# Determining critical points

1. Oct 26, 2007

### wu_weidong

1. The problem statement, all variables and given/known data
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?

3. 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
$$\left[ \begin{array}\\ \frac{\partial P}{\partial x} & \frac{\partial P}{\partial y} \\ \frac{\partial Q}{\partial x} & \frac{\partial Q}{\partial y} \\ \end{array} \right]$$

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.