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Math Help - Poincare Map

  1. #1
    Super Member Showcase_22's Avatar
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    Poincare Map

    I've got a rather long question, and i've done a lot of it, but i've hit a small hurdle and i'm not really sure what it wants me to work out.

    I have:

    \dot{x_1}=x_1(x_2-x_3)
    \dot{x_2}=x_2(x_3-x_1)
    \dot{x_3}=x_3(x_1-x_2)

    and i'm considering it's behaviour on the plane x_1+x_2+x_3=1.

    The question is this:

    On each line x_i=0, i=1,2,3 the dynamics can first be reduced to two equations in two variables (since x_i is identically equal to 0), and then by using x_1+x_2+x_3=1, to one equation in one variable.

    Write down the equation and solve it.
    Find the Poincare map for a time step of t=1.
    Describe the dynamics on such a line.
    I decided to set x_3=0 so I would just be working in the x_1, x_2 plane.

    Substitutiing x_3=0 into the equations gives:

    \dot{x_1}=x_1 x_2
    \dot{x_2}=-x_1 x_2
    \dot{x_3}=0

    Then, using x_1+x_2+x_3=1 we get:

    \dot{x_1}=x_1(1-x_1)
    \dot{x_2}=-x_2(1-x_2)
    \dot{x_3}=0

    This shows that x_3=0 for all values of x_1 and x_2 (since it start at 0 and stays where it is).

    We can solve x_1 and x_2 to get:

    x_1(t)=\frac{Ae^t}{1-Ae^t} and x_2(t)=\frac{-Be^t}{1-Be^t} where A and B are constants.

    So my solution is:

    \begin{pmatrix} {x_1} \\ {x_2} \end{pmatrix}=\begin{pmatrix} {\frac{Ae^t}{1-Ae^t}} \\ {\frac{-Be^t}{1-Be^t}} \end{pmatrix} <----- Is this right?

    So now I move on to try and find the Poincare map. I know that:

    \begin{pmatrix} {\dot{x_1}} \\ {\dot{x_2}} \\ \dot{s} \end{pmatrix}=\begin{pmatrix} {x_1(1-x_1)} \\ {-x_2(1-x_2)} \\ 1 \end{pmatrix}

    Now I have to make it 2 \pi periodic so I get:

    \begin{pmatrix} {\dot{x_1}} \\ {\dot{x_2}} \\ \dot{s} \end{pmatrix}=\begin{pmatrix} {x_1(1-x_1)} \\ {-x_2(1-x_2)+f(cos(2 \pi s))} \\ 1 \end{pmatrix}

    From here I would just integrate the LHS, but I run into problems with \dot{x_2}. I tried choosing f(cos( 2 \pi s))=cos(2 \pi s) but I still couldn't integrate that.

    Am I approaching the Poincare map in the correct way?
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  2. #2
    Newbie
    Joined
    Dec 2010
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    Modelling Nature's Nonlinearity?

    Hi, i couldn't help but notice this is part of the modelling nature's nonlinearity assignment?

    I was just wondering how far along you were and if you could help me because im struggling on it!

    Also, in regards to your question, i think you need ONE equation in ONE variable, not TWO as you have with x1 and x2? i hope this helps, you probably have your answer already anyway,

    If you could help, i would really appreciate it.

    Thank you!
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  3. #3
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    Jan 2011
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    Tbh mate I haven't got a clue. Can you do the first question? All the notation is confusing me
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  4. #4
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    Dec 2010
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    I cant do it either, its so much effort for what it is, im going to drop it for an unusual option or something! i mean if its this hard, i wonder how hard the next assignment is! ah well! sorry mate
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