329_pdfsam_math 54 differential equation solutions odd

329_pdfsam_math 54 differential equation solutions odd - 5...

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Exercises 5.7 EXERCISES 5.7: Dynamical Systems, Poincar` e Maps, and Chaos, page 301 1. Let ω =3 / 2. Using system (3) on page 294 of the text with A = F =1, φ =0,and ω =3 / 2, we deFne the Poincar´ emap x n =s in(3 πn )+ 1 (9 / 4) (4 / 4) =s in(3 πn )+ 4 5 = 4 5 , v n = 3 2 cos(3 πn )=( 1) n 3 2 , for n =0 ,1 ,2 ,... . Calculating the Frst few values of ( x n ,v n ), we Fnd that they alternate between (4 / 5 , 3 / 2) and (4 / 5 , 3 / 2). Consequently, we can deduce that there is a subharmonic solution of period 4 π .Le t ω =3 / 5. Using system (3) on page 294 of the text with A = F =1, φ =0 ,and ω =3 / 5, we deFne the Poincar´ emap x n =s in ± 6 πn 5 ² + 1 (9 / 25) 1 =s in ± 6 πn 5 ² 1 . 5625 , v n = 3 5 cos ± 6 πn 5 ² =(0 . 6) cos ± 6 πn 5 ² , for n =0 ,1 ,2 ,... . Calculating the Frst few values of ( x n ,v n ), we Fnd that the Poincar´ emap cycles through the points (
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Unformatted text preview: , 5 , 10 , . . . , ( − 2 . 1503 , − . 4854) , n = 1 , 6 , 11 , . . . , ( − . 6114 , . 1854) , n = 2 , 7 , 12 , . . . , ( − 2 . 5136 , . 1854) , n = 3 , 8 , 13 , . . . , ( − . 9747 , − . 4854) , n = 4 , 9 , 14 , . . . . Consequently, we can deduce that there is a subharmonic solution of period 10 π . 3. With A = F = 1, φ = 0, ω = 1, b = − . 1, and θ = 0 (because tan θ = ( ω 2 − 1) /b = 0) the solution (5) to equation (4) becomes x ( t ) = e . 05 t sin √ 3 . 99 2 t + 10 sin t. Thus v ( t ) = x ( t ) = e . 05 t . 05 sin √ 3 . 99 2 t + √ 3 . 99 2 cos √ 3 . 99 2 t + 10 cos t 325...
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