$\displaystyle x' = 2*x - y - 2*x^3 - 3*x*y^2$
$\displaystyle y' = 2*x + 4*y - 4*y^3 - 2*(x^2)*y $
Use V (x, y) = $\displaystyle 2x^2+ y2$
to show that there must be a limit cycle.
Hey flaming. Nice system. I'd like to know how but I'm not real familiar with that approach. Maybe someone here can show us. Anyway, here's one of the nicest phase portraits I've ever created using your system for the plot (I'm getting better with the coding of it). As you can see, it clearly shows the limit cycle about the equilibrium point at the origin. I can at least linearize it to show that the linearized version has a spiral source with eigenvalues $\displaystyle (3+i, 3-i)$ but don't know how to show the system has a limit cycle using your expression.