# stability in the periodic orbit and in the singular point <mover> x &#x02D9;<!-- ˙ --

stability in the periodic orbit and in the singular point
$\stackrel{˙}{x}=-y+\lambda x\left(36-9{x}^{2}-{y}^{2}\right)\phantom{\rule{0ex}{0ex}}\stackrel{˙}{y}=9x+\lambda y\left(36-9{x}^{2}-{y}^{2}\right)\phantom{\rule{0ex}{0ex}}\stackrel{˙}{z}=-6z-{\lambda }^{2}{x}^{2}{y}^{2}{z}^{3}$
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?
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scoseBexgofvc
It appears that you have an issue with the Jacobian and hence the eigenvalues.
There is one critical point at $\left(x,y,z\right)=\left(0,0,0\right)$
Evaluating $J\left(x,y,z\right)$ at this critical points yields the eigenvalues:
${\lambda }_{1}=-6,{\lambda }_{2}=36\lambda -3i,{\lambda }_{3}=36\lambda +3i$