I'm trying to solve the following problem:
Use the Poincaré-Bendixson's criterion to show that the system has a periodic orbit $$ \dot{x}_1 =x_2 \\ \dot{x}_2=-x_1+x_2-2(x_1+2x_2)x_2^2 $$
The unique equilibrium is (0,0), and the reduced Jacobian matrix $$ \begin{matrix} 0 & 1 \\ -1& 1 \end{matrix} $$
And the eigenvalues $1\pm\frac{1}{2}i\sqrt{3}$ meaning the first part of the lemma is achieved.
However, I am unsure as how to do the the second part of the lemma which states "Every trajectory starting in M stays in M for all future time". If I understand it correctly the term $$ f(x)\cdot \nabla V(x)=\frac{\partial V}{\partial x_1}f_1(x) +\frac{\partial V}{\partial x_2}f_2(x)\leq0 $$ I am not sure how to obtain $V(x)$ though. In my book they grab it out of the blue air for one example as $V(x)=x_1^2=x_2^2$ and $c>0$ and $V(x)=c$ on the surface.
Can you help me how to determine $V(x)$ and whether or not it has a periodic orbit?