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Help me learn to sketch phase portraits!

  1. Jan 16, 2012 #1
    Hey guys,

    Been struggling with this all day. I've got the following situation:

    x1*=-x1+x2

    x2*=-2x1-x2

    Where the 1&2 after the x are subscripts.

    Now i can find the equilibrium no trouble: (0,0). I also know, by looking at the eigenvalues that it is a sink.

    But how on earth do you sketch it? I've found nullclines:

    -x1+x2 = 0

    -2x1-x2 = 0

    But i cannot, for the life of me, tell which way these lines are going.

    Help really appreciated.
     
  2. jcsd
  3. Jan 16, 2012 #2
    You want to bring it into a cannonical form, this system will be a spiral sink with eigenvalues - 1 +/- sqrt( 2 i ) so it will be a spiral sink. The cannonical form for a case like this, is a matrix like:
    a -b
    b a

    The lines will go inward since it is a sink, and if a > 0 then the spiral goes counterclockwise
     
  4. Jan 16, 2012 #3
    Thanks for your reply. What i posted above is just a preliminary example that i cannot do, i've asked my professor too.

    I've never seen ( i dont think) the method you have mentioned?

    Are there any other ways? I had in mind something like:

    dy/dx=f(x,y)/g(x,y)

    (Of course, where my x and y are x1 and x2, and i could look at any point x1,x2 and see its direction).

    I really need quite a bit of intuition in this as i can not see what is going on.
     
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