give you the love and care that they never did. Narcissists are known for pushing and provoking their victims until they finally react and then they play the victim. This is what they are using against you. This is what they are saying about you to other people. But they don't mention the fact that they pushed and provoked you until you finally reacted to them. They tell people that you are having these emotional outbursts for no apparent reason or they will make up something to go along with it. And then they play the victim as though they can't cope with your aggressive behaviour. They paint this picture as though you are the one who is abusing them. They start talking about you behind your back and asking people what you went through with them and how they dealt with it. It creates a relatable situation. It shows a connection between your reactions to the narcissist behaviour and any arguments or disagreements that you may have been through with a family, friends or associates. It is something where they are able to feel sympathy for the narcissist or identify with them especially if you have had disagreements with family members, friends or associates or if they were dysfunctional as well. It creates a relatable situation, a connection between them and they will talk about how they respond to you and how they are able to maintain control over you. That's how it can often seem as though they are all used in the same tactics or they all behave in the same way. Your family, friends or associates are able to relate to what the narcissist is saying about you. They are able to establish a bond or a connection over dealing with you and responding to you in a way where they are able to maintain control. The narcissist can become ingrained or enmeshed within your family or social circle. And it becomes very difficult to get them out of there. It's very difficult to get rid of them when they have involved themselves with so many people who you believed were close to you or on your side, especially when they have managed to brainwash those people and get them to turn against you. When you go to your family, friends or associates and try to tell them your side of the story, they are not going to hear it because the narcissist has already infected them with their distorted way of thinking, their ideas and beliefs and if you have ever confided in the narcissist about a certain situation that you have dealt with with a family member, friend or associate, they will have spoken about back to them as well to make it seem as though you are against them or you are talking about them. This is how they get people who you're close with to turn against you. Anything that you share with a narcissist will be used against you later down the line. They will use it to turn people against you. They enjoy it. They love turning people against each other. They love to cause harm or trouble to people who are once close together. They love to cause conflict and drama amongst people and then sit back and watch it play out. They make some feel powerful and as though they have the ability to control people's view of things and their emotions, they love to cause these kinds of problems for people, especially if they can't get their way and have it play out as they would like. Once they have infected the people around you, it becomes like a contagious disease which spreads from one person to another. Their emotions, feelings and attitudes spread and affect other people. It transmits or transfers. It goes from one person to the next person until everyone around you is infected by this and now you become the one who is different. You become the one with a different or opposing feeling or attitude regarding the situation. This is why it is so difficult to remove them once they have infected the people around you. Try to avoid explaining or defending yourself to these types of people.  Be yourself. Continue living your life as you normally would. Talk to someone who you can trust about the situation. Talk to someone who knows you. Someone who is going to understand and be sure to tell them everything that you experienced. Don't let your shame or embarrassment over the situation stop you from revealing everything that the narcissist has put you through. The narcissist is counting on your shame and embarrassment about them and the situation to prevent you from revealing what you had to go through. Talk to someone who you can trust. Talk to someone who knows you and be sure to tell them everything. Thank you for watching. I hope this video resonated with you. Please like, comment, share and subscribe. Click the bell icon to receive notifications for my future videos. If you're delighted tonight, my PayPal link is in the video description. Coaching Inquiries, you can email me at narxfathercoaching.com. Check out the immersion dice in the Narx Father store where you can purchase your own Narx Father t-shirt, tank top or mug. The link is in the video description. Thank you for watching and I'll talk to you soon.


What's up everybody welcome back to the channel I'm EvilRabbit here joined by MK6 and we are in the 2000s GTR Skylines here to do the Forza Thon because it is Forza Thon time once again I haven't done a Forza Thon in a while so we're here trying to get the splash down Danger Sign 3 stars to unlock the Nissan Skyline GTR Horizon Edition which I actually don't have one so I'm kind of excited to actually get one I've ever actually owned one.


ee of five judge patch awesome patch box two That's a nice card jumbo jersey auto of flattie migrero jr to 15 That's a nice hit Look you autograph von grissom out of 50 Local kid from ovito Will clark jersey auto out of 40 Game used jersey auto will clark Next up is a robin yount auto number to 25 There's a redemption coming up Guy rookie auto number to 10. Nolan gorman. That's a nice gorman there Started hitting better towards the end of the year This one's gross another awesome redemption dual auto green parallel mike schmidt brice harper Nice, that'll be nice green dual.  I don't know what player that is, but I'll show you Show hey, oh, tony what of one That's crazy way Nameplate relic. That's awesome. Come on, tom man. That's some uh, I'm not gonna sleep tonight Great Too amped up now Big show one of ones.  Tom is hitting monsters when he rips this stuff All right That does it man Two nights in a row just an insane card or in a few or a few in Uh definitive appreciate you sir the ghost's first name is probably Francis That sounds like a ghost name. Why are we talking to us because we make Puns and stuff that's impossible. Maybe Maybe he's not a fan of it. Maybe this ghost doesn't have a sense of humor What I call that Didn't think about that sorry I don't know. Let's think about this. I'll park up this Always thinking of this that far. It's too funny. It's true is I like the one where they play uh That one's always great. I can play a lot of these songs in a real guitar. Why don't we teach you kids how? This is real So I forgot to tell you donnie But speaking of this ghost Jake and I had an incident. We were like talking one of the studios I think it was I think it was the other studio. Okay, and those doors are heavy, right? They're like they're like a soundproof type door, right? So We're talking And I'll show you exactly what the door did Dude, it's it was so weird So the door might be like this And we're just like standing there.  I swear to god because I was sitting there and I'm like Jake and he turns around it's like where it's like right there Yeah, I just got chills thinking about that. That's that was the middle of the day Like yeah, that's insane I think it was the channel two one the thing that door that you have to move those doors They do not move these are not light No, like if if augusta win like hits that it's not really gonna blow if you walk into the door and it's kind of open It'll like hurt because of how heavy it is. That's true. You know what I mean? That's insane. Yeah, I I just I just remember that jakes. There's like a There's one of those text chats that jake showed the clip in and he was like Here we're running me at the the break the breaker studio door. Good lord That's scary. Not not great. Not good donny We've had a lot of experiences like I want I want to do some background Checking on on this property. Should we uh, should we hire somebody? Hire go sonny's How long is I am gonna stay healthy for? I hope forever, but I don't know you came out in 2020, right? 2019. Yeah, it's 2019 draft So he's probably like Like 24 25 He's still pretty young. Yep I mean his numbers when he's healthy are crazy like very very good But We'll see. Yep Everybody forced here ripping two boxes of 2023 Origins international blasters here for Matthew F. Let's go Matthew Let's find a uh big bang or one of the pedigrees the other one. Oh, yes. There it is. Perfect And startups and exclusive gridiron Blaster exclusive inserts. All right, here we go like a low-numbered rookie in here. That'd be cool Turkey so far you think Other than Stroud. Yeah, other than Stroud. Other than Stroud. Ah, this has been really good. I probably go ahead. Pijon needs the ball more Yeah, I don't know what they're doing in Atlanta. Nobody knows what they're doing in Atlanta Not even in Atlanta. They truly don't know what they're doing. You'd say rookie or she rice Uh Quentin Johnston On the uh original gridiron And him I don't know offensive rookie and it's tough Stetson Bennett to 299. It was the variation like the foil I think that's what that is. Looks like a nice Bennett With the jigba wicks and mims Are we saying Addison or Zay flowers downs has them good had downs is a good games I think ours been more consistent tank del tank del's been I think came out of nowhere So we are Gibson la porta are gonna kind of move up that way too. Give just scoring touchdowns now Yeah, brace young. I think that's a variation the black and white. I think probably Yeah, oh, yes his variation Okay, then tune scott startups of Addison vikings Fensely what like jail card Carter, uh, who is the weather spoon? Yeah, he's the goshdowns last grave do spawn original grad I don't know connell four to raiders Well connell, I think well could be okay, but he just needs more flying time New office. Yeah The bonnet channel 299 If he was healthy, he might be the the guy we were talking about.  I think so I forget what the injury was. I don't think it's serious, but I think it might be a couple games Burns swift garrots Johnston spears and startups of boot day All right, here we go. Let's get that sp He's a h-hanger, is he number? What's his number? Wait 28, right If it's 28, then yes, here's your number that Richardson's a 99. There we go colts. Nice, which is matthew Nice Richardson a couple boxes here. Yeah, h-hanger is the number Gibbs B. John and carter I thought it was 24 for some reason jsn on the variation Mayer mckay startups downs Richardson base josh downs Jake hanner original gridiron another number here is rookie dorian thompson robinson 149 not a bad time to get a dtr Yeah, that is first career one. I think stroud clifford and jaden reed Jaden reed the past like four or five weeks has been really good. Yeah, he's been salad I don't know. He was like what like second second rounder. Yeah So it's not like super surprising, right? Addison rookie vikings abanaconda start-up save flowers. I don't know where you went to college I can't remember The draft was like six years ago at this point Tillman tank del and cleaton tune All right, there we go matthew pretty good stuff man some great numbered rookies Get those out to you And that does it for tonight everybody What a spooky yeah the ghost that's for sure if you build late sports cards the ghosts will will come out All right, that'll do for tonight y'all 10 box tuesday start at uh 4 p.m. Eastern tomorrow Uh, wednesday's museum release day as well as hoops and nt collegiate football So get your spots everybody. We will see you tomorrow afternoon. Bye


What's going on everybody? I'm Sinaya and I'm here to bring you three quick picks for game one of the NBA finals between the Golden State Warriors and the Boston Celtics. So first and foremost you can give me Boston plus four on the spread. I love this number because the Celtics love playing in other people's arenas. They've actually won eight of their last 10 away games and four of the last five matchups against the Golden State Warriors. So of course you can go with the money line if you want a little bit of risk but I'm going to go a little safer and say plus four. Boston they're going to go into the Oracle Center on fire lit and ready to show not only why they deserve to be in the finals but why they deserve to be the champs. So my second pick for this game one is Marcus Smart over 5.5 assists. Not only has he hit this mark in each of the last two matchups against the Golden State Warriors but the Warriors are also giving up an average of seven assists to opposing point guards. Now he's got plenty of options for shooters Tatum Brown Horford Williams like the list goes on from the starters to the bench. So I think he's going to be playing a beautiful facilitating role in this game one and passing the ball out to some shooters who are going to make it happen. For my final pick I'm going with the Celtics team total over 104.5. Y'all know me I love a good trend and over the last five matchups the Warriors have not been able to hold the Celtics to anything less than 107. So with both of these teams being well rested they're going to come into this match fired up and it's going to be a very fast paced game. I see them hitting this mark pretty easily. Anyways take a look at my picks tail them or fade them best of luck on all of your bets and let us see some green flips.


If you have young kids, they are not protected unless you have a will and you have a living trust. Why do you need both? That's because each one does a different job in protecting your children. The will is where you name guardians for your children. The people that you select that you know will care for your children in the way you would. The trust spells out where the money is used to raise those children. In other words, two separate functions. One, who watches the children. Two is how is the money spent on the children? The guardians and the trustees can be the same people. So you could say, I have a trusted sister and I'd also like her to be the guardian. You know the qualities of the individuals that you want to choose. Is there one that handles finances better? Is there one that gives living care better? Maybe it's the same people. But again, the choice is yours. But you're not protected unless it's spelled out in a living trust-centered estate plan.