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lecture2A_PL_2005

lecture2A_PL_2005 - Poincar-Lindstedt Method e CDS140B...

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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 ω ( ); Introduce a new time-like variable θ such that the periodic solution is 2 π -periodic θ = ωt, ω - 2 = 1 - η ( ) Rewrite the equation using θ as the independent variable x + x = [ ηx + (1 - η ) f ( x, (1 - η ) - 1 / 2 x , )] = g ( x, x , , η ) 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 ( θ ) = a (0) cos θ + n =1 n γ n ( θ ) . To determine γ n ( θ ), we substitute the series into the rescaled equation, collect terms which are coefficients of equal powers of and produces equations for γ n ( θ ).
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