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Unformatted text preview: values x ( t ) = Î·, x i ( t ) = 0 , i = 1 , ..., m produces an approximation of x ( t ): || x ( t )-( x ( t ) + Â±x 1 ( t ) + ... + Â± m x m ( t )) || = O ( Â± m +1 ) on the time-scale 1. 2 3 The PoincarÂ´ e Expansion Theorem Theorem 9.2 (PoincarÂ´ e Expansion Theorem) Consider the initial value problem Ë y = F ( t, y, Â± ) , y ( t ) = Î¼, with | t-t | â¤ h, y â D â R n , â¤ Â± â¤ Â± , â¤ Î¼ â¤ Î¼ . If F ( t, y, Â± ) is continuous w.r.t. t, y and Â± and can be expanded in a convergent power series w.r.t. y and Â± for || y || â¤ Ï, â¤ Â± â¤ Â± , then y ( t ) can be expanded in a convergent power series w.r.t. Â± and Î¼ in a neighborhood of Â± = Î¼ = 0, convergent on the time-scale 1. Remark: PoincarÂ´ e Expansion Theorem is a part of the preparation to ï¬nd convergent series approximation of periodic solutions. 3...
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- Fall '10
- Taylor Series, Boundary value problem, Lecturer, Lipschitz continuity, PoincarÂ´ Expansion Theorem