lecture1B_ch9_2004

# lecture1B_ch9_2004 - ± , equating corresponding...

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Lecture 1B: Introdction to Perturbation Theory CDS140b Lecturer: Wang Sang Koon Winter, 2004 1 Basic Ideas Perturbed and unperturbed problems: To ﬁnd approixmated solutions to a perturbed problem, one will always assume that one have suﬃcient knowledge of the solutions of the unperturbed problem. Time scale: In construting approximated solutions, one has to indicated on what interval of time (time-scale) one is looking for an approximation. Example (Duﬃng equation): ¨ q + q + ±q 3 = 0 . 1

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2 Naive Expansion Theorem 9.1: Consider the initial value problem ˙ x = f 0 ( t, x ) + ±f 1 ( t, x ) + ... + ± m f m ( t, x ) + ± m +1 R ( t, x, ± ) with x ( t 0 ) = η and | t - t 0 | ≤ h, x D R n , 0 ± ± 0 . Assume that in this domain 1. f i ( t, x ) , i = 0 , ..., m continuous in t and x , ( m + 1 - i ) times continuously diﬀerentiable in x ; 2. R ( t, x, ± ) continuous in t, x and ± , Lipschitz-continous in x . Subsitituting in the equation for x the formal expansion x 0 ( t ) + ±x 1 ( t ) + ... + ± m x m ( t ) , Taylor expanding w.r.t. powers of
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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 time-scale 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 | t-t | ≤ 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 time-scale 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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lecture1B_ch9_2004 - ± , equating corresponding...

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