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Nonlinear Oscillations

  1. Jan 19, 2010 #1
    1. The problem statement, all variables and given/known data

    http://img51.imageshack.us/img51/853/39983853.jpg [Broken]


    2. The attempt at a solution

    Q3.1

    I get the general solution as

    [tex]x(t) = Ae^{3t}+Be^{-t} + cost - 2sint[/tex] .

    Q3.2

    Letting

    [tex]y=\dot{x}[/tex]

    and using the general solution, we get

    [tex]y=3Ae^{3t}-Be^{-t}-sint-2cost[/tex] .

    Therefore the solution in the form they ask for is

    [tex](Ae^{3t}+Be^{-t} + cost - 2sint, 3Ae^{3t}-Be^{-t}-sint-2cost, t)[/tex] .

    Or am I misunderstanding?

    Q3.3

    [tex]x_{0}=x(0)=A+b-1 , y_{0}=y(0)=3A-B-2[/tex] .

    Solving these simultaneously gives

    [tex]A=0.25(x_{0}+y_{0}+1) , B=0.25(3x_{0}-y_{0}-5)[/tex]

    so the Poincaré mapping is

    [tex](0.25(x_{0}+y_{0}+1)e^{3t}+0.25(3x_{0}-y_{0}-5)e^{-t}+cost-2sint, 0.75(x_{0}+y_{0}+1)e^{3t} - 0.25(3x_{0}-y_{0}-5)e^{-t}-sint - 2cost, t)[/tex]

    Is that correct so far?

    To find the fixed points, do I let x(2pi)=x(0), y(2pi)=y(0) and solve for x(0) and y(0)? I tried this but I get something ridiculously complicated, so I'm worried i'm not understanding the question correctly at all...

    Please help! :-( Thanks.
     
    Last edited by a moderator: May 4, 2017
  2. jcsd
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