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38 examples of nonlinear systems, ecosystems, conservation laws

38 examples of nonlinear systems, ecosystems, conservation laws

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18.03 Lecture #38 Dec. 9, 2009: notes Please fill out the online course evaluation! The address is http://web.mit.edu/subjectevaluation and the deadline is Monday morning December 14 at 9:00. Today we continue with autonomous nonlinear systems of differential equations dx dt = F ( x, y ) , dy dt = G ( x, y ) . (Automonomous system ODE) Last time we looked at critical points , points ( x 0 , y 0 ) where both F and G are zero. These are the constant solutions of (Autonomous system ODE): x ( t ) = x 0 , y ( t ) = y 0 . (Critical point solution of automonomous system ODE) Near a critical points, the differential equations (Autonomous system ODE) can be approxi- mated by d ( x - x 0 ) dt ( x - x 0 ) ∂F ∂x ( x 0 , y 0 ) + ( y - y 0 ) ∂F ∂y ( x 0 , y 0 ) , d ( y - y 0 ) dt ( x - x 0 ) ∂G ∂x ( x 0 , y 0 ) + ( y - y 0 ) ∂G ∂y ( x 0 , y 0 ) . (Crit pt approx) (I’ve written this in terms of new variables ( x - x 0 ) and ( y - y 0 ).) This is a constant coefficient linear system, with matrix A = ∂F ∂x ( x 0 , y 0 ) ∂F ∂y ( x 0 , y 0 ) ∂G ∂x ( x 0 , y 0 ) ∂G ∂y ( x 0 , y 0 ) , the Jacobian matrix for ( F, G ) at ( x 0 , y 0 ). Example: undamped pendulum. Last time we looked at the undamped pendulum x ′′ = - sin( x ), corresponding to the first-order system x = y y = - sin( x ) (undamped pendulum) with y ( t ) = x ( t ). We found critical points at x 0 = sin( ) , y 0 = 0 . The matrix for the linear approximation at a critical point is A = parenleftbigg 0 1 ( - 1) k +1 0 parenrightbigg . 1
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If k is even, the matrix A = parenleftbigg 0 1 - 1 0 parenrightbigg has eigenvalues ± i ; the solutions to the approximation are circles around the critical point. The question I left unanswered last time was whether the solutions near these critical points with k even—that is, solutions where the pendulum is not far from hanging straight down, and not moving too fast—are actually (distorted) circles around the critical point, or whether they might spiral in or out. The best way I know to see that there is no spiraling is to use physical intuition: in an undamped pendulum, energy should be conserved. A little more physical thinking leads to defining an energy function E ( x, y ) = y 2 / 2 - cos( x ) . The first term y 2 / 2 is kinetic energy, and the second - cos( x ) (the height of the mass above the pendulum axis) is potential energy. Using the chain rule to differentiate this as a function of t gives dE dt = yy - sin( x ) x = - y sin( x ) + y sin( x ) = 0; so E
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