In this optional sub unit I'll present the bifurcation diagram for a different differential equation and this will lead us to the phenomenon of Hysteresis or path dependence. Well see that in a second. We'll start with this differential equation so dx/dt, Ill use x this time instead p because this doesn't really represent a population, is rx plus x cubed minus x to the fifth so r is now our parameter before it was h. This time we'll use r. So well build up the bifurcation diagram piece by piece by letting r be different values, plotting the right hand side of this and seeing what the I'm function looks like and making a phase line. So here's a we have when r equals one. Down up and down. So there are three fixed points cause the line crosses the x-axis three times: here , here and here. So three fixed points: one, two, three. Make a note this is for r equals 1. When this function is negative (this is a derivative), when the derivative is negative X is decreasing, when positive we're increasing, negative decreasing, positive increasing. So this function has 3 fixed points. There's an unstable fixed point at 0 and there are two stable fixed points out here a little bit more than one away from the origin. So that's the situation when r equals 1. If I decrease r and make it a little bit negative this curve gets a little wiggle in it and it starts to look like this So the curve gets steeper but then it acquires a little wiggle in here. So let's calculate - figure out - the phase line for this. Here we have five fixed points that's a new record for us. 1,2,3,4,5 equilibria to classify and they kinda scrounge together. Thats gonna be a little bit challenging for me to draw. Ok. So there there are fixed points 1,2,3,4,5. The function is positive that means we are moving to the rightI, negative in here then, positive in this region, negative, positive, negative. This is r equals (-) 0.2. Okay so I see three stable fixed points: here, here and here in the middle. So you probably noticed a stable fixed point occurs when the line crosses across from top to bottom. So that happens here, here and here and then we have these two unstable fixed points in between: here and here when the line goes from below to above. So 5 fixed points, 3 are stable and two are unstable. So this is the story for r equals, that should be -0.2. The last r will look at is r=-0.4. So r iss a little bit more negative here and what happens is that these bumps straighten out, so this bump in this bumb get pulled up or down and we end up with this. So here the phase line is kinda simple almost boring again. We have just one fix point. So we had five but four of them disappeared and were left just with this one at the origin and it keeps its stability. So we had, its a little hard to see, we had four and then here we have one but this one, the one at the origin, remains. Okay so we have three phase lines and then we can uh- connect them sort of glue them all together and see what the bifurcation diagram might look like. So as before I'm going to slice off these phase lines and let's take a look so here's r equals 1, here is r equals -0.2 and I should have written here  this was r equals minus 0.4. So here is what we have. So from these three phase lines it might not be immediately clear what the uhm- what the entire bifurcation diagram looks like. We might wanna do a few more phase lines for intermediate r values try an r of 0, an r of minus .1, an r of plus point one and so on. But rather than take the time to do that let me sort of scetch what this looks like and then I'll show you a neater drawing of this bifurcation diagram since the main goal is, is to get this bifurcation diagram and then look at it and learn about hysteresis. So let me just draw a few things on here. So let me just draw a few things on here. I am going to use blue for an unstable fixed point and so it turns out we have a line of unstable fixed points here. Wait those are stable Oh dear, can I recover from this? This was gonna be red or its red and blue (maybe it looks purple) or it looks mostly red, all right, so these are stable (just the wrong color) they are stable, the arrows are going in and then we also have some stable fixed points here in here. Here and here, and these are gonna look like this and this one's going to come down like this sand then we'll have unstable fixed points here, here and this line connects up here. So that's our bifurcation diagram um it's not the best picture in the world. To me it kinda looks like a fish, like a salmon, let's throwing up which, you know, is not what I intended. Okay, but this is the bifurcation diagram so we have uhm- stable points in red and unstable points in blue and hopefully you can see how these lines the blue and red lines line up with these fixed points in this vomiting fish looking thing. Okay so let me draw a nicer version of this diagram and then we'll analyze that and uhm- learn about Hysteresis. So here's a slightly neater version of the bifurcation diagram from the previous screen and I'll be focusing just on the positive x values, I have only drawn arrows on here. So we have a line of stabble fixed points, attractors, and then we have here in blue a line of unstable fixed points, repellers, then unstable here and then stable here. So let's imagine sort of talk through a scenario like this that the parameter, for whatever reason, starts of somewhere out here and we have positive x value. So then we're gonna get pulled to this attractor and now imagine that the parameter starts decreasing who knows what the parameter is in this case. I don't know that there's a clear physical or uhm- anaolog or something but whatever r is, it decreases and so as r decreases then the equilibrium value begins to decrease and we decrease r some more the equilibrium value decreases further and we move down along here. And this looks a lot like what happened when we were increasing the fishing rate in the logistic differential equation. So we moved down here, we move down here. R continues to decrease, r continues to decrease, r continues to decrease until you get here and then this uhm- fixed point, this attractor here disappears, is gone, and so then if we decrease r a little bit more the quantity X, whatever it is, is gonna get pulled down here to 0. And so then perhaps we like this, this positive thing is good, zero is bad, maybe this is the growth rate in the economy or some fishing number of fish or something, we zip down here and then we might say uh-oh we crashed, we better increase r. And so well increase r butt his red point here 0 is stable, its attracting and so we don't automatically jump back up to here because this a stable. We move a little bit we get pushed back. So then we increase r, well increase r, well increase r , still.. more, until we get a little bit over here and then this fixed point at 0 loses its stability we go from red to blue and then you'll jump back up to here. So again were seeing jumps but this time its uhm- there's a new feature which is as follows: suppose we wanted to know if r was around here, whenever that is  -0.2. What stable behavior would we observe in this model and the answer is that it would depend not just on the r-value but where one came from and this is the idea of Hysteresis. Let me draw a picture to illustrate or outline almost literally the story that I just told. So thinking about this uhm- portion of the bifurcation diagram - I guess Ill make a really rough sketch of this- so I can move down this way, then I come to this collapse point and Ill go down here, then I would increase until here and then I would jump back up and could go in either direction here. So this is to connect up with this would be the r=0 point. So this system now has what you call path dependence. So what would you observe at this r value? Well it depends not just on the r value but on the path you took to get there. If you reach this r-value -the one on my finger is-from above, from the right then you'd be up here, here on this diagram. If you approached this r-value from below having gone beyond this sort of falling off that cliff- then you'll be down here at 0. So this is called Hysteresis or path dependence. So the term for this type of behavior is known as Hysteresis or path dependence, so that the equilibrium property, the observed behavior of this differential equation, this model depends not only on r - it looks like it depends only on r, if you tell me what r is I can solve the differential equation and I can tell you what X would end up being - but in a situation where you have multiple attractors and theyre arranged like this knowing r is not enough. You need to also know where r came from. It depends not just on r but the path that r took. And this is surprising an interesting I think because path dependence is a type of memory. The value that population or whatever this is in a sense remembers where it's been but it's not obvious at all that this equation has memory built into it. This just says that the growth rate, the rate of change of X depends on X and this number r. So it's a type of memory or history that gets introduced into a differential equation as a result of this bifurcation, uhm, this particular structure in a bifurcation diagram like this. I don't know that this is common or ubiquitous in differential equations but it's not uncommon either and you don't need a tremendously complicated differential equation in order to get this behavior. So this is another type I guess of bifurcation, [rather two?} bifurcation here and a bifurcation there and taken together those two bifurcations lead to this path dependence . So uhm- again just to underscore one more time we have a simple differential equation, something that's continuous, smooth, differentiable, doesn't have anymemory built-in and we can have a system that behaves in jumps and that develops a sort of memory or path dependent. So that's the idea behind Hysteresis and path dependence.