Show that the following system of ODEs has a limit cycle $$\dot{x} = 4y^3 + (x+1) - (x+1)[(x+1)^2 + y^4] $$ $$\dot{y} = -2(x+1) + 2y^3 - 2y^3[(x+1)^2 + y^4]. $$
The way the problem is written, it seems pretty suggestive to me to look at the point $(x,y) = (-1,0),$ which turns out to be a critical point. Making the change $x \mapsto x-1,$ we can consider the critical point at the origin for the equivalent system $$\dot{x} = 4y^3 + x - x^3 - xy^4 $$ $$\dot{y} = 2y^3 -2x - 2y^3x^2 - 2y^7. $$ The stability of the origin is given by evaluating $$\begin{pmatrix} 1-3x^2 - y^4 & 13y^2-4xy^3 \\ -2-4xy^3 & 6y^2-6y^2x^2 - 14y^6 \end{pmatrix}$$ at the origin, giving the eigenvalues $\lambda = 1,0.$ Not sure if this is useful in anyway. But from this point, I am stuck.
In general, when trying to show that a limit cycle exists, usually the Poincare Bendixson theorem is invoked. This theorem states sufficient conditions to conclude that a limit cycle exists, but I do not know how to use it rigorously (with a computer, plot the vector field, and if there exists an annulus-like region $R$ called a "trapping region" where the vector field only points into the region, then there exists a limit cycle in $R$).
Perhaps I am overthinking it and there is an easier way to go about this problem. Any suggestions, hints, solutions, all is appreciated. Thanks!