1 00:00:01,599 --> 00:00:12,680 The first bifurcation is the saddle note bifurcation. This is the basic mechanism by which fixed 2 00:00:12,680 --> 00:00:23,630 points are either created or destroyed, essentially as a system parameter is varied. The two fixed 3 00:00:23,630 --> 00:00:36,719 points moves towards each other collide and then mutually annihilate. Prototypical example 4 00:00:36,719 --> 00:00:46,490 is x dot = r + x square where r is a parameter which is greater than 0, is equal to 0 or 5 00:00:46,490 --> 00:00:55,199 less than 0. So, for r less than 0, if you plot x dot versus x, that is the plot for 6 00:00:55,199 --> 00:01:01,890 x dot versus x, we find that we have two fixed points. 7 00:01:01,890 --> 00:01:11,210 So, the system has two fixed points one is stable and the other is unstable, for r is 8 00:01:11,210 --> 00:01:20,719 equal to 0. When we plot x dot versus x, we find that we only have one fixed point. The 9 00:01:20,719 --> 00:01:28,780 fixed point is attracting from the left and repelling from the right. So, the fixed points 10 00:01:28,780 --> 00:01:41,750 actually turn into a half stable fixed points at x* = 0 and when r is greater than 0 plotting 11 00:01:41,750 --> 00:01:52,820 x dot versus x reviles that in fact we have no fixed points. 12 00:01:52,820 --> 00:02:03,860 So, a bifurcation effectively takes place at r = 0 as the vector fields for r greater 13 00:02:03,860 --> 00:02:10,290 than 0 and r less than 0 are qualitatively different from each other. 14 00:02:10,290 --> 00:02:18,170 Now, we go ahead and plot something know as bifurcation diagrams. So, if we show a stack 15 00:02:18,170 --> 00:02:30,530 of vector fields for discreet values of r that is the stack of vector fields, when r 16 00:02:30,530 --> 00:02:38,970 greater than 0 there are no fixed points. When r is equal to 0, there is half stable 17 00:02:38,970 --> 00:02:51,560 fixed point and when r less than 0, we have two fixed points, one stable and other unstable 18 00:02:51,560 --> 00:03:02,590 and that is what happen when r varies, so we have stable and unstable branches. 19 00:03:02,590 --> 00:03:16,840 So, a common way is to treat the parameter r as playing the role of an independent variable 20 00:03:16,840 --> 00:03:26,170 in the limit of a continuous stack of vector fields. What we get is the following, so we 21 00:03:26,170 --> 00:03:38,390 plot x versus r as the stable branch and that is the unstable branch. So, we can also include 22 00:03:38,390 --> 00:03:53,761 arrows in the diagram, so including arrows in the bifurcation diagram we get. So, this 23 00:03:53,761 --> 00:04:03,390 is called the bifurcation diagram for the saddle node bifurcation. So, this is the first 24 00:04:03,390 --> 00:04:11,510 bifurcation that we have dealt with and diagram on the left is referred to as its bifurcation 25 00:04:11,510 --> 00:04:16,130 diagram. 26 00:04:16,130 --> 00:04:28,500 Let us now conduct a linear stability analysis of x dot = r - x square, we first identify 27 00:04:28,500 --> 00:04:38,920 the fixed points, so x dot = f(x) = r - x square, yields(gives)x* = plus minus square 28 00:04:38,920 --> 00:04:50,260 root of r. For r greater than 0, we get two fixed points and to establish linear stability, 29 00:04:50,260 --> 00:05:01,370 we get f prime of x* = -2 times x*. So x* = plus the square root of r yields a stable 30 00:05:01,370 --> 00:05:10,730 fixed points x* = - square root of r is unstable. For r less than 0, there are actually no fixed 31 00:05:10,730 --> 00:05:24,020 points, for r = 0, we get f prime of x* = 0 and so the linearized term actually vanishes. 32 00:05:24,020 --> 00:05:35,150 Let us consider an examples, consider the first order system x dot = r - x -e to the 33 00:05:35,150 --> 00:05:47,750 -x, show that it under goes a saddle node bifurcation as r varies, further finding the 34 00:05:47,750 --> 00:05:58,800 value of r at the bifurcation point. First, we plot r - x and e to the -x on the same 35 00:05:58,800 --> 00:06:11,180 graph, where the line r - x intersects with the curve e to the -x, we get r - x = e to 36 00:06:11,180 --> 00:06:24,050 the -x and so f(x) which is r - x e to the -x = 0. So, intersections of the line and 37 00:06:24,050 --> 00:06:31,330 the curve actually correspond to the fixed points of the system. 38 00:06:31,330 --> 00:06:39,180 Now let us go ahead and make some plots, that is the line r - x that is the curve e to the 39 00:06:39,180 --> 00:06:54,370 -x. So we note that this system has two fixed points, one stable and the other one is unstable. 40 00:06:54,370 --> 00:07:03,270 Note that the flow is to the right where r - x is greater than e to the -x and so x dot 41 00:07:03,270 --> 00:07:14,320 is greater than 0. Now let us go ahead and decrease the parameter r little bit, so that 42 00:07:14,320 --> 00:07:22,790 is the line r - x, that is curve e to the -x and note that in this case we have one 43 00:07:22,790 --> 00:07:24,260 fixed point. 44 00:07:24,260 --> 00:07:32,730 The fixed point turns out to be actually a half stable fixed point. So, at some critical 45 00:07:32,730 --> 00:07:44,280 value r = rc the line actually becomes tangent to the curve. So that is the saddle node bifurcation 46 00:07:44,280 --> 00:07:59,820 point, for r less than rc that is r - x that e to the -x and note that they do not actually 47 00:07:59,820 --> 00:08:05,479 intersect at all, so there are no fixed points. 48 00:08:05,479 --> 00:08:15,949 We still need to find the bifurcation point rc. So, we impose the condition that the graphs 49 00:08:15,949 --> 00:08:30,270 of r - x and e to the -x intersect tangentially. We require equality of the functions and their 50 00:08:30,270 --> 00:08:45,940 derivatives. So, e to the -x = r -x and d dx of e to the -x is d dx of r - x which gives 51 00:08:45,940 --> 00:09:01,340 us -e to the -x = -1, which yields x = 0, so substituting x = 0 into e to the -x = r 52 00:09:01,340 --> 00:09:15,399 - x gives r =1. So, the bifurcation point is at r critical =1 and the bifurcation occurs 53 00:09:15,399 --> 00:09:20,170 at x =0. 54 00:09:20,170 --> 00:09:26,080 In this lecture, we introduced the saddle node bifurcation. This is the basic mechanism 55 00:09:26,080 --> 00:09:32,829 through which fixed points can be created or destroyed. We introduced prototypical examples 56 00:09:32,829 --> 00:09:39,730 for the saddle node. Now essentially what it showed us was the following, we had a model, 57 00:09:39,730 --> 00:09:45,910 where we actually had two fixed points, one of the fixed point was stable and one of the 58 00:09:45,910 --> 00:09:53,250 fixed points was unstable. And then a parameter in the system changes as the parameter changes, 59 00:09:53,250 --> 00:09:56,839 both of these fixed points actually came close to each other. 60 00:09:56,839 --> 00:10:04,639 And at particular point of time they actually merged and became one half stable fixed point 61 00:10:04,639 --> 00:10:10,640 and as the parameter change even more, in fact the fixed point in the system vanished. 62 00:10:10,640 --> 00:10:17,449 So, we had a scenario, where we had two fixed points stable and an unstable. Then we had 63 00:10:17,449 --> 00:10:24,129 one fixed point which was half stable fixed point and as the parameter varied all fixed 64 00:10:24,129 --> 00:10:25,350 points in the system actually vanished. 65 00:10:25,350 --> 00:10:32,040 So that is why fascinating because when you modelling the real world, you have the real 66 00:10:32,040 --> 00:10:38,910 world of which you abstract about a simplified model and that model will have some parameters 67 00:10:38,910 --> 00:10:47,300 in it. And so the lesson here is that as parameters varies you would have fairly serious qualitative 68 00:10:47,300 --> 00:10:54,569 changes in the underline dynamics and we illustrated that through the saddle node bifurcation in 69 00:10:54,569 --> 00:10:55,060 this lecture.