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Unformatted text preview: Poincar´ e-Lindstedt Method CDS140B Winter, 2004 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 00 + 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 + ∞ X 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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- Fall '10
- Periodicity, Period, Periodic function, periodic solutions, periodic solution