Critical points

Hi all,
I have a question about critical points.

Consider the plane dynamic system x' = P(x,y) and 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?

I think functions that "fit" P are functions like the sine function,
and functions that "fit" Q are cosine function and polynomials
involving even powers of x.

Also, if I can somehow combine P and Q into A*(x,y)^T + g(x,y) where T
is the transpose, A is the coefficient matrix and g(x,y) consists of
the higher order terms where lim (x,y)->(0,0) {Norm(g) / Norm((x,y))}
= 0, and let lamda_1 and lamda_2 be the 2 eigenvalues of A, then if I
can show trace(A) = 0 and det(A) > 0, or lamda_1+ lamda_2 = 0 and
lamda_1 * lamda_2 > 0, then (0,0) is a canter. Otherwise, it is not.
However, I don't know how to show that.

Please advise.

Thank you.

Regards,
Rayne

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