fa_09lecture_set5a

fa_09lecture_set5a - Periodic Solutions (Jordan and Smith,...

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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 + = && 2 3 cos . (5.14) x x x + - = && & Assuming a power series solution equation (5.14) gives Collecting terms of powers of & gives O( ): O( ): O( ): 1 2 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 two-term approximation is: The above is called the non-resonant 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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fa_09lecture_set5a - Periodic Solutions (Jordan and Smith,...

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