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Unformatted text preview: ± , equating corresponding coeﬃcients and applying the initial values x ( t ) = η, x i ( t ) = 0 , i = 1 , ..., m produces an approximation of x ( t ):  x ( t )( x ( t ) + ±x 1 ( t ) + ... + ± m x m ( t ))  = O ( ± m +1 ) on the timescale 1. 3 The Poincar´ e Expansion Theorem Theorem 9.2 (Poincar´ e Expansion Theorem) Consider the initial value problem ˙ y = F ( t, y, ± ) , y ( t ) = μ, with  tt  ≤ h, y ∈ D ⊂ R n , ≤ ± ≤ ± , ≤ μ ≤ μ . If F ( t, y, ± ) is continuous w.r.t. t, y and ± and can be expanded in a convergent power series w.r.t. y and ± for  y  ≤ ρ, ≤ ± ≤ ± , then y ( t ) can be expanded in a convergent power series w.r.t. ± and μ in a neighborhood of ± = μ = 0, convergent on the timescale 1. 2...
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This note was uploaded on 01/04/2012 for the course CDS 140b taught by Professor List during the Fall '10 term at Caltech.
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