Is the Critical Point (0,0) a Center in This Plane Dynamic System?

[wu_weidong]

Homework Statement

Consider the plane dynamic system $$\dot{x} = P(x,y), \dot{y} = Q(x,y)$$ 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?

The Attempt at a Solution

I know that for (0,0) to be a centre, the eigenvalues of A should satisfy
$$\lambda_1 + \lambda_2 = tr(A) = 0, \lambda_1 \lambda_2 = det(A) > 0$$

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

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

 

[verafloyd]

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.