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Unformatted text preview: Poincar eLindstedt 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: Nonlinear perturbation terms alters the period and the frequency of the unperturbed linear problem and we now have T ( ) and ( ); Introduce a new timelike 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 nonzero, 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.
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