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Homework Help: Stability of system

  1. Dec 14, 2011 #1
    For the system

    [itex]\dot{x}[/itex]=y2
    [itex]\dot{y}[/itex]=x2

    Both the eigenvalues are zero. How do I
    find the eigenvectors so that I can sketch
    the phase portrait and how do I classify
    the stability of the fixed point (0,0)?
     
  2. jcsd
  3. Dec 15, 2011 #2

    HallsofIvy

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    Science Advisor

    Well, obviously, both [itex]x^2[/itex] and [itex]y^2[/itex] are positive for all non-zero x and y so (0, 0) is unstable.
     
  4. Dec 18, 2011 #3
    Yes, that is true. Thank you. How do I find the eigenvectors though?
     
  5. Dec 19, 2011 #4

    mbp

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    It is not necessary to compute eigenvectors. This system is Hamiltonian (conservative). On dividing one equation by the other you get
    \begin{equation}
    \frac{dx}{dy} = \frac{y^2}{x^2}
    \end{equation}
    Separating variables and integrating you find the Hamiltonian
    \begin{equation}
    H(x,y) = \frac{1}{3} (x^3-y^3)
    \end{equation}
    The level sets \begin{equation}H = constant\end{equation} define the phase portrait.
     
  6. Dec 21, 2011 #5
    oh my god, that make life so easy. Thank you!
     
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