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Unformatted text preview: Poincar e-Lindstedt Method CDS140B Winter, 2005 1 Periodic Solutions of Autonomous 2nd Order Equations Consider x + x = f ( x, x, ) . Assumptions: periodic solutions exist for small, postive ; requirements of the Poincar e expansion thoerem have been satisfied. Basic Ideas: Non-linear perturbation terms alters the period and the frequency of the unperturbed linear problem and we now have T ( ) and ( ); Introduce a new time-like variable such that the periodic solution is 2 -periodic = t, - 2 = 1- ( ) Rewrite the equation using as the independent variable x 00 + x = [ x + (1- ) f ( x, (1- )- 1 / 2 x , )] = g ( x, x , , ) with initial values x (0) = a ( ) , x (0) = 0. If the Jacobian of the periodicity conditions is non-zero, the corresponding periodic solution of the perturbed equation can be represented by the convergent series x ( ) = a (0) cos + X n =1 n n ( ) . To determine n ( ), we substitute the series into the rescaled equation, collect terms which are coefficients of equal powers of and produces equations for n (...
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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.
- Fall '10