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Unformatted text preview: 2 Stability of solutions to ODEs How can we address the question of stability in general? We proceed from the example of the pendulum equation. We reduce this second order ODE, ¨ g β + sin β = 0 , l to two first order ODE’s. Write x 1 = β , x 2 = β ˙ . Then x ˙ 1 = x 2 g x ˙ 2 = sin x 1 − l The equilibrium points, or fixed points , are where the trajectories in phase space stop , i.e. where ψx ˙ = x ˙ 1 = ψ x ˙ 2 For the pendulum, this requires x 2 = x 1 = ± nα, n = 0 , 1 , 2 , . . . Since sin x 1 is periodic, the only distinct fixed points are β β α β ˙ = and β ˙ = Intuitively, the first is stable and the second is not. How may we be more precise? 2.1 Linear systems Consider the problem in general. First, assume that we have the linear system u ˙ 1 = a 11 u 1 + a 12 u 2 u ˙ 2 = a 21 u 1 + a 22 u 2 14 or ψu ˙ = Aψu with u 1 ( t ) a 11 a 12 ψu ( t ) = and A = u 2 ( t ) a 21 a 22 Assume A has an inverse and that its eigenvalues are distinct....
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This note was uploaded on 02/24/2012 for the course MECHANICAL 12.006J taught by Professor Danielrothman during the Fall '06 term at MIT.
- Fall '06