Lecture 35. Nonlinear Systems1. Introduction to Nonlinear Systems2. Linearization Around Critical PointsDifferential EquationsLECTURE 35Nonlinear SystemsIt’s time to turn our attention to systems of equations which actually may show up in appli-cations. These systems, unlike the ones we’ve been discussing are nonlinear; that is, x01and x02aren’t just linear combinations of x1and x2. As we’ve seen when looking at single differentialequations, nonlinear equations can be difficult to general to deal with. In fact, we won’t usually beable to obtain solutions to these systems. Instead, we’ll focus more on qualitative analyses of thesesystems.We could easily spend an entire semester on this topic. Instead, we’ll try to get a bit of theflavor of how these systems differ from the linear systems we’ve been learning about.1. Introduction to Nonlinear SystemsThe general form of a nonlinear two-dimensional system of differential equations isx01= f1(x1, x2)x02= f2(x1, x2)We could rewrite this more compactly in vector notation asx0= f(x),where x =x1x2and f (x) =f1(x1, x2)f2(x1, x2). For systems like this, there is generally no hope offinding trajectories analytically, as we did for the linear systems we discussed earlier. Thus, asmentioned earlier, our attention will be focused on the qualitative behavior of these solutions.There are some features of nonlinear phase portraits that are especially salient:(1) The fixed or critical points, which are the equilibrium or steady-state solutions. Thesecorrespond to points x satisfying f(x) = 0; in other words, x1and x2are zeroes for bothf1and f2. In this course, we will mostly be focused on learning about the role fixed pointsplay in determining the phase portrait of a nonlinear system.(2) The closed orbits, which correspond to solutions that are periodic for both x1and x2.We’ll briefly discuss some techniques here, but for the most part this is a topic that willbe focused on in a future differential equations course you might take.(3) How trajectories are arranged near fixed points and closed orbits; again, we’ll primarilylook at what happens near fixed points.(4) The stability or instability of fixed points and closed orbits; which of these attract nearbytrajectories, and which repel them?How do we even know we have solutions to our general nonlinear system? As we’ve seen,existence and uniqueness questions can be tricky for nonlinear equations.Theorem 35.1 (Existence and Uniqueness). Consider the initial value problemx0= f(x)) x(0) = x0.If f is continuous and so are its partial derivatives∂fi∂xion some region in the plane containing x0,then the initial value problem has a unique solution x(t) on some time interval near t = 0.1Differential Equations Lecture 35: Nonlinear SystemsThe upshot for our purposes? If we have nice enough f1and f2, so that they and their partialderivatives are continuous for all x1, x2, then any point can be taken as an initial condition for oursystem.An important consequence of the existence and uniqueness theorem is that, for the nice systemswe’ll be considering, different trajectories can never intersect. If they did, that point of intersectionwould be an initial condition corresponding to two different solutions, which can’t happen. Thismeans that phase portraits look very polished: they just seem to fit together.2. Linearization Around Critical PointsThe starting point for just about any qualitative analysis of nonlinear systems is determiningthe critical points. These are points (x∗1, x∗2) that correspond to equilibrium solutions x1(t) = x∗1and x2(t) = x∗2. As we’ve discussed, if our system is linear, the only critical point is the origin,(0, 0). Nonlinear systems, however, can have many fixed points, and our goal is to try to determinewhat we can about the trajectories close to these points.Consider the systemx0= f(x, y)y0= g(x, y).How do we find these critical points? As they’re constant solutions for both x and y, they’repoints where both x0= 0 and y0= 0. Thus ”all” we have to do is to find the values of x and y thatare zeroes of both f and g. For the examples we’ll be looking at, this will be fairly straightforward.Now suppose (x0, y0) is a fixed point. Thus we know thatf(x0, y0) = g(x0, y0) = 0.The goal of the linearization technique is to use our knowledge of linear systems to try to concludewhat we can about the phase portrait near (x0, y0). To do this, we’ll try to approximate ournonlinear system by a linear system, which we can then classify as we’ve been discussing. Since(x0, y0) is a fixed point, and the only fixed point of a linear system is the origin, we’ll want tochange variables so that (x0, y0) becomes the origin of the new coordinate system. Thus, letu = x − x0v = y − y0.We need to rewrite our differential equations in terms of u and v.u0= x0= f(x, y)= f(x0+ u, y0+ v)The natural thing to do here is to Taylor expand f near (x0, y0).= f(x0, y0) + u∂f∂x(x0, y0) + v∂f∂y(x0, y0) + higher order terms= u∂f∂x(x0, y0) + v∂f∂y(x0, y0) + H.O.T.To simplify notation, we’ll sometimes suppress writing explicitly that∂f∂xand∂f∂yare evaluatedat the point (x0, y0), but it’s important to keep this in mind. For our purposes, these partialderivatives are numbers, not functions.Another important observation is that, as we’re considering what happens very close to ourfixed point, u and v are both small; hence the higher order terms are smaller still and will bedisgarded in our computations.2Differential Equations Lecture 35: Nonlinear SystemsNow, by a similar computation we havev0= u∂g∂x+ v∂g∂y+ H.O.T.Ignoring these very small higher order terms, we can write this system of rewritten differentialequations in matrix form. The linearized system near (x0, y0) isu0v0= ∂f∂x(x0, y0)∂f∂y(x0, y0)∂g∂x(x0, y0)∂g∂y(x0, y0)!uv.We will use, from this point on, the notation fx=∂f∂x. The matrixA =fx(x0, y0) fy(x0, y0)gx(x0, y0) gy(x0, y0)is called the Jacobian matrix at (x0, y0) of the vector-valued function f (x) =f(x1, x2)g(x1, x2). Inmultivariable calculus, the Jacobian matrix is the appropriate analogue of the single-variable de-rivative.We can then study this linear system using the standard
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