0 00:00:04,823 --> 00:00:09,333 That brings us to a crucial new idea, which is one of the most important 1 00:00:09,333 --> 00:00:13,020 qualitative ideas that we'll use in the course. 2 00:00:13,020 --> 00:00:16,730 And I hope I've motivated effectively, let's introduce the definition and 3 00:00:16,730 --> 00:00:19,810 then go back and see how it's motivated, okay. 4 00:00:19,810 --> 00:00:25,210 So, we're going to say that a vector field is hyperbolic at a fixed point, 5 00:00:26,250 --> 00:00:30,590 exactly when its linearization, that is it's Jacobian evaluated 6 00:00:30,590 --> 00:00:35,480 at that fixed point, has no purely imaginary eigenvalues. 7 00:00:38,040 --> 00:00:40,300 Let's give some examples of this. 8 00:00:40,300 --> 00:00:44,870 In the scalar damped mass case, if I look 9 00:00:44,870 --> 00:00:49,800 at my damped vector field FD And I expressed the Jacobian, 10 00:00:49,800 --> 00:00:53,870 the Jacobian d by dv of f is just the constant negative b. 11 00:00:53,870 --> 00:00:58,660 And that Jacobian vanishes exactly 12 00:00:58,660 --> 00:01:02,750 when b is not equal to 0, when b goes to 0. 13 00:01:02,750 --> 00:01:10,310 The only way for a one by one matrix, that's what b is, it's a 1 by 1 matrix. 14 00:01:10,310 --> 00:01:14,660 Why because it's the Jacobian of a scalar value function of a scalar so 15 00:01:14,660 --> 00:01:15,680 it's a 1 by 1 matrix. 16 00:01:15,680 --> 00:01:21,030 The only way for the eigenvalues of a 1 by 1 matrix 17 00:01:21,030 --> 00:01:26,250 to be imaginary is if they are exactly 0 right? 18 00:01:26,250 --> 00:01:30,600 Because the only intersection of the imaginary axis 19 00:01:30,600 --> 00:01:34,070 with the real axis in the complex plane is at 0. 20 00:01:34,070 --> 00:01:38,060 Okay, so by saying that I didn't want this thing to be zero, I was really 21 00:01:38,060 --> 00:01:43,110 saying apparently that I didn't want it to have any purely imaginary eigenvalues. 22 00:01:43,110 --> 00:01:47,645 In contrast, let's go to that vector case that we worked on. 23 00:01:47,645 --> 00:01:53,405 You'll see that the skewed symmetric matrix J has purely imaginary eigenvalues. 24 00:01:53,405 --> 00:01:58,455 Alternatively, go to the damped pendulum, and you'll see that the fixed point 25 00:01:58,455 --> 00:02:04,175 at zero velocity if you take either the up pointing pendulum or 26 00:02:04,175 --> 00:02:10,030 the down pointing pendulum at either 0 PI or negative PI and plug that in. 27 00:02:10,030 --> 00:02:17,020 You'll see again that this term doesn't depend upon theta dot at all. 28 00:02:17,020 --> 00:02:18,250 This term depends on theta. 29 00:02:18,250 --> 00:02:22,810 If I plug in n pi over here you can see that this is either going to + or- 1. 30 00:02:22,810 --> 00:02:28,840 This determinant is not going to 0, 0 times whatever this quantity is doesn't 31 00:02:28,840 --> 00:02:33,296 vanish but mine is minus this is not equal to 0 so this guy is never 0, right? 32 00:02:33,296 --> 00:02:39,830 F DP determinant 33 00:02:39,830 --> 00:02:45,230 is never equal to 0 As long as these terms are not equal to 0. 34 00:02:45,230 --> 00:02:48,230 But if I let b go to 0, if I let b go to 0, 35 00:02:49,440 --> 00:02:52,480 what you'll see is you get a skew-symmetric matrix. 36 00:02:52,480 --> 00:02:57,210 And its eigenvalues are also purely imaginary, and so 37 00:02:57,210 --> 00:02:59,990 it fails the hyperbolicity case, okay? 38 00:02:59,990 --> 00:03:05,890 So what we're saying is, the hyperbolicity condition is 39 00:03:05,890 --> 00:03:11,430 inherited from our realization that even if the Jacobian matrix is non-singular, 40 00:03:11,430 --> 00:03:18,250 even when it's non-singular, if it's got purely imaginary values. 41 00:03:18,250 --> 00:03:22,820 THat says that there's some inability to fight with the higher order of terms as 42 00:03:22,820 --> 00:03:26,770 we explored in the previous scaled example and the vector example from before. 43 00:03:27,920 --> 00:03:33,090 Let's talk about the properties of the normal form near a fixed point 44 00:03:33,090 --> 00:03:35,590 under the hyperbolicity assumption, okay? 45 00:03:36,830 --> 00:03:41,200 In general if a fixed point of some vector fields, 46 00:03:41,200 --> 00:03:44,900 some general vector field x dot equals f general of x. 47 00:03:44,900 --> 00:03:51,660 If xe, is a fix point, the equilibrium state of this vector field f-gen. 48 00:03:51,660 --> 00:03:59,970 If it's hyperbolic, namely, if when I take the Jacobian of the vector field, right? 49 00:03:59,970 --> 00:04:04,570 So the Jacobian of the vector field evaluated at the equilibrium state. 50 00:04:04,570 --> 00:04:08,230 Let me look at the Jacobian evaluated at equilibrium state. 51 00:04:08,230 --> 00:04:11,180 Let me compute the eigenvalues of that Jacobian. 52 00:04:11,180 --> 00:04:14,380 Let me take the real part of the eigenvalues of the Jacobian, 53 00:04:14,380 --> 00:04:16,720 evaluated at the equilibrium state. 54 00:04:16,720 --> 00:04:20,260 If the real part of those eigenvalues doesn't vanish, 55 00:04:21,280 --> 00:04:24,310 then that's the definition of hyperbolicity, okay? 56 00:04:24,310 --> 00:04:28,950 So if the fixed point of a vector field is hyperbolic, then 57 00:04:30,020 --> 00:04:35,100 the linearized dynamics, that is the approximated 58 00:04:35,100 --> 00:04:39,950 dynamics where the approximation the new coordinates is just 59 00:04:39,950 --> 00:04:46,990 the Jacobian times the dummy variable u, okay. 60 00:04:46,990 --> 00:04:52,340 I'm using a new system of coordinates u, because I want to emphasize that the old 61 00:04:52,340 --> 00:04:57,480 equilibrium state, xe, is now plugged into the Jacobian to give 62 00:04:57,480 --> 00:05:02,920 me the linearized, to give me a linear vector field f had. 63 00:05:02,920 --> 00:05:07,110 Under these conditions, under these hyperbolicity conditions It turns out that 64 00:05:07,110 --> 00:05:12,760 the linearized dynamics is locally conjugate to the original linear dynamics, 65 00:05:12,760 --> 00:05:17,070 no matter what that nonlinear general vector field is, okay? 66 00:05:17,070 --> 00:05:19,410 So this is a very, very powerful result. 67 00:05:19,410 --> 00:05:22,850 There's a local conjugacy via a sum continuous 68 00:05:22,850 --> 00:05:26,430 change of coordinates defined in some neighborhood under the fixed point. 69 00:05:26,430 --> 00:05:28,710 We don't know what that change of coordinates is. 70 00:05:28,710 --> 00:05:32,010 And we don't know exactly what the neighborhood is but we're guaranteed that 71 00:05:32,010 --> 00:05:36,720 not only in the hyperbolic case, not only is the linearized field a good numerical 72 00:05:37,865 --> 00:05:42,235 approximate, but the behavior qualitatively is identical. 73 00:05:42,235 --> 00:05:46,135 Because any behavior that I see in the x-coordinates system 74 00:05:46,135 --> 00:05:49,285 has to have come from a linear behavior 75 00:05:49,285 --> 00:05:52,665 in the u-coordinate system at least in that local neighborhood. 76 00:05:52,665 --> 00:05:56,900 So this is a very strong and wonderful property. 77 00:05:56,900 --> 00:05:58,970 These ideas are a little bit more complicated, and 78 00:05:58,970 --> 00:06:01,040 you have to go to a somewhat more complicated book. 79 00:06:01,040 --> 00:06:06,560 The best reference that I know is the Bible of Applied Dynamical Systems, 80 00:06:06,560 --> 00:06:09,350 that many of us call it, by Guckenheimer and Holmes. 81 00:06:09,350 --> 00:06:14,450 It's a book that's now quite old, the latest printing I think was in 2003, 82 00:06:14,450 --> 00:06:17,670 but it's still the most important book for 83 00:06:17,670 --> 00:06:22,070 people who are interested in applying dynamical systems to engineering settings. 84 00:06:22,070 --> 00:06:24,130 So, it's a more complicated and more advanced book, 85 00:06:24,130 --> 00:06:27,200 but I strongly recommend it to you if you're interested in understanding these 86 00:06:27,200 --> 00:06:31,330 ideas in a bit more generality and detail. 87 00:06:31,330 --> 00:06:35,250 Let's try to make sense intuitively of well, 88 00:06:35,250 --> 00:06:37,590 all these ideas that we just introduced. 89 00:06:37,590 --> 00:06:40,100 Let's go back to the pendulum, 90 00:06:40,100 --> 00:06:43,840 which who's orbits I'm plotting in red on these graphs. 91 00:06:43,840 --> 00:06:50,250 And let's compare the pendulum to the linearized versions of the pendulum, or 92 00:06:50,250 --> 00:06:55,270 the linearized approximation of the pendulum down or up at these bottom or 93 00:06:55,270 --> 00:06:58,870 top equilibrium states that we explored in the previous section. 94 00:06:58,870 --> 00:07:02,610 Okay, so, what are we looking at? 95 00:07:02,610 --> 00:07:07,970 We're looking at the orbits cut out on the plane 96 00:07:07,970 --> 00:07:14,310 by this damped pendulum vector field, which has the effect of gravity 97 00:07:16,250 --> 00:07:21,080 I'm sorry which has the effect of the damping term resisting of velocity and 98 00:07:21,080 --> 00:07:23,380 the effect of the gravity term which is trying to force us down. 99 00:07:25,220 --> 00:07:29,770 And you'll recall that we've computed the conditions for 100 00:07:29,770 --> 00:07:34,910 fixed points and the fixed points occur at qb the bottom. 101 00:07:34,910 --> 00:07:37,070 Which is 0,0 in this plot. 102 00:07:38,370 --> 00:07:40,700 0,0 on this plot. 103 00:07:40,700 --> 00:07:47,130 And at bottom if I compute the Jacobian you'll see that 104 00:07:47,130 --> 00:07:53,260 I get this expression where cosine at bottom evaluates to 0, okay? 105 00:07:53,260 --> 00:07:55,610 So I bottom my value weight to 0. 106 00:07:55,610 --> 00:07:59,390 And the, you can 107 00:07:59,390 --> 00:08:04,330 see if you do the exercise that this Jacobean matrix is actually hyperbolic. 108 00:08:04,330 --> 00:08:07,150 Its eigenvalues are not purely imaginary. 109 00:08:07,150 --> 00:08:13,120 And so we're guaranteed that not only is the numerical approximation good, 110 00:08:13,120 --> 00:08:17,120 the gray approximation is gonna be good for the original red flow, but 111 00:08:17,120 --> 00:08:20,510 qualitatively the behavior has to be the same and 112 00:08:20,510 --> 00:08:23,000 not unsurprisingly as you reduce the scale. 113 00:08:23,000 --> 00:08:26,580 Here I'm going to reduce the scale by one-third to go the right hand picture. 114 00:08:26,580 --> 00:08:30,880 I'm getting very very near the bottom, here's the bottom 0,0 and you can see 115 00:08:30,880 --> 00:08:35,790 that numerically as well as qualitatively the orbits are lining up in red and gray. 116 00:08:35,790 --> 00:08:39,230 And that has to be true numerically because as you get closer and closer and 117 00:08:39,230 --> 00:08:40,410 closer to the origin, 118 00:08:40,410 --> 00:08:45,000 you can't even see the higher order terms in the vector field anymore. 119 00:08:45,000 --> 00:08:47,740 The first order Taylor series approximation is doing a really really 120 00:08:47,740 --> 00:08:52,530 really good job of nailing what the flow is numerically as well as qualitatively. 121 00:08:52,530 --> 00:08:56,550 But even considerably father away, we don't know exactly how much farther away 122 00:08:56,550 --> 00:09:00,960 this qualitative conjugacy will hold, you can see that qualitatively, the red and 123 00:09:00,960 --> 00:09:05,240 the gray orbits are very very similar, as theory guarantees they must be. 124 00:09:06,550 --> 00:09:13,820 Let's do the same kind of analysis at the unstable top fixed point of the pendulum. 125 00:09:13,820 --> 00:09:19,300 Here, we're using the exact same pendulum dynamics only I'm going to look at 126 00:09:19,300 --> 00:09:24,230 the equilibrium state, which is the fixed point at the top, q sub t where I'm at pi. 127 00:09:24,230 --> 00:09:28,070 So I'm now at 3.14159 at 0 velocity. 128 00:09:28,070 --> 00:09:31,520 Okay, and then I'm in both pictures on there. 129 00:09:31,520 --> 00:09:35,890 And now the cosine at the top changes sign, so 130 00:09:35,890 --> 00:09:38,130 it will turn out that if you look at this matrix, 131 00:09:38,130 --> 00:09:43,500 it will turn out that it's actually changed the sign of the eigenvalues. 132 00:09:43,500 --> 00:09:46,150 But still, I don't have purely imaginary eigenvalues. 133 00:09:46,150 --> 00:09:49,020 The sign of the real part has changed, and 134 00:09:50,170 --> 00:09:54,740 when I look close enough to the fixed point, 135 00:09:54,740 --> 00:09:59,120 I don't see that the linearized orbit is giving a terrific 136 00:09:59,120 --> 00:10:03,090 good approximation to the nonlinear coordinates except very very close. 137 00:10:03,090 --> 00:10:07,860 On a very close they give me a really good numerical approximation, 138 00:10:07,860 --> 00:10:11,850 the orbit of red and the orbit of gray aligns it very nicely 139 00:10:11,850 --> 00:10:15,220 even when I'm a little bit further away I'm still guaranteed, at least for 140 00:10:15,220 --> 00:10:18,830 some neighborhood, that qualitatively red and grey are the same. 141 00:10:18,830 --> 00:10:21,490 And I hope you can see that intuitively from this picture. 142 00:10:21,490 --> 00:10:25,510 Because there's a change of coordinates between the red and the grey orbits. 143 00:10:26,650 --> 00:10:31,210 Let's, look at the unstable 144 00:10:31,210 --> 00:10:36,990 pendulum when we destabilize it through anti-damping. 145 00:10:36,990 --> 00:10:40,580 And what I mean by anti-damping is that we'll encounter situations later on. 146 00:10:40,580 --> 00:10:44,350 We're actually trying to destabilize otherwise stable fixed points to get 147 00:10:44,350 --> 00:10:47,530 a good locomotion gate so we pump energy here. 148 00:10:47,530 --> 00:10:51,640 So instead of some kind of power dissipated mechanism 149 00:10:51,640 --> 00:10:55,440 I'm imagining some kind of energy adding mechanism that I've introduced or 150 00:10:55,440 --> 00:11:00,210 we've introduced in order to get this boring bottom state not to be stable. 151 00:11:00,210 --> 00:11:05,250 When you introduce that thing, then you can see that local to the origin 152 00:11:05,250 --> 00:11:10,340 you get good numerical approximations and even a little bit further away 153 00:11:10,340 --> 00:11:16,235 you get qualitatively good but still conjugate behaviors. 154 00:11:16,235 --> 00:11:21,135 All right, and so we'll give you plenty of computational experience with these 155 00:11:21,135 --> 00:11:25,745 ideas and manipulations in the exercises for this segment. 156 00:11:28,175 --> 00:11:29,755 Before we move ahead, 157 00:11:29,755 --> 00:11:34,815 let's just pause and think about the implications of fixed-point hyperbolicity. 158 00:11:34,815 --> 00:11:36,555 What it's telling us, what we get out of this thing. 159 00:11:37,910 --> 00:11:40,400 We're realizing that the linearization 160 00:11:40,400 --> 00:11:44,190 of a hyperbolic fixed point predicts the local nonlinear behavior. 161 00:11:44,190 --> 00:11:46,010 It predicts it in two different senses. 162 00:11:46,010 --> 00:11:47,890 Numerically, it predicts it, 163 00:11:47,890 --> 00:11:51,450 it approximates it, because the linear term is dominating the Taylor expansion. 164 00:11:52,610 --> 00:11:57,000 Formally, we're guaranteed that, at least in some neighborhood There's a conjugacy. 165 00:11:57,000 --> 00:12:01,020 There's a change of coordinates to a linear system, which we understand very, 166 00:12:01,020 --> 00:12:01,790 very well. 167 00:12:01,790 --> 00:12:03,860 And those qualitative properties are the same. 168 00:12:05,270 --> 00:12:09,400 From our point of view the most important qualitative property that we care about is 169 00:12:09,400 --> 00:12:13,880 the one that we're going to introduce in this next segment, 170 00:12:13,880 --> 00:12:16,350 which rounds out the unit. 171 00:12:16,350 --> 00:12:17,299 The notion of stability.