cds140b-plvdp

cds140b-plvdp - The Poincar e-Lindstedt Method: the van der...

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Unformatted text preview: The Poincar e-Lindstedt Method: the van der Pol oscillator Joris Vankerschaver jv@caltech.edu The purpose of this document is to give a detailed overview of how the Poincar e-Lindstedt method can be used to approximate the limit cycle in the van der Pol system x + ( x 2- 1) x + x = 0 where > 0 is small. 1. Introduce a new time scale = t so that the new period becomes 2 . The van der Pol equation becomes 2 x 00 + ( x 2- 1) x + x = 0 , where x = - 1 x represents the derivative of x with respect to the new parameter , and similar for x 00 . 2. Substitute series expansions for x ( ) = x ( ) + x 1 ( ) + and = + 1 + into the equation. Note that = 1 since the solution has period 2 when = 0. Substitute the same expansions into the initial conditions and find the resulting initial conditions for x i ( t ). For the van der Pol equation we have hence (1 + 1 + ) 2 ( x 00 ( ) + x 00 1 ( ) + )+ (1 + 1 +...
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cds140b-plvdp - The Poincar e-Lindstedt Method: the van der...

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