Okay, so now we are starting to continuing the actions by Professor Diaz Ramona. Okay, thanks. We're going on, but now turn to the opposite task. Where were we yesterday? Do you remember? We were talking about the linen cycles. Yesterday, the linen cycles, the topic was how to rule out that some dynamical signal has a good cycle. Vote tasks are not easy for the general system at all. So I just showed you yesterday some most frequently used methods to try to rule out some cycle from some dynamical system. So now we will see how we do the opposite task. So, to establish that there is an encyclical system. One of the most important results concerning the existence of the linen cycle, but as you will see at the end of this chapter, this is very important. More general sense, the result called Poincaré-Bendixson theorem. This theoretically very important result in the nonlinear dynamics, because due to lack of Poincaré-Bendixson theorem, there is no dimension. Then in plain, we are linked to the health theories that generally are the points. So in contrast to the process of linearization and apartment theorem, which I said, we have quite an empty dimension. So no matter what's the dimension of your learning system, you're always talking about meditation and art, you are always talking about the application of Poincaré-Bendixson theorem is already in the play. You don't have even in the space. So what is generally the Poincaré-Bendixson theorem it says. If we have a region which is closely bound and subset of the play, our vector field is, as always, continuously different. And Fort is the most important assumptions. It says that this region does not contain any fixed point. Okay. So inside the region, there is no fixed point and port. If there exists some trajectory that starts in that region. Then it will remain in that region for all the future time. Then either this is itself a closed orbit, or its spiral approach to some closed orbit. So either way, surely have the closed orbit. Okay. First, how we, in practice, applied in this theorem. So the first three assumptions are quite easy. So the first two just constructs a closed bounded region in the plane. The vector field as we discussed earlier is usually continuously fresh or we don't have problems in that direction. You exclude the fixed point from that region. But the most tricky one is how to track your trajectory inside some region. How to be sure that trajectory will not leave the region. So if it starts there, it will remain in that region for all the future time. That is basically what should be done. So in practice, we are constructing so-called the tracking region. And how to check that something is the trapping region. Yes, or try to construct it. But how to conclude that trapping region. So by plotting the vector fields on the boundary. Okay, that is one more reason why we at the beginning. I show you how to plot the vector field. Of course, by yourself, we use a computer software. And you also saw yesterday, according to some examples, we also use the vector field directions in order to rule out the close orbit. But also here, in order to prove the existence of the close orbit also the vector field directions are important. You check on the boundary if the direction field point inwards. Okay. On the both boundaries. So the trajectories cannot point that region. It's quite clear. Okay. So let's see. The examples. So you remember our first two example from the yesterday. When I plot you do the limit cycle. It was the basic examples. In all the books. Yesterday we concluded about the existence of the limit cycle by transforming the system to the power coordinate and then discuss. So on the left. The system. Our limit cycle here. And the vector field. This is the table that we used yesterday to establish the stability of the limit cycle but it also helps in constructing the trapping region. Yeah, because now you know that between zero and one the vector field is pointing to the right. So it's enough to take some cycle here example with radius one half. That is what I plotted there. You have this green green field and inside this edge to take one radius bigger than one. That is what I wanted here. As you see the vector field. The ones that the trajectory enters in this region from this side. Always remain there and also once the trajectory enters region from the other boundary also remains trapped in this region. So this green ring is now the trapping region. And by application of current band exon theorem. We can conclude that there is surely limit cycle. The other picture is for. That second example when I just constructed so let's go on some example. My next example which is not so obvious. So consider this non linear system and prove that prove that it has a cycle. Let's start with the fixed point. The origin is the only fixed point and by linearization. We now conclude that it is unstable folks. I construct the rectangle. The axis between minus one and one and absolutely between minus two and two. So, on the left, just plotted to the face portrait. Oh, it is quite obvious that we have the lemon cycle cycle here, but let's prove it. Consider the boundaries of this. This alliance. And here is the calculus. So let's take, for example, absolute to and see what's the science is usually what's the sign of X prime and absolute prime. So the sign of X prime is going to change. What absolute prime is less than zero. Okay, what's that mean that absolute prime is less than zero. We are going down. Yeah, surely, but maybe X will be there, but anyway, this direction. So anyway, the vector field points like this. Okay, we are absolutely equal to so this boundary. So here we either go there or there. The vector field point in one. So the trajectory will never leave this region crossing and pretty much the similar. Okay, we are pretty much the single similar calculation on each on the on the boundary. So on the left as you see we also calculated points here. We also come to the conclusion by calculating the the first derivative is going here and from here, definitely this rectangle is the tracking region. Yeah. The trajectory start in this region. What my question is, can we apply the point curve band extend here? Not yet. Why? Because now we deal with the fourth assumption. Nicely, we construct the trapping region, but the third assumption remember, third and fourth is what's important. Not supposed to have a fixed point inside, because if we have a fixed point inside, then we cannot come to the conclusion that there is some attractor inside maybe the fixed point will attract the trajectories. If there is no fixed point inside the region, then surely something will attract in that region. The trajectories that has to be non-linear, the nine experiments. In the play. Okay. So what do we do? Luckily, in this situation, our fixed point is unstable. What that means? That means that we can construct a neighborhood like a small circle around the fixed point. And since it is unstable by the formal mathematical definition by the intuition, what means the unstable fixed point? That the trajectory which starts close to the fixed point will leave the neighborhood after some time because it is unstable. This small region, like the circle around the fixed point, we are sure by the definition of the stability that the trajectory will leave that neighborhood enter into the now boundary region, which this is yellow one, yeah, without that circle, which is now the trapping region on which we can look by the Poincaré-Bandit's theorem and conclude that there is a fixed limit. Of course, in the application, usually have the parameters and usually the stability of the fixed point depends on that parameter. If you want to prove that for some parameters, there is a limit cycle, then first you have to assure that your fixed point inside is unstable. So, to repeal the trajectory, then to construct the trapping region because if the, remember yesterday at the end, I told you, let's keep this part to the lack of time, that by the index theory, come to a very important conclusion that the close arbitrary must recycle at least one fixed point. Okay, so the trapping region and the inside cycle always have a fixed point inside, but that fixed point has to be unstable. Now, to repeal the region, you construct the small circle, then you construct the trapping region on which boundary trajectories, the vector field is too large. Yeah. Why should it be unstable? Unstable. Why should it be unstable? The earlier example, it's not because you have an institutional answer. Yes, but I will come to that point. Yeah, we can just, if you go back for time, then you can make another. So, okay, first to answer to this question, why the fixed point has to be unstable? That's what you said. Yeah. And why so? The fixed point is because if not, then you cannot make a boundary. Of course, you can make another, but in this case. Yes. Yeah. Yeah. Because it's actually unstable. Yeah, because it's unstable. Yeah. But the other question and other, if we just reverse the time, we can apply it in a reverse time. What means that on the both boundary, the vector will be closed outwards. Yeah. But then we will have the unstable cycle in that conclusion. I think, yeah, I just remember so, yeah, so this is generally the situation in the previous example you see. That's a larger limit cycle is unstable. Yeah. See on the green ring. The vector field points out. And the tracking region for unstable. Okay. So, in caribandis and theorem as a measure is very, very important. The nonlinear dynamics. I didn't say that the nonlinear dynamics is a play and say the compare with a higher dimension. It's quite limited. So if the trajectory strapped the close bounded region. And without any fixed point inside. In fact, we must approach orbit cycle. We will say but as I mentioned in the higher dimension the point caribandis no more applies. And we come to much more complicated discussion and dynamics so the transaction is as seem a wonder around. Or, without into some either to fix point or those orbit has to be a tractor, but as you probably heard, there can be some so called strange attractors, right. So I don't. Just get to the idea. Introduction to house. Mine. Sorry. Somewhere. Share some. You should hear this one. Yeah. I don't know. Maybe it's when he stopped sharing. I'm going to make a comment. That's the basic idea about what my vacation is, because in the application, of course, you have the photo. I think that really I don't know much about the energy. For example, in the, in the system. The vector field is constant. So when you are in that factor, you also are busy. You fall into that factor and use always. So once you start the cycle, you're consuming the energy. Energy while getting close to the inside. Less and less energy. As you say, So you can. No, but you have to be patient. In every place of space. When you are in that factor, maybe you are compensating. Always look at the divergences. So you are, you are losing energy at a constant base. Always like that. I'm not sure each dynamic systems. We don't, we don't see it because we are training or like all this in my part system. Now that the normal system for the beginner, it's one dimensional parameter, but also we have the theory of vitrification and two parametric vitrification and so on and so on. And of course, for the beginning, I decided just to show you the global idea what to do by for a very basic each tax book. The examples of the bar from page. So what is the verification. Asia is simply if we change the parameter. And come to some what you call the critical value. Then the dynamical system will be radical change what that means that mean that two systems will no longer be what we call the topological equivalent. Okay, so we will have the change either in the number of the fixed points. Okay, maybe the fixed point will disappear, or we will have more fixed points. One, then there might be the equal number of the fixed point that they will change this ability. We have one to stable fixed point but one will become unstable point or we have one stable one stable this that is what you say so the change in the number or either the stability of the fixed point. So ironically change the dynamic of the system. If it is happening from some values of the parameter, we call it verification. Also, there is a change in the existence of the cycle. Yeah, so it can be that the system has a limit cycle then the cycle disappear or the cycle appear after some while of the parameter that we call the called the bifurcation critical critical line. So I will represent you three. First three basic bifurcation concerning the number of the fixed points and either their stability. Several nodes bifurcation is the most typical which equilibrium are destroyed or created. Also known as the fault bifurcation and the typical what we call the canonical normal form of the saddle noted bifurcation is the nonlinear system of this form. What is what are the fixed points of the system. Obviously, absolute zero, but depending on me. Yeah, x square x square is me so if me is negative, then we will obviously don't have any fixed point. If M is zero, we have only one fixed point. Yeah, and if M is Um, um, great and zero. Yeah, we will have two solutions of this equation. So definitely the number of the fixed point depending on the parameter. So when me is negative, no fixed point, and the flow is from the right to the left. Okay, here you have the face portrait in all three. In all three cases on the left me is minus two or negative in the middle me is zero and on the right me is great and zero so we have a it is not seen very well because of this color, but there is one fixed point. Okay, the saddle, obviously, and there is another fixed point now is always symmetric fixed points. This is obviously the node. Yeah, this is the note now we have to explain but what is interesting in the case when me is zero, we have only one fixed point at the origin. And it is non hyperbolic. So you remember I told you that I will talk about that case. We said that if by the linearization, we have a hyperbolic point, then we can conclude about the dynamics in the system by the dynamics of the. But if the fixed point is non hyperbolic, then we can come to the conclusion. And yesterday we see the situation about the center, which is one example of the non hyperbolic fixed point, which has complex 18 values with the real part zero. We haven't still considered the cases where there is one agent while you reach zero. And this comes to the situation when we have here we have the fixed point, which is no hyperbolic, you see that one. The reason why there is zero, another reason why there is minus one. So, by the linearization we cannot say anything about the dynamics inside of this, but throughout this big rotation. So, in the case me is greater than zero we have to symmetric fixed point, and by the linearization and usual technique. We either plot agent wires, of course here, you already have a diagonalize so canonical form. Yeah. So, you know, the 18 miles. This is the numbers on the, the main diagonal so we don't have to calculate. So one fixed point is stable nodes and another fixed point is a saddle point. And the saddle node of this location, the number of fixed points is n is negative. No, if m is zero to one, one non hyperbolic fixed point and when the perimeter is positive, we have two fixed points, one stable and one unstable fixed point. The usual way to show the bifurcation is throughout the bifurcation diagram. What is bifurcation diagram so it is plot in me, you are explained. Okay. Just plot this normal form, so the first equation graph, and just plot the vector field, and it is a usual way to denote the stable part with this and unstable part with the dashed line. So here we have when means no fixed points. Okay. As we increase, now we have one fixed point. This is the vector field of this and here, obviously approach to this part is stable fixed point. Because we have one fixed point which is stable and another fixed point which is unstable. So this is the best way to show another type is a transcritical bifurcation. Now first equation is also the second order polynomial but with x next to me. What is now the situation. This equation. Yeah. Always have roots. Yeah. x one and the other is x me. Yeah. So now we will always have two fixed points. Okay. Absolutely zero and x is either zero or me. So in transcritical bifurcation there is no change in the number of the fixed points. Change in their stability. Okay. When means less than zero, we have to fix point one is origin and another is and by the linearization. We show that 18 values are both negative. This is stable notes. Yeah. Both ideas are real and negative. We have the stable note and the other fixed point. Now we have one. A value which is positive. Another is negative. So several. Yeah. One positive one. We have to fix point one is stable. One is unstable. When me is zero obviously we have one. Again. At the bifurcation value, which is critical, we have no hyperbolic fixed points. The bifurcation at critical value always happen in non hyperbolic fixed points. When me is great than zero. Now the origin became the saddle. Okay. The origin is always the fixed point. First it was stable. After the bifurcation the origin becomes unstable. So the saddle point, but another fixed point appear on the right. And it is now a stable note by the linearization. There are the base portrait. This way. Yeah. We have saddle. We have the origin, which is the stable note. As me enlarge these two points are close, close, close, close, close. These two points can become one. Yeah, they approach to each other as we increase the parameter. This is obvious because this coordinated depend on the parameter. Yeah, so as we increase parameter, these two fixed points are closer and closer and closer become one. That point, the origin change the stability. It is no more stable fixed point trajectory. It became the saddle, but another attractor another fixed point appear, which will now attract the directories and we have another. Another stable stable stable fix. So this, this bifurcation is only considered not in a change of the number of the fixed point. The stability of the fixed point. Okay, this is the conclusion about the trust critical bifurcation. The origin is stable. We have another fixed point, which is a saddle. At me, let's do fixed points. Of course, so joined in and became non hyperbolic fix fixed point and after further increase the parameter origin became unstable so settle point and another fixed point appear. So, okay, this is the bifurcation diagram. Here we plot. This is the origin. So you equal to zero. Yeah. And this is the line you equal to these two lines. Okay. So here left so when me is negative. The origin is stable. This is the stable part. But another fixed point, which lies on this. This line, it is unstable. Okay, clear. Yeah, when we increase what's going on the origin become unstable so we now have the dashed lines. Yeah. And another fixed point, which is on this line so me equal to you become the stable. Of course, the critical bifurcation is in each of these four cases. Similarly, the normal formula like this. Okay, we also can minus here or plus here or plus here, minus here so all the combination. This much gives the same dynamics. But maybe before the critical value was here, for example, the only difference is that now for negative parameter the origin is unstable, then becomes stable on the discussion. The dynamics is much the same. Yeah, you have here. Next, the next is the fork, I think that I pronounce it. Correctly. Okay, the normal form is this. Now we have minus me X minus X. What will go on here. How many solution we have here how many roots real roots we have here. X one always, but the other two also depend on the sign of me. Yeah. Zero is always the solution but the other two depends on the sign of me so we will either have one or two. So when we is less than zero, there's only one fixed point at the origin by the linearization we come to the conclusion that is the stable note. We have two negative agent values. In the case means zero again, the situation is the same so we have non hyperbolic x point origin, but with increasing further the parameter, and it becomes positive. Now, two more fixed points against symmetric according to the, to the origin and by the linearization now for the origin. Positive one negative. So saddle. And for other to fix point the agent values are both negative. So, we have now to stable notes. Okay. So, super critical fork be vocation for me negative only one stable origin fixed point. We still fix point for the critical value but non hyperbolic. And when me become positive origin becomes unstable, but to other stable fix points up here, this is stable notes. This is bifurcation diagram on the left. Okay. So we have fixed points here. This is me. Zero, and the parabola. When me is positive, the origin is unstable dashed line, and this is. But now we change in the normal form here this time. We have another type of bifurcation. Why it is so radically different here. The discussion is pretty much the same as before. Okay, but now when me is negative. The origin will be stable. And we will have to unstable fix me is zero. The origin is again non hyperbolic fixed point. But if we increase me after bifurcation value to unstable fixed point disappear. And only stay one unstable fixed point. And you tell me why do you think that this is radically different than the previous case. There is there is always always one, always one. Here we have always one. We have three after the bifurcation value only one that sense doesn't make any difference just before the barcation. What is the situation when we have only one on the left that only one is stable. But on the right when we have only one attractor only one fixed point unstable. It's the main difference. We no longer have anything stable in the system. Now after the bifurcation value, the system, which was for some values of the parameter stable has at least one fixed point which will attract the trajectories. Now after the bifurcation value is increased, we have no longer anything stable. We don't have any stable fixed point. There is much radical change in the behavior of the trajectories. And you can see here also the face portrait. No cases. Obviously here you have three fixed points. No saddle saddle note. The stable. Well, you have also for me zero the saddle point and great. Now the origin is the only fixed point. But as you see it is the saddle point. So unstable. Wherever you the only fixed point is unstable. So whatever the initial value for this system is go the solution to infinity. They will not approach any. They will not approach any finite value throughout the time. Okay. This was so called the normal form of bifurcation. As you see, according to the techniques that we learned before it is very, very simple to analyze this form. The normal form is no problem. Then some examples in some examples before. But of course, the question is when you got some system, which is not in a normal form with arbitrary function. Is there a way to see this occurs some bifurcation and which one. Not not very pleasant calculation. But with course using computers. You have some software which do that. Of course, you can do that for arbitrary nonlinear system. So for some function at one after according you. You have to calculate the partial derivatives of this function. Okay. So the Jacobian matrix is, let's say, first capital D derivative of this vector field, but we also define this expression. So this is the vector field of the second partial derivatives. This is with the third order. What we have to do in order to apply the calculation, we have to use a vector, but the one corresponding to the lambda zero. So to the eigenvalue. Okay. The first assumption is, of course, this is the fixed point. Yeah, and to this fixed point we have the verification. The second assumption is. As you see in all examples, when me is zero. Yeah, at the bifurcation value, as I told, the origin was non hyperbolic fixed point with one eigenvalue to this is necessary. So the first condition that has to be checked in order to have the bifurcation because if the fixed point is at the void, we will not have a problem. After that you calculate the eigenvector v corresponding to zero eigenvalue. And with the w w, the eigenvector corresponding to the transform matrix, which of course again has a value zero. And after that with the calculation, so you multiply w with the vector field partial derivatives according to the parameter. Yeah, this is different from zero. And if the so this is the vector field. If this is different from zero a saddle note bifurcation occurs this fixed point for the bifurcation value and zero. If this first value is zero, then you check further. This is different from zero. This is different from zero. This is the Jacobian now off the vector field. According to the parameter now. So if these two values are different from zero, we have transcritical bifurcation. And in the third case, if the first one is zero, but the third one is also zero. You have to check this to have to be different from zero. And in this case, we have the work bifurcation. So here you have a really one. For example, just maybe the most important steps. Just arbitrary systems. Okay. What you first check, because the origin is the fixed point. Yeah. You calculate the curve in matrix. And how you determine the critical value. Yeah, but yes, but that means that. Determinant is. Yeah, this is a rule. Yeah. Yeah, yeah. Determinant of this. Zero now you have two solutions and I considered here, for example, the bifurcation but I think that also for the other bifurcation value also have what was their transcritical bifurcation if I remember yeah, transcritical bifurcation. So, next, you fixed that bifurcation value. And now you calculate the eigenvector. Yeah, this matrix and transpose matrix. And that just calculate the partial derivatives of this function course for for all the time. The bifurcation values from all the further calculation you took me in the system and calculate the partial derivatives and the fixed point. Also, I said, every point of view is not pleasant with the computer. It's not no problem. And here is, for example, the face portraits. We have the transcritical bifurcation. We have to fix point. I consider the bifurcation for me equal to one. This is for less than bifurcation value. We have to fix point one, the origin is stable, but here the focus. Okay. In the normal form it is a node. We have now the arbitrary system. The other is unstable. It's a saddle point. It will close close close close. Not to the bifurcation value. We have here and this is obviously the note unstable unstable note. But anyway, we have a two fixed point just different difference. And the bifurcation that we're considering existence appearance or disappearance of David cycle is the hopf bifurcation. Here we also have supercritical subcritical. It will be different. Let's look at the normal form which will be more easier. So I put the system in a usual form, but this is also called the normal form of the hopf bifurcation. But in order to analyze the system as you own the previous example and we wanted to do that and once discussion about the limit cycle transform to the polar coordinate. When we do the transformation again we obtain a very, very simple system of differential equation uncoupled the first equation. So now it is like the pitchfork bifurcation for fixed points. So there is zero corresponding to the origin. Yeah, so the origin will always exist. Okay, but depending on me. Yeah, our quadrat error square is me. So if me is positive, you will have two solutions. Of course, one with the positive is a limit cycle. But if me is negative, you will not have any limit cycle here and there is system. So when me is negative. Biggest point is a stable table for zero still stable focus. But as we increase, I thought it to me too, because for as me increase the radius cycle also. Because the equation for the limit cycle is areas equal the square root of me so increasing me the limit cycle radius increase so I thought it for to be obvious. To have a good pitch. So for me, for me too. Yeah, but the before bifurcation for the system in the polar coordinate. Yeah, it's the same. Yeah, so but now that fixed points are actually the limits. Except that, of course, the limit cycle cannot be your equal minus square root, because there is a distance, it has to be positive. Yeah, so only one. That's true, like in pitch for bifurcation, obviously, because every distance. Yeah, that is why only only one actually. Yes. And you get radius is the other side of the cycle. Then we have this stable point. Yes. Yeah, yeah, the stable. Yeah, you can also plot the stability of the length cycle. Yes. Yeah. As you see, now the door is going to become unstable, unstable focus, replying the trajectories, but there exists a limit cycle and obviously limit cycle is stable. So the trajectories, both from outside and from the inside will approach the cycle so come to the conclusion that this table in the cycle up here. The calculation is, as we did it before. Determine the sign of first derivative, positive, positive here so we have the stable in the cycle here. So in the other case, so this is the bifurcation theorem on the left. So when means less than zero on the origin. Okay. Now the parabola but we plot only parabola for the positive values of error. Now the origin become unstable dashed line. And we have the limit cycle, which has the parameter increase the radius of the cycle. So this is super critical of vacation and very similar. We have, we have a subcritical of the vocation, but as in the case of the pitch for bifurcation also quite radical. Now, we have the origin which is the stable, but the limit cycle around that origin is unstable. And as me increasing, so what's going on here, the radius is getting smaller and smaller and smaller and smaller, approaching the fixed point, then on one fixed point and the limit cycle is. That's me increasing, but the origin became stable after the bifurcation value as in the subcritical pitch for now, we also don't have any chapters in the system. We have only one unstable x point in the system. Yeah. Yes, yes, there is another type of vacation also but the vocation of course can occur in just here but represented you and the first equation is one dimensional. Yeah, but you can have the same of course the calculation in the theorem that I will do is a little bit different to establish the vocation and the higher dimension. What is actually the situation. How to explain this. So, any system that occurs to any of the previous bifurcation or the half bifurcation can be reduced to the normal after some transformation that is the point that is why we first to do this normal form and we call it normal. If you have arbitrary system, and if you see that it satisfies condition for some bifurcation, they're the transformation of the function or the time dependent value. So let's put the system in the normal form and after that you discuss that is how the things are done. And in the higher dimension. There is a hop bifurcation but for example if you have the four dimensional course because the hop bifurcation occurs only when the origin is the center. The non hyperbolic fixed points. But with the real power is zero. This bifurcation was also in a non hyperbolic fixed point, but one real. If you have the four dimensional system, the whole bifurcation always occur in the manifolds, which is called center manifolds and which corresponds to the few imaginary, imaginary. So the other two dimension, you can just ignore. You can put in the system in the normal form, and you just discuss the system in the normal form, which the first two equations then will be uncoupled in compare with the rest of the system. You can just ignore the equation third and fourth. Pretty much the calculation will be of course a little different. Okay, yeah, we have now unstable limit cycle. The origin is stable. Bifurcation value origin becomes unstable fixed point. And up to the refrotation of the limit cycle disappear and we have only unstable unstable fixed point. Remember from yesterday we had the problem in the center, but we show that for some systems and hope the bifurcation occur. When the center of the linearism system, the bifurcation value will become. The center of the non-linear system, you don't have bifurcation, because you have already proposed trajectories around, doesn't disappear. Am I clear? So the bifurcation value by the linearization, you will come to the conclusion that the fixed point is the center, it is a linear center. Actually, as you see on the vector field plot is a non-linear box is not the center. This is everything example from the yesterday. When I said that the center can remain the center as in the polynomial system, but the linear center can actually be the non-linear box. That's the point which is critical. And when usually the bifurcation occurs in the system, it's of course changing the appropriate parameter. And this is also sometimes a good way to show the existence. When you notice something like this, just put the parameter inside and show that for this value of parameters. Sometimes if you cannot construct the trapping region, which we see it is not so easy, you can come to the conclusion that there is a limit cycle throughout hope bifurcation in the system. Yeah, again, I put the calculation is, calculation is its technique. The calculation is not pleasant at all, but using the computer, everything can be done. So again, the question considering arbitrary systems, not in a normal form, two dimensional systems. There is a well-famous hope bifurcation theorem, giving the conditions that some system occurs the hope bifurcation. First condition, the origin is the fixed point, but for the bifurcation value, it is a center of the linear system to have pure imaginary eigenvalues. Okay, that is the first condition that has to be satisfied. The real part of the bifurcation value is positive. It means, and the real part is zero. This means that we have a pair of pure imaginary eigenvalues. Now the eigenvalues crossing the pool line with this target, so the derivative of the real part of the bifurcation value is different from zero. And the third task, which is more, most difficult, at least for calculation, is to calculate the so-called, first, the pool of number. We have for two parametric bifurcation, so on, we'll then calculate it also. The second, the pool of number and so on. Is there a table of exponents? No. Just another type. There is too much in mathematics called the opponent. The opponent's stability of the function is from the exponent, the opponent's coefficient. Mixed up, but no. It has to be different from zero. Why? Just because the main reason is, I talked a couple minutes ago, so if you want to change the coordinate and put the initial system in the normal form for the hopf bifurcation. In the calculation, all of this condition pretty much appear and with you in the way to prove the hopf bifurcation here and it is obvious that it has to be satisfied. And then according, because these two numbers has to be zero, according to the sign of these two numbers, is it lesser than zero, everything is described here. So it just depends if we're going to have the subcritical or subcritical value and is it going to happen when we increase the me or we decrease the department. So the conclusion is here. So if this L delta is greater, is less than zero, the unique cycle appears as me increase. This is positive. It decreases value. The stability of the fixed point is determined with delta. So if delta is less than zero, it is stable for great then bifurcation unstable. This is the contrary and always the limit cycle and the fixed point has a difference to build. So if the fixed point is unstable, the limit cycle around is stable. If the limit cycle around is stable, the limit cycle around is pretty much the conclusion. Okay, conclusion again here. So you see, if you put the system in the, okay, what's the steps, just two questions and just to present this more. We translated the fixed point to the origin. That's the first step. The other step to put it in a canonical force that is always done. And after that, your system becomes like this. Okay, the matrix of the system is, you see, for this system, the matrix of the system is zero, zero minus beta, beta, which means that we have a center. Real eigenvalues, real part is zero. So you will always put the system in this one. And after that, you just plot the partial derivative of this one. Mostly up to the third order. So this formula, of course, all the values are in the origin, as the fixed point. So X and I is equal to zero. So this is the number or if you already have the system, which is the functions are expressed using the Taylor expansion in the polynomial form. Then you can calculate it in this way. So, okay, this is the system. This is the inner part. This is from the second. Yeah. Second degree polynomial third degree polynomial and in order to calculate the coefficient, you use the second and coefficient. And you have here the formula. So, as I said, using the computers, it's not the problem to calculate the first coefficient. Of course, you can do software or design for money. So, okay, I think I will stop it there. Any questions? Yeah, yeah, it's quite obvious as I said, so when you start to transform your system in order to put it to the normal form. Then it's obvious. Some points that this coefficient appears coefficient and it has to be, it has to be different from zero. Otherwise, you cannot put the original system that normal form that we discussed at the beginning. So, the system will be in another form, another type of verification. Is it stable or not? This is then the bifurcation, this is the, all this bifurcation is local and it is the global bifurcation. You can have two limit cycles. How is it related to the bifurcation? I mean, not. You cannot decide just for up to, you need the further restructuring. You just now know that there is one limit cycle using the first bifurcation or if all the assumptions on the bifurcation are verified doesn't mean that there will be, maybe there is another way. You'll have to continue with your system in order to see if there is a multiple limit cycle. There is examples of the half bifurcation. There is only one limit cycle appears after some time. Because this will happen because of the rest of the dynamics. Because, for example, I just showed you the good example. Let's imagine that you have a limit cycle around this bifurcus. Let's imagine that you have the saddle point also. That's why we call it the global bifurcation. It's not only concerning this fixed point, but on that kind of bifurcation, another fixed point. Imagine you have the saddle, then what will happen in some dynamical system? That limit cycle goes large, large, large, yeah, closing to the saddle, then at what moment the saddle will attract a homoclinic. The tractor will appeal, you understand me? After some time you increase the parameter, it will rock the homoclinic and nothing else. So no, no homoclinic, no limit cycle, no, no. You can have the situation, it's two limit cycles, but it's another fixed point, which appears in the dynamics, you know, influence. Of course you have also the saddle, for example, saddle nodes bifurcation of the limit cycle. The same as the saddle node bifurcation of the fixed point, so two cycles, which will closely approach, approach, approach each other, become one, and then disappear, but that is what is called the saddle node bifurcation of the limit cycle. Of the limit cycle, yeah, two limit cycle, yeah, yeah. These parameters, they are close close close, one, and then disappear. Just like saddle node bifurcation of the fixed point, you'll have two, they approach each other, there is one, and then everything disappears. This is everything that will want to disappear. How is it possible, if it is really possible to calculate the number of cycles? Of course the calculation is usually not so pleasant, especially if you have a lot of parameters, like the coefficient in dynamics, which I find sometimes you can make it sometimes not. Yes, there is everything pretty much can be calculated. Theoretically, we know what conditions has to be to put the system in the form so that you have. But the question is, are you going, of course with the computer or plot the face portraits and you will see this. Another point is how to obtain the bifurcation value, which is also possible, but to formally prove not only by seeing the face portrait, but sometimes. The model is actually for the lamprope of the system and it's starting for the button on the side, I can identify. My query is, what a new cycle, it is mandatory to have the fixed point in the cycle. What would be the guarantee fixed point in the schedule? What the new cycle, this new cycle you will not get. If it is centered around that, how isolated goes the office? Only for the lamprope of the system, the idea is not in the cycle, the center will be the fixed point. What is going to be the center part? For other systems, what would be the guarantee to send that as a fixed point? You only can have a fixed point. That sends the closed order. The lamprope and my experience with you will happen. You have a center, you have orbit, you circle in a round, you have a lot of limits. That is also one way that I missed, you have an index theory, index of all the fixed points of the saddle is plus one. And the addition of all the fixed points inside the cycle must be one. Also the limit cycle must run several times. I didn't do any system. As a summary, I think that the summary event was that in a two-dimensional system, you can only have fixed points, limit cycles, orbiters, from one point to another or to the same. That's okay. One turning to a limit cycle. Thank you very much for your questions. That is special. I hope you enjoy your stay in the niche. For example, I'm concerned. If you have any point in your wish to all assess the basic knowledge of you, the mathematical point of view is in my best. Thank you very much. Thank you.


I have played a lot of Dragon Ball Fighter Ranked Matches in my time, but none have been as sweaty as some of the sets I played in my last video. These PC Ranked Matches are just different, which is why I decided today that I am coming back to the PC Ranked Environment with my main team, just to see how good these players really are. I had to come out with my main team. After the last time I played PC Ranked, they were working me.  It was really exciting, so I'm back with the main team, Jiren out in front, and I'm just gonna go crazy. I wanna raise my rank and fight even better players on PC. Oh no, 17. Here, come here. Come here, come here, 17.  I will just target 17, although I am getting Jiren right now myself. Ooh, nice extension. The PC player is just a little different. Oh, I didn't even notice he tried. He hit me with the Try Something Counter. That was pretty, dude, the Jiren matchup is so sick. Jiren versus Jiren, this is just a matchup. That was not intentional at all, man. That was supposed to be a normal moment. Let me just float, send him flying. That was supposed to be just like a reflect or a counter. It was, yeah, it was supposed to be a counter, my bad. Here we go, grab. Oh, nice reactions. I thought there was no way he's reacting. Oh, he fumbled. Okay, we're alive, and I got a Sensei bean. No! That's a drag. Wow. No, don't live it. No, in the air, I didn't think you'd be able to jump up that high. See, I'm already so focused, and I'm already losing. I sparked way too early again. Oh, they're so fast now. Prolin buffs are crazy. No, not the Dragon Rush button. Again, I'm losing in scrambles. When I get scared, I just, I start losing it. All right, while we kick them out of the corner. Out of my way, 17. Damn, how many times do I gotta teach you this lesson, old man? 17 keeps popping up with this barrier like I'm trying to talk to him. Get out of my way. All right, 17. How are you doing? You know, I've been waiting for this for a long time, 17. You've been, you've been toying with me for a minute now. Okay, he actually hit the barrier that time. I do like testing 17 players. I want to see if they can actually hit it. I want to see if they can time the barrier. I want to see it. I do. I'm not going to lie to you. I need to see that it's real before I go, go ahead. Okay. All right, put me on, put me on notice, 17. Put me on. Oh, that was a little devious. Okay. He held the bandage there with sparking so that he could dash over me, hit with a mix-up and go into a full combo and get a kill here. He's going to plant me. Oh, never mind. He's getting a level on Gohan, the actual better strategic play. That was pretty good. I like what I'm seeing. That was the wrong move, man. Oh, we tried to time something there. Okay, kicking away. And the 17 assist. Huge hit stun. I feel like I'm playing from behind. I think I can win this set though. Even if I can't win this game, I think this set is very winnable. Nice. Good key blast. I just go into level three here. I definitely didn't mean to Wolfgang fist, but it worked against the barrier, so I'll take it. We're going into this now. I'm worried about the DP, to be honest. Wow, he blocked it. What? He's got the defense. Nice. Well, now I got to throw in some different Yamcha looks.  He's kind of tested me a little bit. Oh, and I missed my level three. Wow, he beat me there. He actually beat me there.  Is that enough to kill? 5 bar, 17, 1, 3? That's probably enough to kill. Hold on, Yamcha! I was asking a lot from Yamcha there. He had great defense there though. Really, really good. And now I know what I'm playing for. Now I know. Hold on. Let me get back in there. Let me get this rematch real quick. Please, please, give me that rematch. Give me that rematch. I need it. I need it. I need a rematch. You're really good. But can I get a rematch? But can I get a... No. Oh, brother. The good luck stamp was actually supposed to be friendly. That was... That's a toxic stamp, man. I've never seen the Gotangs using a not toxic way. Why did he ice me though? I sparked again. Give me that. Okay, kind of an interesting build by me. Let's go. Let's go. Huge start. So I really did mess up though. Thanks. That was the worst route ever, by the way. I just... After I messed it up, I was just like, please tell me I can still find a way to make this kill. So I just had to extend the combo out. We're playing this. Well, I'm playing this pretty slow. Okay, nice. Nice. No way. I threw that. I threw that great corner position. Okay, okay. He had his park eventually, right? Things got really awkward there. I'm not even alive. Oh, let's go, baby. Let's hold this as long as possible. He was waiting for me again to do that. His re... Okay. I have to... I have to respect him. His reaction time is too good to get hit by Jiren's grab. I just have to kind of accept that, and I don't think I can really go for it safely anymore. It's always going to be a risk when I use that move. Normally, I would say normally you can sneak those in pretty easy. Ah, lost that one. Probably should have played it a little safer. 2S. Yes. Yamcha. Thank you, Yamcha. Love it. Love you, bro. Appreciate that. Resetting my combo. Only going to cost me about a buck fifty to get this kill. One. Make it a dollar. Saving fifty cents. I'll take it. Ohan comes out. I told you I thought I could win this set. Nice. Oh, manish. He gets a combo off this, but it's a good one. Wow. Great Key Blast. Put me down. Oh, 2M starter. That was a big mistake. That was a huge mistake. I'm just going to do Krillin. I'm going to keep Krillin out here, even though Yamcha's mix is a little bit better. Because we saw last time, he actually just blocked and respected me. So we knew what we had to do. Let's go. Get into that next game. I want to see him. Is this not over? I'll say it's not over. Oh, God. Oh, God. Please tell me I'm not choking. I don't think destructive disc would have won, so I didn't even want to throw it, to be honest. Wow. Good stuff. Let's go. What a super dash by me. I knew he was looking for that Key Blast zoning. No. No Key Blast. No zoning. I'm not allowing it. It's 1-1 in a very intense first set. Cool. Load my team in. Get this rematch. Load my team in. Get this rematch. Go next. Go next. Go next. That was a good fight. Do I have anything? 1. I forgot I made some toxic stairs. Watch this guy. Nice dragon rush. That time he's not sneaking anything by me. Run by him. Hey. Let's go. Why'd he freeze up? Why'd he freeze up? Let's go. Throw more damage on him. Throw more damage. Just get as much damage as possible. I want him to spark first. I'm feeling dangerous. Okay, there we go. We got the early spark. Oh, Yabcha, sis. Go ahead. Get that Yabcha, sis. Jiren just got out through in that game. It's all about that flow state. I just got to reach my flow state. Let's go into the corner. Jiren. Blow him away. Hit that cutscene. I let my cutscene talk for me. Oh, I wanted to do quarter circle. Hey, man. I wanted to hit that shoulder charge. I don't know what the move's actually called. The purple one. Level three. Is there a reason for this? Oh, what a read. What a read, man. What a read. Oh, the beam because I'm Jiren. Go ahead. Try something for me. I'm asking you to. Oh, my gosh. Oh, my gosh. I'm loving this. Oh, my goodness. Oh, my goodness. I'm having a game. He's having a game. Let's go. Dragomush. Oh, yeah. Oh, yeah. Rock assist. I got to play a little patient. He couldn't get a trade this match. I won every interaction. Jiren. This attack does decide it. That was a great set. Oh, wait. Was that my first win? No, no, no. That was a great set. Yeah, yeah. That was a great set because I remember I did the one stamp accidentally. I got to, I got to throw a thank you stamp. That was a great set.  I got to get a stamp out stamp. I made a couple of toxic stamps by the way. Party match. Beginners welcome. Thank you. I'll just say thank you because I don't want to be toxic. He was very nice. All right. We're back going up against GT Goku. GT Goku hit Janemba in interesting T. I thought it was that dragon rush. I got to test their reflexes nice and early on PC fighters, man. I got to know if they can tech drag rushes if they can't. Nice reflect.  Sadly, I couldn't follow up with Super Dash because I didn't really, I didn't really know what I had until it was too late. But this is good enough for me. GT Goku post spark struggling in the corner. Throw a little bit of this. Get that. I didn't go long enough. Oh, the rock to cover and the side switch into the corner. Wow. Wow. Wow. These are some of the greatest games I've ever played by the way. I am having a great time today. Jiren putting on an absolute try something then. Hey, how's it going? I didn't see you back there. Level three. Go ahead. Let him know. Jiren. He's too strong. Okay. You know what? I'm done by it. Let him know. Oh, he. He waited the perfect amount of time. There we go. Nice. I was trying to get a side switch. I got greedy. I actually just got greedy then grab. Okay. Oh, Chrome was a little too slow. I wish I had his assist for the Jiren grab. It would have been perfect, but it wasn't meant to be. We're still going to take it out. Jiren looking just a little bit injured, but he's doing just fine. Legend number come out. Legend number come out to play. Try something. Oh, by the way. If you're not going to hit me, then Yamcha will hit you. It's only a matter of time. Not going to level three just yet because I want to get this dragon rush. What a tech. He last rock assist. Oh, he stepped on me. Why did he step on me like that? Oh, okay. I'll beat that. Oh, yes. Yes. Never mind. You're better than me. Never mind. Never mind. I don't get to tell you what to do. You tell me what to do. You want me to go check out these mountains? Yes, boss. We'll do. Let me see what's inside these mountains real quick. Oh, is that game? Off one beam. Is that game? Something tells me that isn't game. Oh, I wish it was though. I went blue combo. Wait, you didn't tech? And also I dropped that. Nice tech he blast. Oh, hey, I didn't see you teleporting over there. Except I did. Yamcha. He's just that guy. Wait a minute. All right. That's game one. Can we go ahead and get into another rematch? Nice. I'll take that. I'll take that all day. All right, Jeremy. Show me what you got. Okay. All right. I'll be over here then. I'll be over here. No problem. No problem. Yeah. Just put me down. All right. Never mind. All right. Wow. GT Goku kind of showing up. Let's see if I can get something out of this. Nope. The rock was nice. 2x. Okay. Counter. Oh, I didn't get there in time. No counter means I'm in the quarter. I promise I didn't like switch. Actually, you just deserve to hit me. Just hit me. Okay. Good. I was giving him a free hit because I kind of felt bad there. It lagged and then it threw off his combo. So he had to go back to neutral. I don't think it was my fault. I'm wired in. But, you know, the spirit of the game and the whatnot. Grab. Nice. And with Krillin assist, this is free. All screen conversion. I wanted to grab. I wanted it again. Counter. Yes. Nope. I beat that. Counter. Yes. I've got all the counters in the game. All of them. Counter. Hello. No. You don't get to hit me. I get to hit you. Oh, beautiful. Accidental reflect bait, but I'll say it was on purpose. Brandon, go into the 2x. Do the same Jiren combo you've been looking at. Medium roll. I dropped it again. I should have level one.  Let's go. Get hit out here. He comes flying in. What are we looking at? Do we want to insta counter? Yes. I meant no. That was unfortunate. Let's go. Got hit. Do a full combo. Why not? We're going to get him all the way in the corner just because Krillin has always been cracked. Level three. Hard knockdown. He has to guess what our mix is. I'm feeling like we'll probably just go drag and rush. Except I'll go overhead. And I'll do it again. And I'll do it again. And I'll do it again. Let's go. I faked him out twice. I faked him out twice by accident, by the way. I definitely was just missing my button. Don't got the bar for this. So we're going to set up a rock and a Yamcha assist to double protect. Take out. Oh, nice reaction. 2S. God, that move is so good. 2S is so cheap. Now we're down to just Janemba again. Can we put it away? Can we put it away? No. I don't know why I did that. No combo. The bait. I'm not letting him counter it. Oh, wow. That was great patience. I was trying to fake him out there. I definitely was trying to give him some different looks. I believe that he was trying to get a Key Blast counter. Count on my luck here. No, no. Oh, yes, yes. Oh, yes, yes. I was getting ready to say no. I thought he was going to do it. Fake him out. Oh, don't get killed. I didn't want to bend that up. Oh, Key Blast through the portals, man. That's cheap. All right. We still have Spark though. So I'm very confident we can win this. It would be a great comeback if you want. Oh, it's over. It's over. He shouldn't have stuck a button out. He knew he was risking something. He knew it was a risk, but it was a risk he was willing to take. He just shouldn't have took it. I was hoping to time that with the Yamcha smile, but we skipped it. Damn. True power. Get me into the one more set. Great games. All right. Here we are in the final set. Going against the Super Baby 2 player already throwing some Key Blast out. All right. Let's start swinging then. Oh, the rock assist was not in well time though. I'll be honest. No, Beerus. Poor time rock assist. I'm really trying to get a foot. I want to just make sure that I can. Wow. What a combo. I didn't even see it. Damn. In my face. Oh, don't try that. Come on. Show a little respect at least, bro. Don't try that. Give me. You got to. I forgot he could do that. That is so cheap. That is so cheap. I forgot he could do that. Walk me all the way to the corner and get a free tag out. Insane. And then he's right back in the game. That was insanely good. Great use of that character. That's a lesser known technique for that character too. That's not something you just run into every day. Somebody doing that. But that was really good. All right. We got Super Ruby in the corner. Yamcha assist. They always forget about that. Yamcha assist. Yamcha assist has been humbling people since 2018. Aim. Get the knock down. Let's see our options. Oh, wake up level 3. I can't get no respect on PC. I can't get no respect. Great aim beat down. Come on, man. Give me a little respect. I can't get no respect out here. Wait, Yamcha. Wait, what? Wait, how did he get my... Oh, he has a Yamcha. Never mind.  He can make it so they use their assist. I thought he was using my Yamchas assist. He took control of them. But he just has a different color Yamcha. That actually... I was shocked. I was like, wait, what? Bro actually got my Yamcha while I wasn't looking? But no, we're good. And actually, while I was doing all that talking, we actually secured a two character lead here. What? Wait a minute. Hey, Yamcha, I was talking about you for a minute, bro. I've been waiting for you. I've been waiting. They always forget about the Yamcha assist. Even the Yamcha players forget about Yamcha assist. Let's go. I'll come down. Wow, that has some reach to it, man. Oh, and a guard cancel punish. I just shouldn't have done that. I just didn't want to take the Yamcha mix. I'm afraid of it. He's got combos. He's got actual Yamcha combos. Damn, Yamcha! Yamcha is crazy. You know, I learned those a while ago when they got added to the game and then I forgot them. I got to relearn them. Hey, back scratch. Oh, I thought I had the landing. Even with the mix, it was... It's hopeless in the face of Spirit Ball. No way. Yamcha! He almost killed Yamcha, never! Bro, you get your hands off of him. Ever in your life think you can kill my character when he's on the bench like that. Never again. Destructo disc. That was so cool. That was such a cool set. I feel like there was a lot of interesting interactions there.  Wait a minute. I'm addicted. Wait a minute. Hey, let me get that rematch. Can I get that rematch? Please don't end the video like this. I've been waiting for you. Nice. Oh, he's got the Yamcha stamp ready. Try something. Oh, look at this. Look at what we got here. And with Krillin Assist, this is just easy into the corner. What? How did I fall for that 2H? I had plenty of time to block, too. Nice combo. Just didn't work out for him that time. I'll take it. Okay, how to mix it up a little bit there? No, no, not again. Not again. Way coast to coast. That was good blocks though. That was good blocks. No way. No way, man.  Damn, I can't block my own mix. I can't block Yamcha's mix. I do that every day and I don't even know when. Nice combo. Nice combo.  Wait, whoa, why are you throwing? Why are you throwing? Wait.  Is that what I'm seeing here?  Nah. See, this is what I'm doing. I'm making it seem like he didn't just drop it. I'm turning him into somebody that disrespected me.  Come here. Take this Beyblade. Let it rip. Hit him with the whole Beyblade. I love to see it. He'd disrespect me in some way, bro. I got to meet him with that. That was the meanest 360 Keyblast on me. That was crazy. He hit a mean Keyblast 360. Okay. Okay. All right, let's go. Let's do it then. Dragon Rush. Nice. If you ain't taking that, then I don't want to talk. Baby, what up? Baby, what up? Destructo Disc for the kill on Baby. Beerus comes in full health, but I'm feeling good. I feel like we can take him, especially with that yonk just there. 1v1, I'd be a little more scared because he can cook with Beerus. But it seems that Beerus is usually like, you know, a middle of the road character for him, not necessarily somebody that he would want as an anchor. Maybe I'm wrong on that, but with the way he was playing with Baby, Coast to Coast and getting Beerus out as fast as he did, just something I was feeling, and it's looking like Beerus is on his way out. It's looking like it might be wraps. I am just going to level one into level one, even though I don't think the skills. Come on, Destructo Disc. Oh my God, it actually did kill. That damage had to be exact. I'm not even joking. But with that, that is going to be this Dragon Ball FighterZ rake video. Hopefully you enjoyed it. I know I did. I am addicted to these PC sets, so I got to record some more. Let me know some teams down below in the comments if you want to see that, or let me know if you want to see more fighters by hitting the like button and subscribing, and checking out the videos that should be on your screen right now. IvanDotodoya, and I will see you in the next one.


What's up? How are you guys today? We are back with another Q&A, which we haven't done in a little while So I'm happy to answer some questions for you guys and be sure to keep an eye out in another three or four weeks We'll put up another post on the community page on my youtube channel And you guys can ask your questions again if you missed it this time around So Joseph P. Do you think it's more of a digestion issue when people have a hard time gaining weight? Despite eating a lot rather than just a fast metabolism So instead of trying to troubleshoot all of these things you got to see hey Are you following a high-quality organic diet? Have you reduced the environmental radiation? Do you have to get in the gym to increase your lean body mass? So the solution to that problem regardless of what's causing it is is pretty simple and straightforward So it might take a little while for your lean body mass to improve for you to feel better Once you remove the toxins and increase the animal protein, but it should happen Why is that C? Looking back on your years in the field. What do you think about the health and nutrition information being spread around on YouTube? Channels also has it changed over the years For the most part, I've really done my own thing Maybe only a few times you guys post comments about other people and maybe I'll check them out But I really just do my own research figure things out because you know when I kind of got into this space with the High-quality grass with carnivore diet. No one was really doing that It was kind of based off of health gimmicks conventional stuff just influencers trying to make a profit So then when people started copying me plagiarizing my ideas and all that type of stuff and kind of twisting my Information to sell their own stuff That's where I started looking at it and getting more irritated and just overall disappointed in And how everything else was going but hopefully I can continue to change that over the next few years The real thing that Kind of makes me upset is that I'm really no longer able to just say Everything I know and just give it out there because of what's happened So you know less people are able to improve their health because when I do that then someone just copies it and They censor me and they get all the credit for it. So, you know, it's safe to say this world is not run by people who have your best interest at heart Tarantor Frank, what do you think of things such as ashwagandha for dojah at Gredis Tonkot Alley for testosterone? Also, is there any test supplements you will recommend now that I'm unable to get the bovine powders? so if you can't get the Glangular from Frankie's free-range meat the beef testicles goat sheep testicles that we have you want to go to a local butcher local farm and try to find them because that is 100% Bioactive available testosterone if anyone is telling you like that you can boost your testosterone with anything besides like steroids or Actual like animal testicles then you're wasting your time all that stuff for testosterone specifically You know, I haven't done specific research on other benefits. It's a waste of time John Surgey, I'm finding it hard to reduce sodium since I have a severe potassium deficiency So would you recommend I supplement potassium? I know you have a video on this but reducing sodium is still not helping with potassium deficiency That's kind of hard to say if you're not putting salt on your food and you're eating a lot of potatoes and red meat I mean, it should almost correct itself within like two or three weeks maximum maximum and having coconut water in there with potatoes and meat is such a high potassium intake and The supplements the dosage you have to take so much potassium powder that you would get diarrhea and it would just flush you out That's something I haven't really talked about the potassium concentration and like a synthetic supplement like potassium citrate It's like, you know, it's almost like the volume compared to just eating real food isn't worth or are possible to take so You know, I mean you could send me an email. Maybe I can help you out a little bit, but that doesn't sound it doesn't sound right Cismica How do you get the extremely strong beer like kefir? The one I make at home is carbonated and has the key for flavor But is nowhere close to yours that you sell I use a versus osmosis water organic molasses and organic sugar You could just increase the amount of key for grains and sugar in the relative volume of water. That should help Vessil tea Should we add some salts minerals to our drinking water for proper hydration also our oysters worth a while one week or less for their heavy metals So just eat a high quality diet and drink a quality mineral water You shouldn't have to add anything Especially if you're salting your food and then the mineral content can be obtained in that diet or by using supplements Definite no to oysters. No circumstance. Do I really consume shellfish anymore? I react so poorly to it at this point with my liver function and just in general when you have a really really clean diet And you're doing everything correctly and you go eat some seafood. You might feel like death Gebra GH our will you start supplementing vitamin D is sunlight decreases and do you still recommend 10,000 IU? So I've never really been a fan of vitamin D supplements unless you're incredibly deficient Which is really a one-time thing. So when you start like exploring health and trying to improve your lifestyle Maybe you have to take vitamin D for a few months, but that's it. Then you have to do sun exposure I'll say it again outside of correcting a severe deficiency where your blood levels are like below 10 or 20 nanograms per liter it's really not really not worth it and I haven't brushed up on my measurements in a little while so Below 10 to 20 nanograms per liter or like 40 to 50 nanomoles per deciliter I think that's correct because the units in America versus Europe are different Sir mega what is a good workout routine or die for beginner who's just trying to build strength? I really only recommend my maximum volume hypertrophy routine because you will progressively increase in strength even if you're doing low weight with high volume and even if I was only benching like 20 or 30 pound dumbbells maximum, you know, if I actually wanted to max out on the flat bench press I mean I could rep out, you know 185 for 10 or 15 pretty easily now I don't do that stuff because You know my shoulders aren't as healthy as they were and I don't think it's beneficial for overall aesthetic appearance But you just want to get in the gym Do a amount of high volume follow a good diet and get a lot of rest. I Really like that routine because it's able to build up each muscle, you know Make sure your shoulders are hypertrophy and your chest is hypertrophy. There's a nice balance of Overall muscular development not just going in the gym and benching as much as possible and having imbalances after two or three years of lifting weights Spharo be how should one consider whether or not they should take your b complex if they're drinking large amounts of water Kiefer I remember at one point you were temporarily super dosing be complex alongside Kiefer So a be complex isn't something I like as much as individualized be vitamins the be complex is good for like if you're traveling or an antioxidant protocol, but after you kind of see how you feel after taking the be complex and notice that You know, it's much more energy than you can start isolating individual be vitamins And you know when you take one and you don't feel good, then you know, you want to try another one Victoria already are flavonoids and carotines bad for the liver is a matter of quantity ingested Yeah, most people don't need them because they have too much of the vitamin a and they have a deficiency in the other Like vitamin dk2 all the other be vitamins So, you know, if your liver is overloaded really anything you consume in general needs to be processed So we want to try to minimize that as much as possible Another question from Victoria is semi-complete grains like rice, okay? When you have IBS because I have insulin problems. So I like to stop refined grains plus it depletes minerals and I lack some Yeah, when you have like impaired digestive function and candida yeast overgrowth Sometimes a lot of refined foods can really irritate the gut Especially when you're not on top of your probiotic intake and your antimicrobial protocol and the amount of fiber you're ingesting So until you're able to kind of line that up and do it correctly You want to consume things that don't irritate your stomach as much whether that's Higher like volume foods that have more water content or stuff. That's a more complete fiber source Jake MC can one take enzymes water key for key for grains and mastocom all together with the meal if they have really poor digestive health you can and there is some kind of negative interaction there because the enzymes are pretty caustic and kind of just eat through stuff and Might negate the water key for and key for grains to some degree. So what you're better off doing is At least spacing it out in the meal so you'll have you know the enzymes and Masticum in the beginning of the meal and then you can have the water key for and water key for grains throughout towards the end of the meal and Maybe even put the masticum at the end and have the enzymes in the front or you can try to time it a little differently but the best answer to that question is You try the meal with each of those and see which outcome is the best So you might feel better with just the enzymes you might feel better with just the masticum all all of them might work together too But it can be difficult and it might take some time to get to a point where you can reduce those Can be on productions. How do I reduce EMF exposure if I live in an apartment? We have a lot of products available on Wi-Fi shielding calm The simplest thing is to wear protective clothing like I am right now and then sleep and work in a bed canopy That's the least invasive if you want to start shielding the walls and the floors and all that type of stuff It can get kind of crazy and kind of expensive if you're in an apartment that has other people nearby Mr.. O'nilly boy, how often should you incorporate processed sugars if any I? Mean try not to because they increase the body's need for like vitamin C and anti-scorbutic value But I personally do it like when I'm really hungry and I'm craving caloric density Well, because I still have muscle from bodybuilding, you know My actual lean body mass is not parallel to my digestive system So I kind of overload my body when I'm trying to like satiate my hunger and cravings But if I'm able to control myself I'll just have like a high quality meal with meat and grains and then I'll have an apple for dessert And if I'm still hungry and I just don't care and I want to feel better I'll have like some of the high quality organic snack foods that we have on Frankie's free range foods calm like The white chocolate chip cookie bars or the brown rice crisp bars I'm eating more meat, and it's amazing Why have you never really tried the raw meat or animal food way of eating Daniel Archer? You have not watched the entirety of my youtube channel We did raw primal for a little bit and that was actually what kind of made my health problems even worse From the dairy consumption and getting the H. Pylori from all the raw sheep and raw goat's milk I was drinking I'm not a fan of that diet in any way whatsoever Tea, Culpepper, what are the causes of dandruff in your opinion? So if you're following my type of diet and you reduce the EMF radiation and you're getting plenty of sun Dandruff should really go away. I used to have all sorts of dry skin problems dandruff And now I do like no shampoo. I follow this diet, and it's not something I worry about anymore And that's really gonna be the answer to a lot of problems for you guys is basically just copy what I'm doing, but For me to simplify that we have the like meal protocol video that we did a few months ago that you guys can check out David Salvador My question is do you believe that omega-6 and pork and poultry are really worth worrying about as bad as the same Linoleic acid from seed oils or vegetable sources? Thanks Yes short answer, they're just as bad and they should absolutely be avoided and then if we're gonna get into like the nuances of well Are they as oxidized as the vegetable seed oils? It's it's really two forms of poison that have to be avoided in the diet under no circumstance Should you eat that if you really have to consume conventional meat beef is the least worst culprit Dominic Cruz is cooked ground beef avoidive nutrition. That's what the Baltic psychopath says. I Don't even want to really address this however, we did do a video on Raw versus cooked and I think we might have also included fresh versus frozen just search Frank Tafano raw versus cooked if you want to really in-depth explanation on that and The the short answer is There's a slight Nutrient loss, but an increased bioavailability in some cases and there's also an anti-scorbutic debate that we discussed in that video Julia Turk. Yeah, we get it. I'm really sorry. This is happening I think I said something in that post that you guys can see I do have a question if that's all right What do you suggest in regards to diet for people living in the poverty level $15,000 and below per year? So if you bought a high quality ground beef for seven to ten dollars a pound and Then had rice or a grain that costed you a few dollars a day You're looking at I would say ten to fifteen dollars a day food cost Which oh God forgive my math is probably four thousand dollars a year So it'd be a very boring diet But I think four to five thousand can easily cover all of your food expenses and supplements If you don't mind sticking to that kind of really really strict boring Protocol of grass-fed ground beef and like organic rice. I think it can be done, but You're not gonna be like enjoying and you know, I mean I kind of waste a lot of money on certain things I don't have to eat but It can in my type of diet can easily be done if you're not having a lot of steaks if you're not doing like all of these treats and expensive organic products It's a lot more enjoyable But it's a lot more expensive FSS8 CVC should I eat beef thyroid draw? Should I eat the skin of the testicles or just the inside part concerned about hypothyroidism and testosterone levels? I mean the glandulars have a higher hormone bioavailability when they're raw and I don't think you have to eat the skin of the testicles at all and Yeah, people can be really sensitive to these so you just want to have a very small amount to start Most people react kind of crazy to the adrenals because of their hormone content, but they do feel better overall the next few days for sure John Martinson, do you really think ketosis is something to avoid or cost when it clearly would have been a natural seasonal experience? If not more frequently for our ancestors so the idea that we used to be like hunter-gatherers and That humans were not intelligent enough to harvest grains. It's kind of crazy, but regardless of whether you're debating if grains are a natural part of our diet, which I personally think they are the lifestyle of someone on a keto animal diet is Not something you can really replicate if you want to be in the forest all day in the sun all day Active the the body's nutrient demands are so much higher in that state that You can't really compare it like if you want to go live with the Australian Aborigines you can go do that and live their diet and lifestyle, but There's a lot of things to consider there that is why I currently follow the diet I follow that is very high in grains and things that are still considered natural foods The legit lad the best way to lose weight fast with your better doctrine I tried one small brown rice beans and beef meal plus kefir grains lost four kilograms in a working week Monday to Friday Refeed Friday I am very interested. I mean sounds like my plan is working very well for you four kilograms is around 10 pounds I mean that's really the the protocol if you want to include some days of fasting you can But the real thing that's going to improve your body composition and make you happier is Getting in the gym and doing the resistance training a very high volume to build up the lean body mass base Nixie the onions increase testosterone like the clown and you Tate was like saying if you eat a lot of raw onions every day It increases your testosterone I think that was him just trolling people and he's laughing inside his head at how much everyone's going to be farting if there's any logic to that maybe there's some sort of gut bacteria that likes the type of carbohydrates found in onions which I mean having a healthier gut would increase your testosterone but compared to actual testosterone sources It's comical Hey, Victoria a third question is there Cassine in beef and calves meat? That's an odd question. I think the answer to that is no The Cassine is the protein that some people are allergic to in in the dairy I Think if you're if you're not Tolerating meat and you're having allergic reactions to it the issues a little bigger than Isolating something like like oh is there possibly a casting trace of it in the mean now? That's a little a little crazy Cambian productions best starch sources So really anything organic is necessary and then as you go down that list It gets very very very subjective. I mean, I personally have been eating a lot of organic bread Which is made from? Red winter wheat they haul it and then the pasta to that I eat like the udon noodles That's been a lot of my calories or I'll do like organic white rice organic brown rice And lately maybe we'll do the barley soups again and all that type of stuff, but There are a lot of grains that I do avoid Overall the best from a nutrient and minimally inflammatory perspective is probably like an organic russet potato or Peeled red potato though. Those are probably the best overall starch sources because they have a nice balance of nutrients fiber and Really minimally inflammatory At what do you know about epsom salt bath? So the transdermal absorption of certain things isn't that great? And it's kind of expensive like I tried doing some magnesium bath on occasion. Yeah, I think they're great I think they're amazing if you can afford to do them if you have a bath and you have the time I mean, I'd love to do a magnesium bath every day and make me but it's just for me It's a lot of work. I think the health benefits of them Maybe you'll feel better.  I'm not the guy to ask because I don't go out. I'd like to think I'm a decent-looking guy I'm not tall so I don't really like bothering too much because in my experience the success rate of me with women is just it's It's so it's really really silly because I mean I don't I'm kind of picky I only go up to really pretty girls and those types of girls will have Options of guys that are you know, just as good-looking as me, but they're literally a foot taller. So I deal with that by Focusing on my business doing things. I enjoy helping other people But if you really want female attention, I would just make a lot of money and then you can go to other countries where That's a lot more accessible and easier to do, you know in America Women are stuck up and if you if you're in that type of Area where you want to spend money on women to get their attention. It is a lot more expensive here But you have to find something that you enjoy doing that makes you happy that That isn't dependent on getting attention from females, although if that's something you really crave and desire There's better and easier way to do that Nia Nahi who is your favorite vegan girlfriend? Oh, I have I have to think about that one and go back through my videos a little bit That brunette girl Sophia as forensic is just bad at like she's so bad She's really really hot and pretty there In I think one of the girls from like our hottest vegan girls video is probably higher on that list because a lot of them are really really pretty models I Just I just can't Recall which one I like the most right now. I'll have to let you guys know ask me again Next video and I'll give you guys a better answer Marie Marie Can you consume cod liver the actual liver is not just the oil while pregnant? I don't recommend that to pregnant woman one because any type of seafood is slightly polluted and two most women are Really deficient in the contrasting fat soluble vitamins So they should really focus on reducing the vitamin A and getting more of the the other nutrients Ninja Frank, what is the official diet you now recommend for overall health since you aren't carnivore anymore? No, are you vegan? It's what I've been following and showing you guys for I mean almost the past two years now, so I don't I'm not gonna go super in-depth answering that question. I mean we just did the video Last week two years no longer carnivore where I explain all that stuff so you guys can Go back and watch that if you want to answer But that is all the questions for this Q&A guys, so Thanks again for participating and as always. Thank you guys for continuing to support My YouTube channel and all of my businesses Which you can see on Frank dash to follow calm if you want to read and learn about all the interesting products So You guys can just leave a like on the video put a comment down below and Be sure to subscribe so that YouTube unsubscribes you next week Thanks for joining me guys, and I'll see you for the next video


This video is sponsored by DistroKid. Follow the VIP link in the description down below to get 7% off an already amazing price to distribute your music to the world. Imagine having access to some of the most renowned piano, synths, organs and vintage keys of all time. No, not on your computer in the studio but right there at your fingertips where you need them. Imagine the Arturia Astrolab. Hi folks, I'm Mike and I hope you're well. To celebrate their 25th anniversary Arturia have just released Astrolab, something they've described to me as their biggest flagship product to date. Now I've been playing with it for the last couple of weeks and I have to say thoroughly enjoying it but it's something a little bit different for this channel which is primarily focused on recording in your studio. This is something a little bit outside of that, although you can use it in the studio. So what exactly is Astrolab? In order to understand Astrolab you first need to understand Arturia's Analog Lab, a software plugin with thousands of presets for a really wide range of instruments, including classic analog and digital synths, electric pianos, organs and beautifully modeled acoustic pianos and in more recent times some original instruments including pigments and their augmented series. Now of course in the past you could take all of this out on the road with you, perhaps on a laptop connected to a controller keyboard and attempted to use Analog Lab in this way but it's fraught with problems in a live situation where you really need a lot of stability and really good usability. So what Arturia did is they took Analog Lab and basically put it inside this hardware keyboard with all of the advantages of stability and usability that you get with that. So let's take a look at that hardware. The Astrolab has 61 semi-weighted keys with velocity and after touch. The green or orange LEDs above each key highlight the note being played. The chassis, back and front are all made of metal and although the top is plastic it has a really nice quality to it as do the eight encoders with LED position rings. The various buttons feel firm and solid along with the pitch bend and modulation controls. Star of the show is the large color screen encoder. We use this to make menu selections or to scroll through instruments and we finally press it to make a selection. There's over 1,000 built-in sounds with over 7,000 free ones from Analog Lab Pro. Now although Astrolab does come with a very nice manual I didn't need to refer to it to use these controls. I found them to be very intuitive and well laid out. To the rear we have midi in and out ports as well as two auxiliary pedal ports and ports for expression and sustain pedals. To the right of this interestingly we have two audio ports. Now these are XLR quarter inch combos and they're for use with instruments such as Arturia's vocoder. A nice touch here is that the gain control can be pushed in. This stops it being accidentally adjusted or broken off during gigs. To the right again we have quarter inch ports for our left and right outputs as well as a quarter inch headphone socket. Next to this we have a USB A port. This can be used for storage or for connecting a midi controller which is handy if you say want to use an 88 key weighted keyboard. Next to this we see a USB C port. This is for connection to your computer and then finally we have a screw in port for the supply 12 volt power supply. So although we've got these handy buttons on the top to quickly select types of instruments you'll probably end up using that beautiful large dial with its color display in the middle where you can select by type by instrument even by artists such as say the Beatles or the Daws. But in a live situation you wouldn't want to be scrolling through potentially thousands of presets to find the next one that you want. So you can save favorites but more importantly you can set up playlists. But using the controls on the keyboard is not the only way to interact with Astrolab. Now using your local Wi-Fi network and the Astrolab connect app you can control your Astrolab using your mobile device. My favorite way to do this was with my iPad but you can use your phone as well. Now it's a little bit of a by the way as well as Wi-Fi Astrolab also has Bluetooth connectivity. Now with this one of the things that you can do is stream audio from say Spotify or something from your phone to Astrolab you'll be able to hear it in your headphones and play along with it which is a great tool for learning and practicing songs. Now as you may hope and expect Astrolab comes with a full version of Analog Lab. You can hook up to it on your computer and create things like your own presets or playlists and transfer those to your Astrolab. Now how many presets can you save? Well probably thousands depending on which instruments you're using. There is an internal storage though of around about 23 gigabytes. I think that's going to be more than enough for most of you. Now I should make a couple of quick mentions about compatibility. Astrolab is compatible with Analog Lab version 9 but not the previous versions. They've also said that as new instruments come out those will be usable with Astrolab as well. By the way don't forget to check out our sponsor DistroKid. If you're releasing your music to major platforms like Spotify, Apple, Amazon, Google Play etc they make it very affordable and very easy and if you use that VIP link in the description down below you're going to get further 7% off of your first year. Astrolab also includes some basic recording and MIDI looping functionality. Now as well as of course recording things like the notes and the velocity of those notes that you play it also records changes you make to the sounds via things like the encoders as well and although it's a kind of a basic functionality it is of course just really easy to use. Now I don't think I've come across a keyboard from Arturia without cord and arpeggiator functionality and the Astrolab is no exception to that. There's a classic synth arpeggiator on there. I'm not going to go into detail with that now but it's worth knowing it's there and also of course chord functionality enabling you to play whole chords with just one note. Now although this is primarily a live instrument it does have its use in the studio especially if you're used to recording with analog lab so what you can do is hook up Astrolab use it to control analog lab inside of your door that will be completely synced up and then when you actually record you'll record the sound from analog lab but monitor it from Astrolab meaning you'll get completely latency free monitoring while you're recording. Now as we know that's a huge benefit because we do tend to suffer from latency in complex projects when we're recording virtual instruments. Now of course you can also use it as a MIDI keyboard controller to control other virtual instruments as well. Now I've got to say I've enjoyed using this over the last couple of weeks just because of its sort of immediacy there's something nice about just switching on a keyboard and getting some sound without having to go through a computer and open up software etc. I've really enjoyed using pigments on this or the pigments presets I love the augmented series I really like the modelled acoustic pianos and some of the classic keys as well I just really like things like B3s and you know whirly sort of sounds and all that kind of thing that's just my particular preference let me know what your favorite instruments from the analog collection would be to use on Astrolab. Is this going to revolutionize things for you if you're a live player let me know in the comments down below. As ever don't forget to subscribe and like and I'll see you in the next video.