1 00:00:00,880 --> 00:00:10,360 In this lecture, we focus on linear stability analysis to determine the stability of fixed 2 00:00:10,360 --> 00:00:22,949 points. We had earlier used graphical methods. We now consider linearizing about a particular 3 00:00:22,949 --> 00:00:35,880 fixed point, what really is linearization? Well it is finding a linear approximation 4 00:00:35,880 --> 00:00:50,580 to a function at a given point. So, the linearization of a function is the first order term of its 5 00:00:50,580 --> 00:00:59,020 Taylor series expansion around the given point. 6 00:00:59,020 --> 00:01:11,799 In terms of nonlinear systems linearization allows us to study the local stability of 7 00:01:11,799 --> 00:01:23,870 an equilibrium point of a nonlinear differential equations. And it allows us use tools for 8 00:01:23,870 --> 00:01:37,340 studying linear systems to analyse a nonlinear system near a given special point. 9 00:01:37,340 --> 00:01:49,670 Now consider x dot = f(x) let x* be a fixed point and let x(t) be eta of t plus x* where 10 00:01:49,670 --> 00:02:02,340 eta of t is a small perturbation away from x*. We really want to know if the perturbation 11 00:02:02,340 --> 00:02:10,650 actually grows or decays. So we go ahead and derive a differential equation for eta. So, 12 00:02:10,650 --> 00:02:26,820 eta of t = x of t minus x* so eta dot = dx dt x(t)- x* which = x dot as x* is simply 13 00:02:26,820 --> 00:02:28,300 a constant. 14 00:02:28,300 --> 00:02:41,810 So eta dot = x dot which = f(x*) which = f(x*) plus eta and so now we go ahead and using 15 00:02:41,810 --> 00:02:57,610 a Taylor series expansion we get f(x*) plus eta = f(x*) plus eta times f prime of x* plus 16 00:02:57,610 --> 00:03:07,730 terms which are order eta square. Note that this is the big “O” notation where this 17 00:03:07,730 --> 00:03:19,520 term actually denotes quadratically small terms in eta. 18 00:03:19,520 --> 00:03:32,120 Note that f(x*)=0 since x* is a fixed point. Hence eta dot = eta times f prime of x* plus 19 00:03:32,120 --> 00:03:41,870 terms that are order eta square. If f prime of x* is not equal to 0 the order eta square 20 00:03:41,870 --> 00:03:51,740 terms are in fact negligible and we get the following approximation eta dot = eta times 21 00:03:51,740 --> 00:04:02,740 f prime of x*. Now this is a linear equation in eta which is called the linearization about 22 00:04:02,740 --> 00:04:13,790 x*. And here are some notes the perturbation etas of t grows exponentially. 23 00:04:13,790 --> 00:04:23,060 If f prime of x* is > 0 and decay is f of prime of f star less than 0. If f prime of 24 00:04:23,060 --> 00:04:34,130 x* =0 the order eta square terms cannot be ignored and we will need some sort of nonlinear 25 00:04:34,130 --> 00:04:47,000 analysis for the equations. Now if you consider the magnitude of f prime of x* then this magnitude 26 00:04:47,000 --> 00:04:58,790 plays a role of an exponential growth or decay rate. It is reciprocal one upon f prime of 27 00:04:58,790 --> 00:05:11,560 x* is a characteristic time scale. Now this determines the time required for x of t to 28 00:05:11,560 --> 00:05:20,410 vary significantly in the neighbourhood of the fixed point x*. 29 00:05:20,410 --> 00:05:27,970 Now let us consider this equation x dot = sine x. Now recall that from the geometric view 30 00:05:27,970 --> 00:05:35,180 of thinking essentially, what we said was that when we have equation of form x dot = f(x*) 31 00:05:35,180 --> 00:05:43,260 then what we first do is just plot x dot versus x. So for this particular equation where f(x*) 32 00:05:43,260 --> 00:05:51,020 is sine x. So what we see here is a plot of x dot versus x where using the geometric view 33 00:05:51,020 --> 00:05:57,170 of thinking that we have outlined earlier we were able to classify the stability of 34 00:05:57,170 --> 00:06:04,930 the fixed point noting that we are able to do so without any formal analysis. 35 00:06:04,930 --> 00:06:12,169 And so now we go ahead and use the linear stability analysis to determine the stability 36 00:06:12,169 --> 00:06:22,790 of the fixed points for the equation x dot = sine x. Recall that the fixed points occur 37 00:06:22,790 --> 00:06:34,490 where f(x*) = sine x =0. So the fixed points are at x* = K pi where K is an integer then 38 00:06:34,490 --> 00:06:46,990 f prime of x* = cos of x* which = cos of K pi which = 1if K is even and -1if K is odd. 39 00:06:46,990 --> 00:07:01,640 So when K is even x* is unstable and when K is odd then x* is stable, so this is actually 40 00:07:01,640 --> 00:07:10,490 in complete agreement with the results that we obtained from the geometric view of thinking. 41 00:07:10,490 --> 00:07:17,840 Now, note that with geometric view of thinking, we had no analytic basis really to establish 42 00:07:17,840 --> 00:07:23,370 the stability of the fixed points and it was purely geometric. But now we actually have 43 00:07:23,370 --> 00:07:30,000 an analytical and algebraic technique to actually establish the stability or the instability 44 00:07:30,000 --> 00:07:32,210 of the fixed points. 45 00:07:32,210 --> 00:07:40,139 Now let us consider examples that arises in population growth consider the equation dn 46 00:07:40,139 --> 00:07:55,490 dt = r N times 1- N by K where N of t is the population at time t, r > 0 is the growth 47 00:07:55,490 --> 00:08:05,800 rate and K is carrying capacity of the population. Our objective is that using linear stability 48 00:08:05,800 --> 00:08:17,520 analysis, we wish to classify the fixed points of the model and we also wished to find characteristic 49 00:08:17,520 --> 00:08:36,330 time scale of the system. Now f(N) = r N times 1-N by k. So the fixed points are N*=0 and 50 00:08:36,330 --> 00:08:53,520 N*= K evaluating f prime of N we get r -2rN / K so f prime of N at N*=0 is r. 51 00:08:53,520 --> 00:09:09,360 So N*= 0 is actually a unstable fixed point. And f prime of N evaluated at N*= K is –r, 52 00:09:09,360 --> 00:09:24,399 so N*= K turns out to be a stable fixed point. In both of the above cases the characteristic 53 00:09:24,399 --> 00:09:33,710 time scale turns out be the same, so one upon the absolute value of f prime of N*= one upon 54 00:09:33,710 --> 00:09:42,110 r for both the fixed points. Now let us go ahead and plot N dot versus N by required 55 00:09:42,110 --> 00:09:51,020 familiar with making such plots. So that is the plot for N dot versus N, 0 and K the two 56 00:09:51,020 --> 00:09:59,610 fixed points have been highlighted and K represents the stable fixed point and 0 represents the 57 00:09:59,610 --> 00:10:05,100 unstable fixed point. 58 00:10:05,100 --> 00:10:14,540 So what can we say about the stability of a fixed point when f prime of x* actually 59 00:10:14,540 --> 00:10:26,230 is 0 as a matter of fact in general we cannot say anything stability has to be worked out 60 00:10:26,230 --> 00:10:34,160 and determined on a case by case basis. Now let us go ahead and consider some examples, 61 00:10:34,160 --> 00:10:46,680 x dot = - x cube, x dot = x cube and x dot = x square in all the cases x* = 0 and f prime 62 00:10:46,680 --> 00:10:55,200 of x* =0. Now let us go ahead and consider stability in each of these cases. 63 00:10:55,200 --> 00:11:04,110 So we plot x dot versus x and what we find is that what we have attracting stable fixed 64 00:11:04,110 --> 00:11:12,880 point and similarly plotting x dot versus x in this particular case we find that we 65 00:11:12,880 --> 00:11:21,520 have an unstable fixed point. So the last case actually presents us rather interesting 66 00:11:21,520 --> 00:11:30,589 situation. We go ahead and plot x dot versus x and that is the plot. So we find that the 67 00:11:30,589 --> 00:11:38,640 fixed point is attracting from the left and it turns out be repelling from the right. 68 00:11:38,640 --> 00:11:46,180 So we get the situation where the fixed point turns out to be stable on the one side but 69 00:11:46,180 --> 00:11:54,519 unstable on the other side. So such a situation is referred to as a half stable fixed point 70 00:11:54,519 --> 00:11:59,730 attracting from one end and repelling from the other. 71 00:11:59,730 --> 00:12:06,660 Ok so let us do the quick recap of this lecture. This lecture was essentially about linear 72 00:12:06,660 --> 00:12:13,260 stability analysis. We still dealing with the equations of form the x dot = f(x*) and 73 00:12:13,260 --> 00:12:19,399 essentially the idea was that you identify the equilibrium point. You take a Taylor series 74 00:12:19,399 --> 00:12:26,820 expansion of the original nonlinear system around the equilibrium point. You retain the 75 00:12:26,820 --> 00:12:33,240 linear terms and all higher order terms are actually thrown out. So you are essentially 76 00:12:33,240 --> 00:12:37,980 left with linearized version of the original nonlinear system. 77 00:12:37,980 --> 00:12:43,100 And now you can go ahead and apply all the tools and methods that you learned for linear 78 00:12:43,100 --> 00:12:49,430 system analysis to this particular equation. So that is a big advantage with this advantage 79 00:12:49,430 --> 00:12:54,570 is that you actually thrown out all the nonlinear terms out ok. So if you thrown all the nonlinear 80 00:12:54,570 --> 00:12:59,180 terms out all the fun all the interesting dynamics are essentially been thrown out, 81 00:12:59,180 --> 00:13:05,180 it is a good start, but it is a only a start, so then what we did was we took a few examples 82 00:13:05,180 --> 00:13:11,220 x dot = sine x. We took another example, where which was motivated from a population dynamics. 83 00:13:11,220 --> 00:13:17,370 And then we essentially showed that if you conduct a geometric reasoning around nonlinear 84 00:13:17,370 --> 00:13:23,300 system which is by plotting x dot versus x or if you did a linear stability analysis 85 00:13:23,300 --> 00:13:30,029 then you got essentially the same results. Except that we left you with some food for 86 00:13:30,029 --> 00:13:34,480 fork with last example. The last example was rather interesting one, now essentially what 87 00:13:34,480 --> 00:13:39,990 you actually had was that you had was fixed point, which it turned out be attracting from 88 00:13:39,990 --> 00:13:44,420 one end but repelling from the other end. 89 00:13:44,420 --> 00:13:51,750 So the fixed point could not really be classified has an stable fixed point or an unstable fixed 90 00:13:51,750 --> 00:13:58,320 point and the terminology we use was a half stable fixed point right and I think we were 91 00:13:58,320 --> 00:14:04,510 just we leave this lecture with that particular food for fork with you that you could also 92 00:14:04,510 --> 00:14:09,600 end up you know fixed point would essentially not be just stable or unstable. But you can 93 00:14:09,600 --> 00:14:15,260 an actually have this sort of hybrid cases which arise, which is half stable and half 94 00:14:15,260 --> 00:14:15,620 unstable.