Those of you who have taken multivariable calculus will remember the Multivariable Taylor Series. What is a Taylor Series? Well my immutable colleague professor Rob Griest, who has the world's best calculus MOOC, teaches us that we should think about nice functions as being very, very, very long polynomials, meaning that you can expand nice functions by Taylor Series into higher and higher and higher degree polynomials, and get better and better and better approximations. This is called Taylor's Theorem. And the notion of nice's captured formally by the word analytic. Go take Rob’s course if you’re interested in what this means. From our point of view what it means is that we can expand a functions f of x at any point x. In the neighborhood of some interesting .x* using the value of function adec * plus a linear approximation, achieved by taking the derivative of the function of x*, and multiplying the difference from that x star point by that derivative, and then forgetting about a bunch of remainder terms called the Taylor Remainder, which are guaranteed to get very, very small very quickly. We'll think about this second term as the linearization of the nonlinear function f(x). And let me remind you that capital F at x* is the Jacobean. It's the matrix of partial derivatives evaluated at that point of interest. The linearisation of F at x* turns out to be extremely important and valuable in the world of dynamical systems. If you recall the first segment, we showed that a linear dynamical system could be completely solved. Whereas, in general, nonlinear dynamical systems are unsolvable. However, there's the hope that the linearized dynamics of a non linear system might actually closely describe the properties of that non linear system at least in the neighborhood of a distinguished point. We'll illustrate this again using our damped pendulum back from the previous sections, so I'm plotting for you in equation 6. I'm writing down for you, again, a reminder of what the right hand side of the differential equation looked like. F sub sp the function, the field vector or the vector field, as it's called, of the simple pendulum on the right hand side. At the bottom point, at the bottom equilibrium state, let's evaluate the Jacobean, the matrix of partial derivatives, of little f, sub sp, at q*, q* being 0,0. And let's plug that q* into the Jacobean matrix, capital F. And let's evaluate it, and that's exactly what we've done in this bottom row. That's the linearized dynamics of this nom in your system, the pendulum, at its bottom fixed point. At the top, we have the same formula for the Jacobean, but notice that we're plugging in the top fixed point, which is the angle at theta, 180 degrees above ground, above the downward pointing part, and at 0 velocity. And when we plug that value of theta in, of course, we get a rather different matrix, the same Jacobean matrix, the same linearization function, but it's being evaluated at two very different q*'s. First we did it at the bottom, and the second we did it at the top. We're going to introduce changes of coordinates to talk about the normal forms of these dynamical systems. First, in general, that is away from fixed points. And then in particular, close to fixed points. A change of coordinates is a one to one in smooth continuous function. That is continuous partial derivatives, continuous Jacobean matrix. If it's inverse function is also smooth, then we'll call a change of coordinates. And what we would like to do is, we would like to write the old dynamics in the new coordinates. To do so, if we're given a differential equation, x dot = f(x), and a change of coordinate, y = h(x), where h is one of these continuously invertible and continuous smooth functions. Then we'll take D by Dt of y, and we'll use the chain rule to get our hands on D by Dt of x. Let's do that. We'll see that y dot is in the chain rule. Is the Jacobean of h times times d by dt of x. But x d by dt solves the differential equation. And so that reduces to the Jacobean of h times the vector field f. But we can substitute for x in that equation by using h inverse. And so suddenly we have a new differential equation, y dot equals f tilde of y, where f tilde of y is said to be conjugate to f. It's not the same form as f. F tilde is very, very different. It looks very, very different when you write it down. But it has the same properties and the same behaviors. In exactly analogous way to the diagonalized matrix capturing the linear properties in normal coordinates of the matrix, whose eigenvectors you found to diagonalize it. The normal form of any vector field away from a fixed point is always a constant vector field, simple. It's not to say that the original vector field had this simple behavior, but it's conjugate to it, and so a change of coordinates will make it look simple in that way. We're interested in the normal form at fixed points as well. So what we mean by the normal form is reducing by change of coordinates, a function to it's simplest polynomial conjugate. The Flowbox Theorem, to summarize this idea, says that if you are away from the fixed point, then your dynamics is conjugate automatically. At least in the small neighborhood of any non-fixed point. Your dynamics is conjugate to a boring old constant flow dynamics. The normal form near a fixed point is much more interesting. The Taylor Remainder never is going to be of interest to us in this discussion, but the constant term at a fixed point by definition, vanishes. We suppose that x* is a fixed point, and what that means is that f vanishes at x* as depicted in this equation. Now, our leading term is the linear vector field, capital F evaluated x*. Where remember capital F is the matrix of partial derivatives of the original non-linear field evaluated at the fixed point x*. It dominates. And it turns out not only does it dominate, but there's a theorem the Hartman Grobman Theorem, which I am going to abbreviate HA, and I'm going to call it the Hyperbolic Approximation Theorem. You have to look at more complicated books to really understand this theorem. But what we should understand is that in the neighborhood of a nice fixed point, the non-linear dynamics is conjugate to its linearization. Not only is the linearized dynamics dominating but it actually completely captures the behavior so long as x* has the property of being hyperbolic which is a fancy way of saying it has no purely imaginary eigenvalues. Again, this is a very long story that I'm just giving a small mention of, and I'm referring you to the legendary textbook by Professor John Guckenheimer and Phil Holmes, that is still considered to be the Bible of nonlinear dynamical systems, if you become interested in following these ideas. The fixed point at the bottom, now, we're being told by this theorem, should be captured, behaviorally, by the linearized dynamics at the bottom. Well, the linearized dynamics is governed in turn by the eigenvalues, the roots of the characteristic polynomial of that matrix, that we just calculated in the previous slide. Let's compute the eigenvalues. And we see that the eigenvalues actually are negative, have negative real part. The have nonzero real part assuming assuming we have positive damping. The behavior near a hyperbolic fix point then, including the stability behavior, is governed by the linearized dynamics. And so the change of coordinates preserves the stability properties. Continuous, continuously changeable change of coordinates takes bounded sets, so it preserves stability. It's continuous, and so it preserves limits, so we preserve asymptotic stability. And so the stability properties have to be the same, and so the lineraization governs the stability properties of the hyperbolic fixed points. Taylor says linearization dominates, hyperbolic approximation guarantees that linearization actually determines stability. So that the damped pendulum near the fixed point can be understood behaviorally by looking at it's linearization. And here I depict that by juxtaposing the red nominare flow against the linearized dynamics of the flow near the fixed points for different parametric values of the pendulum down at it's fixed point. Again, we have some exercises where you can do some of these calculations, and do some of your own exploring, but we'll press on. In contrast, near and very near the top fixed point, we once again see that the nonlinear dynamics is closely approximated by linear saddles, and the linear saddles are clearly unstable.
