1 00:00:00,930 --> 00:00:06,960 Fine so we are still dealing with one dimensional flows, but now we focus on an area called 2 00:00:06,960 --> 00:00:08,460 bifurcations. 3 00:00:08,460 --> 00:00:17,420 This lecture is going to be a brief introduction to the area of bifurcation theory. 4 00:00:17,420 --> 00:00:25,830 Now dynamics of vector field on the line are actually not so exciting. 5 00:00:25,830 --> 00:00:35,300 The solutions either go ahead and settle down towards equilibrium or they actually go off 6 00:00:35,300 --> 00:00:38,160 to plus or minus infinity. 7 00:00:38,160 --> 00:00:42,510 So in that sense they are not extremely exciting dynamics. 8 00:00:42,510 --> 00:00:49,030 So, what really is exciting about one dimensional systems? 9 00:00:49,030 --> 00:00:56,710 The answer turns out to be its dependence on parameters. 10 00:00:56,710 --> 00:01:06,590 The qualitative structure of the flow can actually change quite dramatically as system 11 00:01:06,590 --> 00:01:20,260 parameters are varied, for example fixed points can be created or destroyed and the stability 12 00:01:20,260 --> 00:01:27,220 of the fixed point itself may change has system parameters are varied. 13 00:01:27,220 --> 00:01:37,140 So, the qualitative change in dynamics is what we referred to as bifurcations and the 14 00:01:37,140 --> 00:01:46,509 parameter values at which such bifurcations occur are called bifurcation points. 15 00:01:46,509 --> 00:01:52,840 So, we are really delving into this area of applied mathematics called bifurcation theory. 16 00:01:52,840 --> 00:02:05,100 It is really the study of changes in the qualitative or topological structure of dynamical systems. 17 00:02:05,100 --> 00:02:17,020 So, a bifurcation occurs when a small smooth change to some parameter values referred to 18 00:02:17,020 --> 00:02:28,549 as bifurcation parameters of a differential equation causes a sudden qualitative or topological 19 00:02:28,549 --> 00:02:31,120 change in its behaviour. 20 00:02:31,120 --> 00:02:41,700 Now interestingly the name bifurcation apparently was first introduced by Henry Poincare in 21 00:02:41,700 --> 00:02:43,790 year 1885. 22 00:02:43,790 --> 00:02:53,909 So, bifurcations essentially provide us with a way to understand transitions and instabilities 23 00:02:53,909 --> 00:03:00,579 as some system control parameter actually varies. 24 00:03:00,579 --> 00:03:06,330 Consider a simple example bulking of a beam. 25 00:03:06,330 --> 00:03:14,799 So, you have a beam, the beam has a certain weight which is placed on top of it and if 26 00:03:14,799 --> 00:03:21,699 you go ahead and increase the weight beyond certain critical value the beam actually buckles. 27 00:03:21,699 --> 00:03:28,559 So, what we find is that beam as buckled under the weight, when the weight crosses certain 28 00:03:28,559 --> 00:03:29,559 value. 29 00:03:29,559 --> 00:03:41,370 So, with a small weight the beam remains vertical and can comfortably support the load. 30 00:03:41,370 --> 00:03:52,799 As the weight increases the beam then buckles under the weight and we observe a qualitative 31 00:03:52,799 --> 00:03:54,959 change in the system. 32 00:03:54,959 --> 00:04:02,840 So, in this example the weight act as the control parameter and the deflection of the 33 00:04:02,840 --> 00:04:08,540 beam from the vertical is the dynamical variable. 34 00:04:08,540 --> 00:04:15,419 So, this is our first example may be observed bifurcation phenomena happening (()) (04:20), 35 00:04:15,419 --> 00:04:24,660 where there is a qualitative change in the system when a certain parameter varies. 36 00:04:24,660 --> 00:04:29,539 The dynamics of the vector field on real line are not very exciting. 37 00:04:29,539 --> 00:04:35,900 One of two things can actually happen, number one is that the solution can actually blow 38 00:04:35,900 --> 00:04:42,879 of to infinity or solutions which actually converge to some fixed point, so only these 39 00:04:42,879 --> 00:04:44,330 two things that really happening. 40 00:04:44,330 --> 00:04:50,680 You say what so exciting about dynamics or of vector filed on real line? 41 00:04:50,680 --> 00:04:54,680 And the answer turns out to be its dependence on parameters. 42 00:04:54,680 --> 00:05:00,740 When you construct a model of the real world, almost every model that you find will have 43 00:05:00,740 --> 00:05:05,969 some parameter or the other in fact usually they too many parameters. 44 00:05:05,969 --> 00:05:13,740 So the question we are interested in, is that how does changes in the parameter of the underline 45 00:05:13,740 --> 00:05:19,960 model actually induce a qualitative change in the dynamics and that can happen in one 46 00:05:19,960 --> 00:05:21,380 or two ways. 47 00:05:21,380 --> 00:05:29,100 Number one, is that either fixed points can be created or destroyed as parameter varies 48 00:05:29,100 --> 00:05:35,280 or the stability of the fixed points themselves can change. 49 00:05:35,280 --> 00:05:40,220 These are two important qualitative changes that can actually happen in the system as 50 00:05:40,220 --> 00:05:42,659 a parameter actually varies. 51 00:05:42,659 --> 00:05:46,729 And we give a very simple example, you can just get a motivation and get an intuition 52 00:05:46,729 --> 00:05:53,740 in order and that is that imagine you have beam on top the beam you have a certain weight 53 00:05:53,740 --> 00:06:00,189 and the beam can actually hold the weight without actually buckling. 54 00:06:00,189 --> 00:06:05,650 But now what we do is we slowly increase the weight, when the weight increases a certain 55 00:06:05,650 --> 00:06:06,820 threshold. 56 00:06:06,820 --> 00:06:10,110 You will actually find that the beam buckles. 57 00:06:10,110 --> 00:06:17,460 So, this transitions from the beam being straight when the weight is small to the beam buckling 58 00:06:17,460 --> 00:06:18,969 when the weight is big. 59 00:06:18,969 --> 00:06:22,569 Actually, is a qualitative change in the system dynamics. 60 00:06:22,569 --> 00:06:29,539 So, the weight actually acts as a control parameter and as that parameter varies, there 61 00:06:29,539 --> 00:06:32,770 is a qualitative change in the system dynamics. 62 00:06:32,770 --> 00:06:39,120 So, this is just you know brief motivation to bifurcation theory and why we would be 63 00:06:39,120 --> 00:06:45,430 interested in bifurcation in the system and in particular the role that parameters will 64 00:06:45,430 --> 00:06:48,930 play inducing bifurcation phenomena of certain type.