I need to find closed trajectories of a few differential equations, one of the examples is :
$$y''+(2(y')^2+y^4-1)y'+y^3=0$$
I tried to find some information about the trajectories from such a system of equations: \begin{cases} y'=x \\ x'=-2x^3-y^4x+x+y^3 \end{cases} But it seems to be quite complicated and I don't know how to proceed.
The curve $2(y′)^2+y^4=1$ contains a solution. Indeed let $g(y,y')=2(y′)^2+y^4-1$, then the differential equation is $$ \frac{d}{dx}g(y,y')=4y'(y''+y^3)=-4(y')^2·g(y,y'). $$ that is, any solution with $g(y_0,y'_0)=0$ stays on that curve.