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Nonlinear Dynamics: Nullclines and phase plane of a nonlinear system

  1. Oct 30, 2011 #1
    1. The problem statement, all variables and given/known data

    Find the fixed points and classify them using linear analysis. Then sketch the nullclines, the vector field, and a plausible phase portrait.

    dx/dt = x(x-y), dy/dt = y(2x-y)

    2. Relevant equations



    3. The attempt at a solution

    f1(x,y) = x(x-y)

    x-nullcline: x(x-y) = 0 [itex]\Rightarrow[/itex] x = 0

    f2(x,y) = y(2x-y)

    y-nullcline: y(2x-y) = 0 [itex]\Rightarrow[/itex] y = 0

    Fixed point: (0,0)

    J(x,y) =

    (2x-y -x )
    (2y 2x-2y)

    Therefore,

    J(0,0) =

    (0 0)
    (0 0)

    Thus,

    0 = |J(0,0) - λI| = (-λ)(-λ) = λ2 = 0 [itex]\Rightarrow[/itex] λ1,2 = 0
    ___________________________________________________________________

    Now this is where I get stuck; I have no idea where to go from here when λ1 = λ2 = 0.

    I would really appreciate at least a little nudge in the right direction. Thank you.
     
  2. jcsd
  3. Oct 31, 2011 #2
    There are some cases where linearization fails to capture the dynamics of a non-linear system near it's fixed points and I think one case is when the eigenvalues are zero. Need to check out a good reference like "Differential Equations" by Blanchard Hall and Devaney. Also, whenever working on these types of problems, in my opinion, it is essential to have a CAS like Mathematica to help you with it unless you like to suffer. Now unfortunately you can't use StreamPlot with Wolfram Alpha but if you could find a machine at school running Mathematica, you could very easily answer three of the questions with the code:

    Show[{StreamPlot[{x^2 - x y, 2 x y - y^2}, {x, -1, 1}, {y, -1, 1}],
    Plot[{x, 2 x}, {x, -1, 1}, PlotStyle -> Thickness[0.008]]}]

    I don't know how to classify the fixed-point at the origin but if I had to say something, I would say it's some type of sink. Maybe someone else can help us further.
     

    Attached Files:

    Last edited: Oct 31, 2011
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