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Unformatted text preview: Periodic Solutions (Jordan and Smith, Chap. 5) Consider equation For small motions, we can write as ( see how & introduced ): Note that for & =0, 2 sin( ) cos . (5.1) x x τ + Ω = Γ && 2 3 cos . (5.14) x x x ε τ + Ω = Γ && & Assuming a power series solution equation (5.14) gives Collecting terms of powers of & gives O( ): O( ): O( ): ε 1 ε 2 ε For 2 &periodic solutions, which gives Solutions of (5.17a) are (assuming ¡ ¢ integer): & For solutions (5.21) to be 2 &periodic, and Then, (5.17b) becomes: The only 2 &periodic solution is 0, a b = = & Thus, a twoterm approximation is: The above is called the nonresonant case . Example 5.1 : Consider the system We have to introduce a small parameter & to use perturbation methods. So, we embed the system into the equation & Now, assume soln: Since the forcing is 2 ¡periodic, let us seek solutions with this periodicity. Then, and the sequence of problems are: : :...
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 Fall '10
 NA
 Period, Linear system, Nonlinear system, Singular perturbation

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