37 nonlinear systems

37 nonlinear systems - 18.03 Lecture#37 Dec 7 2009 notes...

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Topic for today is autonomous nonlinear systems of diFerential equations , particularly systems of two equations. These are systems dx dt = F ( x, y ) , dy dt = G ( x, y ) . (Automonomous system ODE) The word “autonomous” refers to the fact that the independent variable t is absent on the right. For a system of linear ODE x = a ( t ) x + b ( t ) y, y = c ( t ) x + d ( t ) y, autonomous is the same as “constant coe±cient:” that the functions a , b , c , and d are constants. The program is to use what we know about these (constant coe±cient) linear systems to say something about general autonomous systems. I started with pictures of the solution curves of two-dimensional linear systems, following the discussion in EP, 7.2. I won’t repeat that in detail here. Here are some highlights. Everything depends on the matrix A = p a b c d P ; the system is p x y P = A p x y P . (Linear system) Case of a stable node. This is the case when A has two negative eigenvalues - s < - r < 0 . Write v 1 for the eigenvector for - r , and v 2 for the eigenvector for - s . Of course the general solution is ue rt v 1 + ve st v 2 . If v = 0, this trajectory moves toward the origin on the line through v 1 ; and if u = 0, it moves toward the origin on the line through v 2 . If u and v are both non-zero, then as t + , the second term goes to zero much faster; so the trajectory approaches the origin on a curve tangent to v 1 (the eigenvector for the larger eigenvalue). The word stable refers to the fact that solutions starting close to zero remain close to zero; in fact the node is asymptotically stable , meaning that solutions starting close to zero approach zero as t → ∞ . (In fact all solutions approach zero.) The term sink means the same thing as asymptotically stable . The case of an unstable node (distinct positive eigenvalues) looks identical, but with the arrows reversed on all the trajectories. To identify these cases, you must compute the eigenvalues; to sketch them accurately, you must identify the eigenvectors. Case of an unstable saddle.
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This note was uploaded on 05/06/2010 for the course 18 18.03 taught by Professor Unknown during the Fall '09 term at MIT.

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37 nonlinear systems - 18.03 Lecture#37 Dec 7 2009 notes...

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