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Unformatted text preview: The Poincar´ e-Lindstedt Method: the van der Pol oscillator Joris Vankerschaver [email protected] 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 τ = ωt so that the new period becomes 2 π . The van der Pol equation becomes ω 2 x 00 + ω ( x 2- 1) x + x = 0 , where x = ω- 1 ˙ x represents the derivative of x with respect to the new parameter τ , and similar for x 00 . 2. Substitute series expansions for x ( τ ) = x ( τ ) + x 1 ( τ ) + ··· and ω = ω + ω 1 + ··· into the equation. Note that ω = 1 since the solution has period 2 π 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 + ω 1 + ··· ) 2 ( x 00 ( τ ) + x 00 1 ( τ ) + ··· )+ (1 + ω 1 + ···...
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- Fall '10