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Determining critical points

  1. Oct 26, 2007 #1
    1. The problem statement, all variables and given/known data
    Consider the plane dynamic system [tex]\dot{x} = P(x,y), \dot{y} = Q(x,y)[/tex] with the condition that O(0,0) is a critical point. Suppose P(-x,y) = -P(x,y) and Q(-x,y) = Q(x,y). Is the critical point (0,0) a center? Why?

    3. The attempt at a solution
    I know that for (0,0) to be a centre, the eigenvalues of A should satisfy
    [tex]\lambda_1 + \lambda_2 = tr(A) = 0, \lambda_1 \lambda_2 = det(A) > 0[/tex]

    Also, the matrix A at (0,0) is
    [tex]
    \left[
    \begin{array}\\
    \frac{\partial P}{\partial x} & \frac{\partial P}{\partial y} \\
    \frac{\partial Q}{\partial x} & \frac{\partial Q}{\partial y} \\
    \end{array}
    \right]
    [/tex]

    That's all I've got and I'm not sure how I can make use of the information P(-x,y) = -P(x,y) and Q(-x,y) = Q(x,y) other than that P is an odd function and Q is an even function.

    Please help.

    Thank you,
    Rayne
     
  2. jcsd
  3. Oct 26, 2007 #2
    It provides some hints if you try to find the upper and lower bounds on eigenvalues depending on matrix norm and matrix measure of A.
     
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