stability in the periodic orbit and in the singular point

Question

Again I have this:

$\dot{x}=-y+\lambda x(36-9x^2-y^2)\\\dot{y}=9x+\lambda y(36-9x^2-y^2)\\\dot{z}=-6z-\lambda^2x^2y^2z^3$

I want to analyze the stability in the periodic orbit and in the singular point, so for the singular point I take the derived matrix of the linear part, and I got the eigenvalues, wich are $\lambda_1=-6$ , $\lambda_2=3i$ , $\lambda=-3i$ . I wanted to use Andronov-Vitt but I have two eigenvalues with no real part so I can´t, does anybody can help me with this? Thanks!

Answer 1

It appears that you have an issue with the Jacobian and hence the eigenvalues.

There is one critical point at $(x,y,z) = (0,0,0)$

Evaluating $J(x,y,z)$ at this critical points yields the eigenvalues:

$$\lambda_1 = -6, \lambda_2 = 36 \lambda -3 i, \lambda_3 = 36 \lambda + 3 i$$