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Unformatted text preview: values x ( t ) = , 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. 2 3 The Poincar e Expansion Theorem Theorem 9.2 (Poincar e Expansion Theorem) Consider the initial value problem y = F ( t, y, ) , y ( t ) = , with | t-t | h, y D R n , , . 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 || , , then y ( t ) can be expanded in a convergent power series w.r.t. and in a neighborhood of = = 0, convergent on the time-scale 1. Remark: Poincar e Expansion Theorem is a part of the preparation to nd convergent series approximation of periodic solutions. 3...
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- Fall '10