331_pdfsam_math 54 differential equation solutions odd

331_pdfsam_math 54 differential equation solutions odd -...

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Unformatted text preview: Exercises 5.7 Table 5–F: Poincar´ map for Problem 3. e n 0 1 2 3 4 5 6 7 8 9 10 xn 0 −0.010761 −0.029466 −0.060511 −0.110453 −0.189009 −0.310494 −0.495883 −0.775786 −1.194692 −1.817047 vn 10.998749 11.366815 11.870407 12.559384 13.501933 14.791299 16.554984 18.967326 22.266682 26.778923 32.949532 n 11 12 13 14 15 16 17 18 19 20 xn −2.735915 −4.085318 −6.057783 −8.929255 −13.09442 −19.11674 −27.79923 −40.28442 −58.19561 −83.83579 vn 41.387469 52.925111 68.700143 90.267442 119.75193 160.05736 215.15152 290.45581 393.37721 534.03491 In Table 5-G we have listed the first 21 values of the Poincar´ map. e As n gets large, we see that xn ≈ −(1.092050) sin(1.328172) ≈ −1.060065 , vn ≈ (1.092050) cos(1.328172) ≈ 0.262366 . Hence, as n → ∞, the Poincar´ map approaches the point (−1.060065, 0.262366). e 7. Let A, φ and A∗ , φ∗ denote the values of constants A, φ in solution formula (2), corresponding ∗ to initial values (x0 , v0 ) and (x∗ , v0 ), respectively. 0 (i) From recursive formulas (3) we conclude that xn − F/(ω 2 − 1) = A sin(2πωn + φ), vn /ω = A cos(2πωn + φ), and so (A, 2πωn+φ) are polar coordinates of the point (vn /ω, xn −F/(ω 2 −1)) in vx-plane. ∗ Similarly, (A∗ , 2πωn + φ∗ ) represent polar coordinates of the point (vn /ω, x∗ − F/(ω 2 − 1)). n 327 ...
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