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Direction of trajectory, system of DE's and portrait phase in plane phase

  1. Oct 24, 2012 #1

    fluidistic

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    Hi guys!
    I had the following system of DE's to solve:
    [itex]\alpha '=-2i \alpha[/itex]
    [itex]\beta ' =2i \beta[/itex].
    Where alpha and beta depend on t.
    I solved it by writing the system under matricial form, found the eigenvalues and corresponding eigenvectors.
    The solution is (and I've checked it, it works): [itex]\alpha (t)=c_1e^{-2it}[/itex], [itex]\beta (t) =c_2 e^{2it}[/itex].
    Since the eigenvalues are purely complex the trajectories in the phase plane are either circles or ellipses around (0,0).
    Now I've been reading http://tutorial.math.lamar.edu/Classes/DE/PhasePlane.aspx to check out how to determine the direction of rotation.
    When I pick [itex](\alpha , \beta ) =(1,0)[/itex], I get that [itex](\alpha ' , \beta ' )=(-2i ,0)[/itex]. However on the website I've just linked, there's no example of what happens when you get complex values. I don't know how to sketch the direction of the trajectory at the point (1,0) because of that complex number.
    Is the direction counter clockwise, clockwise, none?!
     
  2. jcsd
  3. Oct 25, 2012 #2

    haruspex

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    You appear to have a 4 dimensional phase space. How do you propose sketching trajectories in that in general?
    OTOH, with beta initially zero, it will always be zero, so that makes it manageable.
     
  4. Oct 25, 2012 #3

    chiro

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    Hey fluidistic.

    Have you checked the orientability of the Jacobian (determinant)?
     
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